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[(est)-337(alors)-337(le)-337(compl)24(\266)469(emen)24(taire)-337(de)]TJ /F8 1 Tf 13.548 0 Td [(B)]TJ /F2 1 Tf 1.1446 0 Td [(dans)]TJ /F8 1 Tf 2.3374 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(.)]TJ -28.048 -1.6024 Td [(Nous)-259(serons)-259(souv)24(en)30(t)-259(amen)24(\266)469(es)-259(\265)499(a)-259(consid)30(\266)469(erer)-259(des)-259(parties)-259(d'un)-265(ensem)24(ble)]TJ /F8 1 Tf 29.06 0 Td [(X)]TJ /F2 1 Tf 1.1627 0 Td [(et)-265(le)-259(compl)24(\266)469(emen)24(taire)]TJ -30.223 -1.1988 Td [(dans)]TJ /F8 1 Tf 2.3976 0 Td [(X)]TJ /F2 1 Tf 1.3012 0 Td [(de)-391(ces)-391(parties.)-620(Si)-397(aucune)-391(confusion)-397(n'est)-391(\265)500(a)-391(craindre,)-409(nous)-391(sous-en)18(tendrons)-391(alors)]TJ -3.6988 -1.1988 Td [(l'ensem)30(ble)]TJ /F8 1 Tf 4.8012 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(et)-325(nous)-325(noterons)]TJ /F8 1 Tf 7.5422 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.129 50.426 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.386 34.828 Td [(le)-331(compl)24(\266)475(emen)24(taire)-325(de)]TJ /F8 1 Tf 9.5602 0 Td [(A)]TJ /F2 1 Tf 1.0723 0 Td [(\(dans)]TJ /F8 1 Tf 2.7229 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\).)-445(P)24(our)]TJ /F8 1 Tf 3.5361 0 Td [(A)]TJ /F2 1 Tf 1.0723 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(B)]TJ /F2 1 Tf 1.1325 0 Td [(parties)]TJ -36.211 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(on)-337(a)-331(donc)]TJ /F8 1 Tf 4.6145 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(n)]TJ /F8 1 Tf 0.72289 0 Td [(B)]TJ /F2 1 Tf 1.0904 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.233 48.711 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.434 33.623 Td [(.)]TJ -14.102 -1.5964 Td [(A)-379(toute)-373(partie)]TJ /F8 1 Tf 6.7169 0 Td [(A)]TJ /F2 1 Tf 1.1205 0 Td [(d'un)-385(ensem)24(ble)]TJ /F8 1 Tf 6.6205 0 Td [(X)]TJ /F2 1 Tf 1.2831 0 Td [(nous)-379(asso)-30(cierons)-379(sa)]TJ /F1 1 Tf 8.8554 0 Td 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1 Tf 0.74699 0 Td [(.)]TJ -33.488 -2.1988 Td [(La)-451(th)24(\266)475(eorie)-451(utilise)-451(fr)24(\266)469(equemmen)30(t)-451(des)-445(limites)-451(de)-445(fonctions,)-475(en)-445(particulier)-451(la)-445(limite)-445(simple.)]TJ 0 -1.1988 Td [(A\257n)-283(d')24(\266)469(eviter)-289(toute)-283(confusion)-289(a)24(v)24(ec)-289(d'autres)-289(notions)-295(de)-283(limite)-289(qui)-289(p)-30(ourraien)24(t)-289(in)24(terv)30(enir)-289(nous)]TJ 0 -1.2048 Td [(utiliserons)-337(le)-337(sym)24(b)-30(ole)-668(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24 37.237 Td [(s)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.813 26.22 Td [(p)-30(our)-337(d)24(\266)475(esigner)-337(la)-337(limite)-331(simple.)]TJ -12.482 -1.5964 Td [(Ainsi)-409(p)-30(our)-409(une)-409(suite)-409(de)-409(fonctions)]TJ /F8 1 Tf 15.277 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.19 35.021 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.048 24.623 Td [(d)24(\266)475(e\257nies)-415(sur)-409(une)-409(ensem)24(ble)]TJ /F8 1 Tf 11.681 0 Td [(X)]TJ /F2 1 Tf 1.3133 0 Td [(l')24(\266)469(ecriture)-409(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.302 34.943 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.283 24.623 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.776 35.021 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.717 24.623 Td [(=)]TJ /F8 1 Tf 1.1807 0 Td [(f)]TJ /F2 1 Tf -38.566 -1.2048 Td [(signi\257e)-331(:)]TJ /F6 1 Tf 4.0602 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(x)]TJ /F6 1 Tf 0.85542 0 Td [(2)]TJ /F8 1 Tf 0.93976 0 Td [(X)-753(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.759 33.297 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.349 23.418 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(!)]TJ /F8 1 Tf 1.2771 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(.)]TJ 5.6747 -19.53 Td [(1)]TJ /F5 1 Tf 6.012 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 28 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 27 0 R /ProcSet 2 0 R >> /Contents 26 0 R >> endobj 27 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R >> endobj 30 0 obj << /Length 11984 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-18024(I.)-385(In)30(tro)-36(duction)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I.1.)-236(Calcul)-381(in\257nit)25(\266)537(esimal,)-376(calcul)-376(in)25(t)25(\266)537(egral.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 60.617 Td [(L')24(\266)469(etude)-337(des)-343(fonctions)-343(de)-337(la)-337(v)54(ariable)-343(r)30(\266)469(eelle,)-343(et)-343(en)-337(particulier)-337(de)-337(leurs)-343(v)54(ariations,)-349(a)-337(amen)24(\266)475(e)-343(\265)500(a)]TJ 0 -1.1988 Td [(d)30(\266)469(ev)24(elopp)-30(er)-337(le)-343(calcul)-331(in\257nit)24(\266)469(esimal.)-445(Sc)30(h)24(\266)475(ematiquemen)24(t)-337(celui-ci)-337(p)-30(eut)-337(se)-331(pr)24(\266)469(esen)24(ter)-331(ainsi)-337(:)]TJ 0 -1.6024 Td [(Etan)24(t)-433(donn)24(\266)469(ee)-433(une)-433(v)54(ariable)]TJ /F8 1 Tf 12.241 0 Td [(y)]TJ /F2 1 Tf 0.95783 0 Td [(fonction)-439(d'une)-433(v)54(ariable)]TJ /F8 1 Tf 10.633 0 Td [(x)]TJ /F2 1 Tf 1.006 0 Td [(\(c'est-)0(\265)500(a-dire)-439(qu'elle)-433(lui)-433(est)-433(reli)24(\266)469(ee)]TJ -24.837 -1.1988 Td [(par)-415(une)-415(relation)]TJ /F8 1 Tf 7.5843 0 Td [(y)]TJ /F2 1 Tf 0.93373 0 Td [(=)]TJ /F8 1 Tf 1.1928 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\)\),)-433(si)-421(l'on)-415(donne)-415(\265)500(a)-415(la)-415(v)54(ariable)]TJ /F8 1 Tf 13.572 0 Td [(x)]TJ /F2 1 Tf 0.98193 0 Td [(un)-415(accroissemen)24(t)-415(\242)]TJ /F8 1 Tf 8.7651 0 Td [(x)]TJ /F2 1 Tf 0.98193 0 Td 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/F8 1 Tf 5.8675 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-415(en)-403(particulier)-403(\265)500(a)-397(comparer)-403(le)-403(signe)-409(de)-397(ces)]TJ -20.416 -1.6446 Td [(accroissemen)30(ts)-283(\(probl)24(\265)475(eme)-283(de)-283(la)-277(monotonie)-283(de)]TJ /F8 1 Tf 19.795 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\))-283(et)-283(\265)499(a)-283(estimer)-283(le)-283(taux)-283(d'accroissemen)24(t)]TJ 16.524 0.6747 Td [(\242)]TJ /F8 1 Tf 0.83133 0 Td [(y)]TJ ET 0.504 w 506.88 631.29 m 523.73 631.29 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.41 51.882 Td [(\242)]TJ /F8 1 Tf 0.83735 0 Td [(x)]TJ /F6 1 Tf 0.96988 0.68675 Td [(\242)]TJ /F2 1 Tf -38.886 -1.8795 Td [(P)30(our)-349(ce)-349(faire)-355(on)-349(c)24(herc)24(he)-349(\265)500(a)-349(remplacer)-349(les)-349(accroissemen)30(ts)-349(\242)]TJ /F8 1 Tf 25.434 0 Td [(x)]TJ /F2 1 Tf 0.92169 0 Td [(et)-349(\242)]TJ /F8 1 Tf 2.012 0 Td [(y)]TJ /F2 1 Tf 0.87349 0 Td [(par)-349(des)-349(accroissemen)24(ts)]TJ -29.241 -1.6687 Td [(in\257nimen)24(t)-361(p)-30(etits)]TJ /F8 1 Tf 7.6145 0 Td [(dx)]TJ /F2 1 Tf 1.4578 0 Td [(et)]TJ /F8 1 Tf 1.1928 0 Td [(dy)]TJ /F2 1 Tf 1.4036 0 Td [(de)-361(telle)-361(fa)-6(\273)439(con)-361(que)-367(le)-361(taux)-367(d'accroissemen)24(t)]TJ /F8 1 Tf 18.855 0.6747 Td [(dy)]TJ ET 428.26 588.88 m 441.36 588.88 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.831 48.334 Td [(dx)]TJ /F2 1 Tf 1.5723 0.68674 Td [(ne)-361(d)24(\266)475(ep)-30(ende)-361(plus)]TJ -32.072 -1.4819 Td [(de)-301(la)-307(taille)-313(de)]TJ /F8 1 Tf 6.1627 0 Td [(dx)]TJ /F2 1 Tf 1.0964 0 Td [(,)-313(c'est)-307(la)-307(d)30(\266)469(eriv)48(\266)475(ee)-307(de)]TJ /F8 1 Tf 8.5964 0 Td [(F)]TJ /F2 1 Tf 1.0904 0 Td [(au)-307(p)-24(oin)24(t)]TJ /F8 1 Tf 3.9398 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(.)-439(Bien)-307(s)-30(^)524(ur)-307(le)-307(discours)-313(pr)30(\266)469(ec)24(\266)475(eden)24(t)-307(doit)-277(^)469(etre)]TJ -21.458 -1.1988 Td [(pr)30(\266)469(ecis)24(\266)475(e)-331(p)-30(our)-331(a)30(v)24(oir)-331(toute)-325(la)-325(rigueur)-331(n)24(\266)469(ecessaire)-325(en)-325(math)24(\266)469(ematique.)-445(On)-331(le)-331(fait)-331(usuellemen)30(t)-331(en)]TJ 0 -1.1988 Td [(utilisan)30(t)-325(la)-319(notion)-325(de)-319(limite)-319(\(on)-325(p)-30(eut)-325(aussi)-319(utiliser)-325(l'analyse)-325(non)-325(standard)-319(p)-30(our)-319(rester)-325(plus)]TJ 0 -1.2048 Td [(pro)-30(c)24(he)-331(de)-337(l'asp)-24(ect)-337(in)24(tuitif)-78(\).)]TJ 0 -1.5964 Td [(Les)-240(accroissemen)24(ts)]TJ /F8 1 Tf 8.3313 0 Td [(dx)]TJ /F2 1 Tf 1.3373 0 Td [(et)]TJ /F8 1 Tf 1.0783 0 Td [(dy)]TJ /F2 1 Tf 1.2892 0 Td [(son)24(t)-246(alors)-253(dits)-246(in\257nit)24(\266)475(esimaux,)-271(par)-240(opp)-30(osition)-253(les)-240(accroissemen)24(ts)]TJ -12.036 -1.2048 Td [(\242)]TJ /F8 1 Tf 0.83133 0 Td [(x)]TJ /F2 1 Tf 0.86145 0 Td [(et)-283(\242)]TJ /F8 1 Tf 1.9518 0 Td [(y)]TJ /F2 1 Tf 0.81325 0 Td [(son)24(t)-289(dits)-289(\257nis.)-427(Le)-289(calcul)-289(de)-289(ces)-283(accroissemen)24(ts)-283(in\257nit)24(\266)469(esimaux,)-301(c'est-)0(\265)500(a-dire)-289(fonda-)]TJ -4.4578 -1.1988 Td [(men)24(talemen)30(t)-325(le)-325(calcul)-319(des)-319(d)24(\266)469(eriv)54(\266)469(ees,)-319(est)-325(le)-325(calcul)-319(in\257nit)24(\266)469(esimal.)-439(L'un)-319(de)-325(ses)-319(buts)-325(est)-325(alors,)-325(\265)500(a)]TJ 0 -1.1988 Td [(l'aide)-319(de)-313(renseignemen)24(ts)-313(sur)-325(les)-319(accroissemen)30(ts)-319(in\257nit)24(\266)475(esimaux)-325(et)-319(donc)-313(sur)-325(la)-313(d)24(\266)469(eriv)54(\266)469(ee)-319(de)]TJ /F8 1 Tf 38.102 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(,)]TJ -38.886 -1.2048 Td [(d'obtenir)-337(des)-337(renseignemen)24(ts)-331(sur)-343(les)-337(accroissemen)30(ts)-337(\257nis.)-451(Le)-337(th)24(\266)469(eor)24(\265)469(eme)-331(des)-337(accroissemen)24(ts)]TJ 0 -1.1988 Td [(\257nis)-331(est)-343(le)-337(protot)24(yp)-30(e)-343(de)-331(tels)-331(r)24(\266)475(esultats.)]TJ 0 -1.6024 Td [(De)-409(son)-409(c)-6(^)500(ot)24(\266)469(e)-409(le)-409(calcul)-409(in)24(t)24(\266)475(egral)-409(a)-403(p)-30(our)-409(ob)-54(jet)-409(de)-403(retrouv)24(er)-415(l'accroissemen)30(t)-409(\242)]TJ /F8 1 Tf 33.018 0 Td [(y)]TJ /F2 1 Tf 0.93373 0 Td [(\265)500(a)-403(partir)-409(des)]TJ -33.952 -1.1988 Td [(accroissemen)30(ts)-355(in\257nit)24(\266)475(esimaux)]TJ /F8 1 Tf 13.235 0 Td [(dy)]TJ /F2 1 Tf 1.0422 0 Td [(.)-500(On)-355(consid)24(\265)475(ere)-355(p)-30(our)-355(cela)-355(que,)-361(si)]TJ /F8 1 Tf 14.416 0 Td [(x)]TJ /F2 1 Tf 0.92168 0 Td [(v)48(arie)-349(de)]TJ /F8 1 Tf 3.7952 0 Td [(a)]TJ /F2 1 Tf 0.87952 0 Td [(\265)499(a)]TJ /F8 1 Tf 0.8494 0 Td [(b)]TJ /F2 1 Tf 0.78313 0 Td [(et)-355(donc)]TJ -35.922 -1.1988 Td [(\242)]TJ /F8 1 Tf 0.83133 0 Td [(x)]TJ /F2 1 Tf 0.86747 0 Td [(=)]TJ /F8 1 Tf 1.0663 0 Td [(b)]TJ /F6 1 Tf 0.66265 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)-343(l'accroissemen)24(t)-343(\242)]TJ /F8 1 Tf 8.3434 0 Td [(y)]TJ /F2 1 Tf 0.8012 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-343(corresp)-36(ondan)24(t)-343(est)-349(la)-343(somme)-343(d'une)-349(in\257nit)24(\266)475(e)]TJ -20.078 -1.1988 Td [(d'accroissemen)24(ts)-240(in\257nit)24(\266)475(esimaux)]TJ /F8 1 Tf 13.843 0 Td [(dy)]TJ /F2 1 Tf 1.0482 0 Td [(.)-415(La)-246(notation)-246(p)-30(our)-240(d)24(\266)475(esigner)-246(cette)-240(somme)-246(in\257nie,)-259(obten)24(ue)]TJ -14.892 -1.8976 Td [(en)-355(\266)469(etiran)24(t)-391(l'initiale)-385(S)-385(de)-385(somme,)-397(est)-391(alors)]TJ /F9 1 Tf 18.705 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.828 43.435 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.602 29.238 Td [(dy)]TJ /F2 1 Tf 1.4337 0 Td [(,)-397(c'est)-391(la)-385(somme)-385(in)24(t)24(\266)469(egrale)-385(ou)-391(l'in)24(t)24(\266)475(egrale)]TJ -21.705 -1.7048 Td [(des)-337(accroissemen)30(ts)]TJ /F8 1 Tf 8.4458 0 Td [(dy)]TJ /F2 1 Tf 1.0482 0 Td [(.)]TJ -9.494 -1.6024 Td [(P)30(our)-240(les)-240(fonctions)-234(usuelles,)-265(en)-234(faisan)24(t)-240(in)24(terv)30(enir)-240(la)-234(fonction)]TJ /F8 1 Tf 25.325 0 Td [(f)]TJ /F2 1 Tf 0.83133 0 Td [(d)24(\266)469(eriv)54(\266)469(ee)-240(de)]TJ /F8 1 Tf 4.5361 0 Td [(F)]TJ /F2 1 Tf 1.0181 0 Td [(on)-240(a)]TJ /F8 1 Tf 2.0301 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39156 0 Td [(dx)]TJ /F2 1 Tf -38.072 -1.9518 Td [(et)-391(donc)]TJ /F8 1 Tf 3.6747 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-379(=)]TJ /F9 1 Tf 1.9157 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.198 35.909 Td [(b)]TJ -0.62931 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.313 23.979 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)-620(La)-397(th)24(\266)469(eorie)-397(se)-391(confond)-397(alors)-397(a)30(v)24(ec)-397(celle)-391(des)-397(primitiv)24(es)]TJ -15.187 -1.7048 Td [(mais)-331(nous)-337(v)24(errons)-337(que)-337(la)-331(th)18(\266)475(eorie)-337(de)-337(l'in)24(t)24(\266)469(egration)-337(est)-337(de)-337(p)-24(ort)18(\266)475(ee)-337(bien)-337(plus)-337(g)30(\266)469(en)30(\266)469(erale.)]TJ 0 -2.2952 Td [(Au)-349(d)30(\266)469(ebut,)-343(le)-349(probl)18(\265)475(eme)-343(du)-349(sens)-343(pr)24(\266)475(ecis)-349(\265)500(a)-343(donner)-343(au)-349(sym)24(b)-30(ole)-349(d'in)24(t)24(\266)475(egration)]TJ /F9 1 Tf 32.03 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.905 30.185 Td [(b)]TJ -0.63793 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.765 19.979 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4398 0 Td [(et)-343(en)]TJ -36.988 -1.7108 Td [(particulier)-325(celui)-325(de)-331(l'existence)-325(des)-331(primitiv)24(es)-325(des)-331(fonctions)-331(consid)30(\266)469(er)24(\266)469(ees)-325(n')24(\266)475(etait)-325(pas)-331(soulev)48(\266)475(e.)]TJ 0 -1.1988 Td [(La)-307(n)30(\266)469(ecessit)24(\266)475(e)-307(de)-307(donner)-307(plus)-301(de)-301(rigueur)-307(\265)500(a)-307(ces)-301(notions)-307(a)-301(conduit)-307(\265)500(a)-301(d)24(\266)475(ev)24(elopp)-30(er)-313(une)-301(th)24(\266)469(eorie)-307(de)]TJ 0 -1.1988 Td [(l'in)24(t)24(\266)475(egration.)]TJ /F5 1 Tf 0 -11.982 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.5 0 Td [(2)]TJ ET endstream endobj 32 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 31 0 R /ProcSet 2 0 R >> /Contents 30 0 R >> endobj 31 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R >> endobj 36 0 obj << /Length 16690 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-18024(I.)-385(In)30(tro)-36(duction)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I.2.)-236(La)-381(m)20(\266)542(etho)-35(de)-376(de)-381(Riemann)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 59.973 Td [(Comme)-325(nous)-319(l'a)24(v)24(ons)-325(vu)]TJ /F9 1 Tf 10.693 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.362 87.418 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.422 59.973 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4157 0 Td [(repr)24(\266)475(esen)24(te)-325(in)24(tuitiv)24(emen)30(t)-325(la)-325(somme)-319(d'une)-331(in\257nit)24(\266)475(e)-331(d'ac-)]TJ -15.62 -1.759 Td [(croissemen)30(ts)-337(in\257nit)24(\266)469(esimaux)]TJ /F8 1 Tf 12.253 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(dx)]TJ /F2 1 Tf 1.4277 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(x)]TJ /F2 1 Tf 0.90361 0 Td [(v)54(arian)24(t)-337(de)]TJ /F8 1 Tf 4.7289 0 Td [(a)]TJ /F2 1 Tf 0.86144 0 Td [(\265)500(a)]TJ /F8 1 Tf 0.83133 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(.)]TJ -28.127 -1.753 Td [(L'id)30(\266)469(ee)-355(de)-349(Riemann)-349(consiste)-349(alors)-355(\265)500(a)-349(sub)-30(diviser)-349(le)-355(segmen)24(t)-355([)]TJ /F8 1 Tf 25.578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])-349(en)-349(sous-segmen)24(ts)-349([)]TJ /F8 1 Tf 8.7108 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.466 80.581 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.91 56.461 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.802 80.581 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 56.461 Td [(])]TJ -38.886 -1.1988 Td [(p)-30(our)-475(1)]TJ /F11 1 Tf 3.5181 0 Td [(6)]TJ /F8 1 Tf 1.2892 0 Td [(i)]TJ /F11 1 Tf 0.86145 0 Td [(6)]TJ /F8 1 Tf 1.2892 0 Td [(n)]TJ /F2 1 Tf 1.0783 0 Td [(\(a)24(v)24(ec)]TJ /F8 1 Tf 2.7229 0 Td [(a)]TJ /F2 1 Tf 1.0422 0 Td [(=)]TJ /F8 1 Tf 1.2952 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.121 78.866 Td [(0)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 19.916 55.262 Td [(6)]TJ /F8 1 Tf 1.2952 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 31.103 78.866 Td [(1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 22.699 55.262 Td [(6)]TJ /F6 1 Tf 1.2952 0 Td [(\242)-162(\242)-168(\242)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F8 1 Tf 1.2892 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.345 78.866 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.548 55.262 Td [(=)]TJ /F8 1 Tf 1.2952 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))-475(;)-548(p)-30(our)-475(c)24(haque)]TJ /F8 1 Tf 7.6687 0 Td [(i)]TJ /F2 1 Tf 0.81927 0 Td [(on)-475(c)18(hoisit)]TJ /F8 1 Tf 4.8193 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 78.866 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 54.063 Td [(compris)-313(en)30(tre)-313(la)-313(b)-30(orne)-313(inf)24(\266)475(erieure)-319(et)-313(la)-313(b)-30(orne)-313(sup)0(\266)469(erieure)-319(de)]TJ /F8 1 Tf 25.633 0 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(sur)-319([)]TJ /F8 1 Tf 1.9337 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.138 77.15 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.693 54.063 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.474 77.15 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37 54.063 Td [(])-313(\(usuellemen)30(t)-319(on)]TJ -31.669 -1.1988 Td [(prend)-367(p)-30(our)]TJ /F8 1 Tf 5.259 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.991 75.435 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.873 52.864 Td [(la)-361(v)48(aleur)-361(de)]TJ /F8 1 Tf 5.512 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(en)-367(un)-367(p)-30(oin)30(t)]TJ /F8 1 Tf 5.4819 0 Td [(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.724 75.435 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.964 52.864 Td [(de)-361([)]TJ /F8 1 Tf 1.6386 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.828 75.435 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.488 52.864 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.164 75.435 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.795 52.864 Td [(])-367(ou)-361(encore)-373(la)-361(b)-30(orne)-361(inf)24(\266)469(erieure)-367(ou)]TJ -24.464 -1.2048 Td [(sup)0(\266)475(erieure)-289(de)-289(ces)-283(v)54(aleurs\).)-433(On)-283(remplace)-289(alors)-283(les)-289(accroissemen)30(ts)-289(in\257nit)24(\266)475(esimaux)]TJ /F8 1 Tf 34.392 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(dx)]TJ /F2 1 Tf 1.3795 0 Td [(par)]TJ -37.717 -1.1988 Td [(les)-228(accroissemen)24(ts)-228(\257nis)]TJ /F8 1 Tf 9.9699 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.733 71.995 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.223 50.461 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.526 71.995 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.488 50.461 Td [(\241)]TJ /F8 1 Tf 0.78916 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.914 71.995 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.163 50.461 Td [(\))-234(et)-234(la)-228(somme)-228(in\257nie)-234(de)-234(ces)-228(accroissemen)30(ts)-234(in\257nit)24(\266)475(esimaux)]TJ -14.831 -2 Td [(par)-331(la)-337(somme)-337(\257nie)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.328 71.133 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 13.729 49.406 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 20.397 67.676 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.343 48.461 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.793 69.133 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.259 48.461 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.578 69.133 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.735 48.461 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.569 69.133 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.62 48.461 Td [(\))-337(app)-30(el)24(\266)469(ee)-337(somme)-331(de)-337(Riemann.)]TJ -15.289 -2.6145 Td [(Cette)-391(somme)-391(de)-385(Riemann)-391(est)-391(consid)24(\266)469(er)24(\266)475(ee)-397(comme)-391(une)-385(b)-30(onne)-391(appro)24(ximation)-391(de)-391(l'in)24(t)24(\266)475(egrale)]TJ -0.030121 -1.1988 Td [(\266)469(etudi)30(\266)469(ee)-307(dans)-307(la)-307(mesure)-313(o)-30(\265)530(u)-307(les)-313(accroissemen)30(ts)-307(\257nis)]TJ /F8 1 Tf 21.789 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.595 63.676 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.006 44.647 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.388 63.676 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.422 44.647 Td [(\241)]TJ /F8 1 Tf 0.95181 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.216 63.676 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.259 44.647 Td [(\))-307(son)24(t)-313(pro)-30(c)24(hes)-313(des)-307(accroisse-)]TJ -26.928 -1.1988 Td [(men)24(ts)-433(in\257nit)24(\266)469(esimaux)]TJ /F8 1 Tf 9.6084 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)-753(Cela)-439(supp)-30(ose)-439(que)-439(le)-439(pas)-439(de)-433(la)-439(sub)-24(division,)-463(c'est-)0(\265)500(a-dire)-439(le)]TJ -12.645 -1.2048 Td [(maxim)24(um)-361(des)]TJ /F8 1 Tf 6.4458 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.612 60.237 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.88 42.244 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.647 60.237 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.783 42.244 Td [(,)-367(soit)-361(su\261sammen)24(t)-361(p)-30(etit)-355(p)-30(our)-361(que)-361(\242)]TJ /F8 1 Tf 16.187 0 Td [(x)]TJ /F2 1 Tf 0.89157 0 Td [(=)]TJ /F8 1 Tf 1.0964 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.353 60.237 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 35.06 42.244 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.388 60.237 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.319 42.244 Td [(soit)-361(pro)-30(c)24(he)-361(de)]TJ -32.988 -1.1988 Td [(l'accroissemen)24(t)-283(in\257nit)24(\266)469(esimal)]TJ /F8 1 Tf 12.452 0 Td [(dx)]TJ /F2 1 Tf 1.3795 0 Td [(et)-283(que)-289(la)-283(fonction)]TJ /F8 1 Tf 7.8072 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(soit)-283(su\261sammen)24(t)-289(r)30(\266)469(eguli)24(\265)475(ere)-289(p)-30(our)-289(v)54(arier)]TJ -22.518 -1.1988 Td [(tr)24(\265)475(es)-343(p)-30(eu)-343(sur)-343([)]TJ /F8 1 Tf 5.8434 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.75 56.806 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.06 39.846 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.086 56.806 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.367 39.846 Td [(])-337(a\257n)-343(que)]TJ /F8 1 Tf 4.4398 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.741 56.806 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.06 39.846 Td [(soit)-343(une)-343(b)-30(onne)-337(appro)24(ximation)-343(de)]TJ /F8 1 Tf 14.801 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)-469(Ce)-343(discours)-343(p)-30(eu)]TJ -31.084 -1.2048 Td [(rigoureux)-331(conduit)-337(toutefois)-337(\265)500(a)-337(p)-30(oser)-331(la)-337(d)30(\266)469(e\257nition)-337(suiv)54(an)24(te)-337(:)]TJ /F12 1 Tf 1 -1.747 Td [(Av)30(ec)-301(les)-289(notations)-295(pr)24(\266)469(ec)24(\266)475(eden)24(tes,)]TJ /F8 1 Tf 13.813 0 Td [(f)]TJ /F12 1 Tf 0.88554 0 Td [(est)-301(dite)-295(in)24(t)24(\266)475(egrable)-295(\(au)-295(sens)-295(de)-295(Riemann\))-295(sur)]TJ /F2 1 Tf 19.416 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])]TJ /F12 1 Tf 0.57229 0 Td [(si)-295(et)]TJ -37.367 -1.1988 Td [(seulemen)30(t)-361(si)-367(les)-361(sommes)-361(de)-361(Riemann)-361(on)30(t)-361(une)-361(limite)-361(lorsque)-361(le)-367(pas)-361(de)-361(la)-361(sub)-24(division)-361(tend)]TJ 0 -1.8976 Td [(v)24(ers)-337(0.)-445(Cette)-337(limite)-331(est)-337(alors)-343(l'in)24(t)24(\266)475(egrale)-337(de)]TJ /F8 1 Tf 18.783 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(sur)]TJ /F2 1 Tf 1.6807 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(,)-331(not)18(\266)475(ee)]TJ /F9 1 Tf 3.2771 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.164 49.961 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.361 33.798 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dx)]TJ /F12 1 Tf 1.4277 0 Td [(.)]TJ /F2 1 Tf -31.572 -3 Td [(On)-343(p)-30(eut)-343(in)24(terpr)24(\266)469(eter)-343(la)-343(m)24(\266)475(etho)-36(de)-343(de)-337(Riemann)-343(en)-343(consid)24(\266)475(eran)24(t)-343(que)-349(l'on)-343(remplace)-343(la)-343(fonction)]TJ /F8 1 Tf 0 -1.1988 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(par)-373(une)-379(fonction)-379(en)-373(escalier)-373(constan)24(te)-379(sur)-379(c)24(hacun)-379(des)-373(in)24(terv)54(alles)-379(])]TJ /F8 1 Tf 29.042 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.336 42.142 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.229 29.599 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.672 42.142 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.536 29.599 Td [([)-373(de)-373(v)48(aleur)]TJ /F8 1 Tf 5.0422 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 42.142 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 28.4 Td [(\(la)-349(v)54(aleur)-349(aux)-349(p)-30(oin)24(ts)-349(de)-343(sub)-30(division)-343(n'a)24(y)24(an)24(t)-349(aucune)-349(imp)-30(ortance\).)-487(La)-349(somme)-343(de)-349(Riemann)]TJ 0 -1.2048 Td [(est)-391(alors)-385(l'in)24(t)24(\266)469(egrale)-391(de)-385(la)-385(fonction)-391(en)-385(escalier)-385(et)-391(l'on)-385(esp)-6(\265)475(ere)-391(qu'en)-385(dimin)24(uan)24(t)-385(le)-391(pas)-391(de)-385(la)]TJ 0 -1.1988 Td [(sub)-24(division)-325(cette)-325(fonction)-325(en)-319(escalier)-325(sera)-319(une)-325(appro)24(ximation)-319(su\261san)18(te)-319(de)]TJ /F8 1 Tf 32.596 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(p)-30(our)-319(que)-325(son)]TJ -33.512 -1.1988 Td [(in)24(t)24(\266)475(egrale)-337(soit)-337(une)-337(b)-30(onne)-331(appro)24(ximation)-337(de)-331(celle)-343(de)]TJ /F8 1 Tf 22.41 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)]TJ -23.006 -1.747 Td [(Cette)-337(m)24(\266)475(etho)-30(de)-337(a)-331(l'a)24(v)54(an)24(tage)-337(de)-331(la)-331(simplicit)24(\266)469(e,)-337(de)-331(suivre)-337(l'in)24(tuition)-331(que)-337(l'on)-337(p)-30(eut)-337(a)30(v)24(oir)-337(de)-331(la)]TJ 0 -1.2048 Td [(notion)-391(d'in)24(t)24(\266)469(egrale)-397(et)-391(app)-30(orte)-397(une)-391(solution)-397(satisfaisan)24(te)-391(au)-397(probl)24(\265)475(eme)-391(p)-30(os)24(\266)469(e.)-620(Elle)-391(a)-391(p)-30(ermis)]TJ 0 -1.1988 Td [(en)-415(particulier)-421(de)-415(prouv)24(er)-421(que)-421(toute)-421(fonction)-421(con)24(tin)24(ue)-415(sur)-427(un)-421(in)30(terv)54(alle)-421(de)]TJ /F3 1 Tf 33.325 0 Td [(R)]TJ /F2 1 Tf 1.1386 0 Td [(admet)-421(une)]TJ -34.464 -1.1988 Td [(primitiv)30(e)-343(sur)-337(cet)-331(in)24(terv)54(alle.)-445(Elle)-337(pr)30(\266)469(esen)24(te)-331(cep)-36(endan)24(t)-337(deux)-331(incon)24(v)48(\266)475(enien)24(ts.)]TJ 0 -1.747 Td [(T)78(out)-397(d'ab)-36(ord)-397(le)-397(fait)-397(que)-397(les)-403(fonctions)-391(en)-397(escaliers)-403(constituen)30(t)-397(de)-397(b)-30(onnes)-397(appro)24(ximations)]TJ 0 -1.2048 Td [(de)]TJ /F8 1 Tf 1.4217 0 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(supp)-30(ose)-427(une)-421(grande)-427(r)30(\266)469(egularit)30(\266)469(e)-427(de)]TJ /F8 1 Tf 15.223 0 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(et)-427(la)-421(classe)-427(des)-427(fonctions)-427(accessibles)-421(par)-427(cette)]TJ -18.681 -1.1988 Td [(m)24(\266)475(etho)-30(de)-337(est)-337(relativ)24(emen)24(t)-337(restrein)30(te)-337(et)-337(p)-30(eu)-331(stable)-337(par)-331(passage)-337(\265)500(a)-331(la)-337(limite.)]TJ 0 -1.747 Td [(P)30(ar)-343(ailleurs,)-349(la)-349(m)24(\266)475(etho)-36(de)-343(utilise)-349(une)-343(sub)-30(division)-343(de)-349(la)-343(source)-349(de)]TJ /F8 1 Tf 27.855 0 Td [(f)]TJ /F2 1 Tf 0.94578 0 Td [(en)-343(in)24(terv)54(alles)-349(ce)-343(qui)-349(n'a)]TJ -28.801 -1.1988 Td [(de)-361(sens)-355(que)-367(p)-30(our)-361(une)-361(v)54(ariable)-367(r)30(\266)469(eelle.)-524(De)-367(ce)-361(fait)-367(elle)-361(se)-361(pr)24(^)469(ete)-361(mal)-361(\265)500(a)-361(des)-361(g)30(\266)469(en)30(\266)469(eralisations)-361(ne)]TJ 0 -1.2048 Td [(serait-ce)-331(qu')0(\265)500(a)-337(l'in)24(t)24(\266)475(egration)-343(de)-331(fonctions)-337(de)-331(plusieurs)-343(v)54(ariables.)]TJ 0 -1.747 Td [(Ce)-337(son)24(t)-337(ces)-337(incon)30(v)48(\266)475(enien)24(ts)-337(qui)-331(v)24(on)24(t)-307(^)469(etre)-331(con)24(tourn)30(\266)469(es)-337(par)-331(la)-337(m)24(\266)475(etho)-36(de)-331(de)-337(Leb)-30(esgue.)]TJ 19.163 -5.512 Td [(3)]TJ /F5 1 Tf 6.012 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 38 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 37 0 R /ProcSet 2 0 R >> /Contents 36 0 R >> endobj 37 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R 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[(C'est)-355(cette)-355(m)24(\266)475(etho)-30(de)-355(que)-355(nous)-349(allons)-349(d)24(\266)469(ev)30(elopp)-36(er)-355(dans)-349(ce)-355(cours.)-499(En)-349(v)18(oici)-349(une)-349(pr)24(\266)469(esen)24(tation)]TJ 0 -1.2048 Td [(succincte.)-421(Dans)-259(un)-259(but)-259(de)-259(simpli\257cation)-259(supp)-30(osons)-259(que)-265(la)-259(fonction)]TJ /F8 1 Tf 28.91 0 Td [(f)]TJ /F2 1 Tf 0.8494 0 Td [(\265)500(a)-259(in)24(t)24(\266)469(egrer)-259(soit)-259(b)-30(orn)24(\266)475(ee,)]TJ -29.759 -1.1988 Td [(plus)-427(pr)30(\266)469(ecis)30(\266)469(emen)24(t)-427(supp)-30(osons)]TJ /F8 1 Tf 12.645 0 Td [(m)-433(<)-433(f)]TJ /F2 1 Tf 3.1145 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.81928 0 Td [(<)-427(M)]TJ /F2 1 Tf 2.2892 0 Td [(.)-716(Con)24(trairemen)30(t)-433(\265)500(a)-421(la)-427(m)24(\266)475(etho)-36(de)-421(adopt)24(\266)469(ee)-427(par)]TJ -19.831 -1.1988 Td [(Riemann,)-343(on)-349(ne)-343(v)54(a)-343(pas)-349(consid)30(\266)469(erer)-343(une)-349(sub)-24(division)-343(de)-349(la)-343(source)-343(de)]TJ /F8 1 Tf 29.151 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(mais)-343(de)-343(son)-349(but.)-475(On)]TJ -30.09 -1.2048 Td [(in)24(tro)-30(duit)-228(donc)-234(des)-234(p)-30(oin)24(ts)-228(de)-234(sub)-30(division)]TJ /F8 1 Tf 17.307 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.31 75.219 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.843 52.714 Td [(\(0)]TJ /F11 1 Tf 1.1687 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(i)]TJ /F11 1 Tf 0.62048 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(\))-234(a)24(v)24(ec)]TJ /F8 1 Tf 2.7169 0 Td [(m)]TJ /F2 1 Tf 1.1566 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.526 75.219 Td [(0)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 34.639 52.714 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52 75.219 Td [(1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 37.06 52.714 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(:)-162(:)-168(:)]TJ /F11 1 Tf 1.4398 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.043 75.219 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.084 52.714 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(.)]TJ -38.886 -1.1988 Td [(P)30(our)-337(1)]TJ /F11 1 Tf 3.2169 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(i)]TJ /F11 1 Tf 0.62651 0 Td [(6)]TJ /F8 1 Tf 1.0663 0 Td [(n)]TJ /F2 1 Tf 0.93373 0 Td [(on)-337(note)-343(alors)]TJ /F8 1 Tf 6.0181 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.198 73.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.675 51.515 Td [(l'ensem)30(ble)-337(des)]TJ /F8 1 Tf 6.5482 0 Td [(x)]TJ /F2 1 Tf 0.90964 0 Td [(o)-24(\265)524(u)-337(l'on)-337(a)]TJ /F8 1 Tf 4.1747 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.716 73.504 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.584 51.515 Td [(<)-289(f)]TJ /F2 1 Tf 1.6627 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.6747 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.207 73.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.91 51.515 Td [(.)-457(Ces)-337(parties)]TJ /F8 1 Tf -33.578 -1.1988 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 71.788 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7289 50.316 Td [(formen)24(t)-325(une)-319(partition)-319(de)-319(la)-319(source)-319(de)]TJ /F8 1 Tf 16.53 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(ce)-325(qui)-319(p)-30(ermet)-319(de)-319(d)24(\266)475(e\257nir)-325(une)-319(fonction)]TJ /F8 1 Tf 16.44 0 Td [(')]TJ /F2 1 Tf 0.96988 0 Td [(qui)-319(est)]TJ -36.253 -1.2048 Td [(constan)24(te)-319(sur)-331(c)24(haque)]TJ /F8 1 Tf 9.4518 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.233 70.073 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.193 49.111 Td [(o)-24(\265)524(u)-319(elle)-331(prend)-325(une)-319(v)48(aleur)]TJ /F8 1 Tf 10.807 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.474 70.073 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.241 49.111 Td [(comprise)-319(en)24(tre)]TJ /F8 1 Tf 6.6867 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.897 70.073 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.247 49.111 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.871 70.073 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.373 49.111 Td [(.)-439(Il)-325(est)-325(clair)]TJ -34.042 -1.1988 Td [(que)]TJ /F8 1 Tf 1.8916 0 Td [(')]TJ /F2 1 Tf 1.0181 0 Td [(est)-367(une)-367(b)-30(onne)-367(appro)24(ximation)-367(de)]TJ /F8 1 Tf 14.584 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)-373(plus)-367(pr)24(\266)475(ecis)24(\266)475(emen)24(t)-367(l')24(\266)475(ecart)-367(en)24(tre)]TJ /F8 1 Tf 13.97 0 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-367(et)]TJ /F8 1 Tf 1.9518 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-367(est)]TJ -37.181 -1.1988 Td [(ma)-54(jor)18(\266)475(e)-289(par)]TJ /F8 1 Tf 5.0422 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.767 66.633 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.47 46.714 Td [(\241)]TJ /F8 1 Tf 0.90361 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.621 66.633 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.651 46.714 Td [(si)]TJ /F8 1 Tf 0.95783 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.974 66.633 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.488 46.714 Td [(.)-427(On)-283(p)-30(ourra)-289(donc)-283(le)-289(rendre)-283(aussi)-289(p)-30(etit)-283(que)-283(l'on)-289(v)24(eut)-289(si)-283(le)-289(pas)]TJ -13.157 -1.1988 Td [(de)-313(la)-319(sub)-30(division)-319(par)-319(les)]TJ /F8 1 Tf 10.831 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.043 64.918 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.452 45.515 Td [(est)-319(su\261sammen)24(t)-325(p)-24(etit.)-445(L'id)30(\266)469(ee)-319(est)-325(alors)-319(d'appro)-30(c)24(her)-319(l'in)24(t)24(\266)475(egrale)]TJ -12.12 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(par)-331(celle)-343(de)]TJ /F8 1 Tf 5.3373 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(.)]TJ -8.259 -1.747 Td [(Dans)-253(le)-259(cas)-259(particulier)-253(o)-24(\265)524(u)-253(l'on)-259(prend)-253(p)-30(our)]TJ /F8 1 Tf 18.006 0 Td [(f)]TJ /F2 1 Tf 0.85542 0 Td [(une)-253(fonction)-259(monotone,)-271(on)-253(remarque)-253(que)-259(les)]TJ /F8 1 Tf 19.217 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 60.693 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 40.611 Td [(son)24(t)-313(des)-307(in)24(terv)54(alles,)-313(par)-313(suite)]TJ /F8 1 Tf 12.934 0 Td [(')]TJ /F2 1 Tf 0.96386 0 Td [(est)-313(une)-307(fonction)-313(en)-307(escalier)-307(don)24(t)-313(l'in)24(t)24(\266)475(egrale)-313(est)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.474 59.909 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 39.687 41.563 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.543 56.452 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.301 40.611 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.94 57.909 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.217 40.611 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.638 57.909 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 40.611 Td [(\))]TJ -38.777 -2.1145 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.4337 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.914 54.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6566 38.497 Td [(\))-385(est)-385(la)-379(longueur)-385(de)-379(l'in)24(terv)54(alle)]TJ /F8 1 Tf 14.024 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.526 54.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.759 38.497 Td [(.)-590(Cette)-385(somme)-379(est)-385(une)-385(somme)-379(de)-379(Riemann)-385(et,)]TJ -18.428 -1.2048 Td [(dans)-331(ce)-337(cas,)-337(il)-337(n'y)-331(a)-331(pratiquemen)24(t)-337(pas)-337(de)-337(di\256)30(\266)469(erence)-337(en)30(tre)-337(les)-337(deux)-331(m)24(\266)469(etho)-30(des.)]TJ 0 -1.747 Td [(Dans)-289(le)-289(cas)-289(g)24(\266)475(en)24(\266)475(eral)]TJ /F8 1 Tf 8.5361 0 Td [(')]TJ /F2 1 Tf 0.93976 0 Td [(est)-295(plus)-283(compliqu)24(\266)469(ee,)-301(il)-289(s'agit)-283(de)-289(ce)-289(qu'on)-289(app)-30(elle)-295(une)-289(fonction)-259(\266)469(etag)24(\266)475(ee.)]TJ -9.4759 -1.9458 Td [(Elle)-307(p)-30(eut)-307(s')24(\266)469(ecrire)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.241 49.866 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 12.97 34.545 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.31 46.409 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.584 33.599 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.707 47.866 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.5 33.599 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.983 47.866 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 33.325 66.692 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.337 33.599 Td [(o)-24(\265)524(u)-307(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.56 47.866 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 39.723 66.692 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.536 33.599 Td [(est)-313(la)-307(fonction)-307(indicatrice)-307(de)]TJ /F8 1 Tf 12.627 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.534 47.866 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.554 33.599 Td [(et)-307(l'in)24(t)24(\266)469(egrale)-307(de)]TJ /F8 1 Tf 7.0602 0 Td [(')]TJ /F2 1 Tf 0.96385 0 Td [(v)54(aut)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3103 45.547 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 31.527 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3793 42.081 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 30.575 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.767 43.538 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 8.0241 31.936 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.716 43.538 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 20.373 60.656 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.554 30.575 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)-439(Le)-301(probl)24(\265)469(eme)-301(est)-307(alors)-307(de)-301(d)24(\266)475(e\257nir)-307(les)-301(in)24(t)24(\266)469(egrales)-301(du)-307(t)24(yp)-30(e)]TJ /F9 1 Tf 24.175 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.672 43.538 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.554 30.575 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3916 0 Td [(p)-30(our)]TJ /F8 1 Tf -37.133 -2.0602 Td [(A)]TJ /F2 1 Tf 1.1566 0 Td [(partie)-415(de)]TJ /F3 1 Tf 4.3855 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-674(P)30(ar)-409(d)24(\266)469(e\257nition)-409(cette)-415(in)24(t)24(\266)475(egrale)-415(est)-421(la)-409(mesure)-415(de)]TJ /F8 1 Tf 21.831 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(.)-674(Dans)-415(le)-415(cas)-409(o)-30(\265)530(u)]TJ /F8 1 Tf 7.9277 0 Td [(A)]TJ /F2 1 Tf 1.1566 0 Td [(est)]TJ -37.934 -1.2048 Td [(un)-343(in)30(terv)54(alle,)-343(cette)-343(mesure)-343(est)-349(la)-337(longueur)-343(de)]TJ /F8 1 Tf 20.078 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)-463(Il)-343(s'agit)-343(donc)-337(de)-343(g)30(\266)469(en)30(\266)469(eraliser)-343(la)-337(notion)-343(de)]TJ -20.825 -1.1988 Td [(longueur)-337(\265)500(a)-331(des)-337(parties)-337(qui)-337(ne)-331(son)24(t)-337(plus)-337(des)-337(in)30(terv)54(alles.)]TJ 0 -1.753 Td [(Nous)-271(v)24(o)24(y)24(ons)-277(donc)-277(que)-277(la)-271(m)24(\266)475(etho)-36(de)-271(de)-271(Leb)-36(esgue)-277(supp)-30(ose)-271(la)-277(d)30(\266)469(e\257nition)-277(pr)30(\266)469(ealable)-277(de)-271(la)-271(mesure)]TJ 0 -1.1988 Td [(de)-427(parties)-433(de)]TJ /F3 1 Tf 6.241 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-734(La)-433(m)24(\266)475(etho)-30(de)-433(a)-427(donc)-433(s)24(\266)475(epar)24(\266)475(e)-433(le)-439(probl)24(\265)475(eme)-433(de)-427(l'in)24(t)24(\266)469(egration)-433(en)-427(deux)-433(sous-)]TJ -6.9639 -1.1988 Td [(probl)24(\265)469(emes)-373(:)-530(d'une)-385(part)-379(celui)-373(de)-379(la)-373(mesure)-385(\(c'est-)0(\265)500(a-dire)-379(l'in)24(t)24(\266)469(egration)-379(des)-379(fonctions)-379(indica-)]TJ 0 -1.1988 Td [(trices\),)-343(d'autre)-337(part)-343(celui)-343(de)-337(l'extension)-343(de)-337(l'in)24(t)24(\266)469(egrale)-343(\265)500(a)-337(des)-343(fonctions)-337(plus)-343(g)30(\266)469(en)30(\266)469(erales)-337(gr)-6(^)500(ace)]TJ 0 -1.2048 Td [(\265)499(a)-331(l'appro)24(ximation)-337(par)-337(des)-337(fonctions)-307(\266)469(etag)30(\266)469(ees.)]TJ 0 -1.747 Td [(L'appro)24(ximation)-457(par)-457(des)-463(fonctions)-427(\266)469(etag)24(\266)475(ees)-433(\266)475(etan)24(t)-463(meilleure)-457(que)-463(l'appro)24(ximation)-457(par)-463(des)]TJ 0 -1.2048 Td [(fonctions)-283(en)-289(escalier,)-295(les)-289(fonctions)-289(relev)54(an)24(t)-289(de)-283(cette)-289(appro)-30(c)24(he)-289(p)-30(euv)24(en)30(t)-265(^)475(etre)-289(moins)-289(r)30(\266)469(eguli)30(\265)469(eres)]TJ 0 -1.1988 Td [(que)-259(celles)-259(concern)30(\266)469(ees)-253(par)-259(la)-259(th)24(\266)469(eorie)-259(de)-259(Riemann)-253(et)-259(formen)24(t)-259(une)-259(classe)-259(a)24(y)24(an)30(t)-259(de)-259(grandes)-259(pro-)]TJ 0 -1.1988 Td [(pri)24(\266)475(et)24(\266)469(es)-325(de)-325(stabilit)24(\266)469(e.)-445(Mais)-331(en)-325(fait,)-331(l'in)24(t)24(\266)469(er)24(^)469(et)-325(essen)24(tiel)-331(de)-325(cette)-331(m)24(\266)475(etho)-36(de)-325(est)-331(qu'elle)-331(s'applique)]TJ 0 -1.2048 Td [(\265)499(a)-283(bien)-283(d'autres)-283(th)24(\266)469(eories)-283(de)-283(la)-283(mesure)-289(que)-283(celle)-289(issue)-283(de)-277(la)-283(notion)-283(de)-283(longueur.)-427(Sa)-283(p)-30(ort)24(\266)469(ee)-283(est)]TJ 0 -1.1988 Td [(de)-385(ce)-391(fait)-391(consid)24(\266)475(erablemen)24(t)-391(plus)-391(grande.)-614(En)-391(particulier)-385(elle)-397(p)-30(ermet)-391(de)-385(d)24(\266)469(e\257nir)-391(l'in)24(t)24(\266)475(egrale)]TJ 0 -1.1988 Td [(de)-355(fonctions)-355(d)24(\266)475(e\257nies)-361(sur)-361(n'imp)-30(orte)-355(quel)-361(ensem)24(ble)]TJ /F8 1 Tf 21.964 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(d)30(\265)469(es)-355(lors)-361(qu'on)-355(a)-355(su)-361(d)30(\266)469(e\257nir)-355(la)-355(mesure)]TJ -23.229 -1.1988 Td [(de)-331(su\261sammen)24(t)-337(de)-337(parties)-331(de)]TJ /F8 1 Tf 13.331 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(.)]TJ /F5 1 Tf -14.235 -5.5181 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.5 0 Td [(4)]TJ ET endstream endobj 42 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 41 0 R /ProcSet 2 0 R >> /Contents 40 0 R >> endobj 41 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F13 39 0 R >> endobj 44 0 obj << /Length 11186 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-18024(I.)-385(In)30(tro)-36(duction)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I.4.)-236(Le)-376(probl)25(\265)537(eme)-376(de)-381(la)-376(mesure.)]TJ 0 -2.5226 Td [(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.259 59.195 Td [(1.)-289(L'in)36(tuition.)]TJ /F2 1 Tf -1.9277 -1.6024 Td 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[(L'additivit)24(\266)475(e)-343(d)30(\266)469(enom)24(brable)-337(exig)24(\266)475(ee)-337(p)-30(our)-337(une)-337(mesure)-337(implique)-337(alors)-337(:)]TJ 0 -1.5663 Td [({)-421(si,)-445(utilisan)30(t)-427(l'op)0(\266)469(eration)-427(\(A\),)-421(une)-427(partie)]TJ /F8 1 Tf 18.627 0 Td [(U)]TJ /F2 1 Tf 1.2169 0 Td [(est)-421(construite)-421(comme)-427(r)30(\266)469(eunion)-421(d'une)-427(famille)]TJ -19.843 -1.1988 Td [(d)30(\266)469(enom)24(brable)-427(de)-427(parties)]TJ /F8 1 Tf 10.825 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.095 58.193 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.596 40.816 Td [(deux)-427(\265)500(a)-427(deux)-427(disjoin)30(tes)-433(don)24(t)-433(on)-427(sait)-427(d)24(\266)475(e\257nir)-433(la)-427(mesure)]TJ /F8 1 Tf 24.229 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.638 58.193 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 40.816 Td [(\))]TJ -38.777 -1.3976 Td [(alors)-337(la)-331(mesure)-343(de)]TJ /F8 1 Tf 8.241 0 Td [(U)]TJ /F2 1 Tf 1.1265 0 Td [(doit)-307(^)475(etre)-337(la)-331(somme)]TJ /F9 1 Tf 8.512 0.94579 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94579 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.31 56.193 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.108 39.418 Td [(\))-337(;)]TJ -21.777 -1.7711 Td [({)-325(si,)-331(utilisan)30(t)-331(l'op)-6(\266)475(eration)-331(\(B\),)-331(une)-331(partie)]TJ /F8 1 Tf 17.904 0 Td [(W)]TJ /F2 1 Tf 1.4036 0 Td [(est)-331(construite)-331(comme)-325(le)-331(compl)24(\266)469(emen)30(taire)-331(dans)]TJ /F8 1 Tf -19.307 -1.1988 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(d'une)-325(partie)]TJ /F8 1 Tf 5.5904 0 Td [(V)]TJ /F2 1 Tf 1.1265 0 Td [(et)-325(l'on)-325(sait)-325(d)30(\266)469(e\257nir)-325(les)-325(mesures)]TJ /F8 1 Tf 13.295 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-325(et)]TJ /F8 1 Tf 1.8675 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F2 1 Tf 0.80121 0 Td [(\))-325(alors)-331(la)-319(mesure)-325(de)]TJ /F8 1 Tf 8.9096 0 Td [(W)]TJ /F2 1 Tf 1.4036 0 Td [(doit)]TJ -37.47 -1.1988 Td [(^)469(etre)]TJ /F8 1 Tf 1.9759 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))]TJ /F6 1 Tf 0.50602 0 Td [(\241)]TJ /F8 1 Tf 0.89157 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F2 1 Tf 0.80723 0 Td [(\))-283(\(rapp)-24(elons)-289(qu'il)-283(s'agit)-277(de)-283(parties)-283(b)-30(orn)30(\266)469(ees)-283(et)-283(que)-283(ces)-277(mesures)-283(son)24(t)-283(\257nies\).)]TJ -7.4819 -1.5663 Td [(On)-379(con)0(\273)439(coit)-385(alors)-379(que,)-397(connaissan)24(t)-385(la)-379(mesure)-379(des)-385(segmen)24(ts)-379(\()24(\266)469(egale)-379(\265)500(a)-379(la)-379(longueur\),)-391(on)-385(p)-30(eut)]TJ 0 -1.2048 Td [(d)30(\266)469(e\257nir)-259(inductiv)30(emen)24(t)-259(la)-253(mesure)-265(de)-253(toute)-259(partie)-259(bien)-259(d)30(\266)469(e\257nie)-259(\(b)-30(or)24(\266)475(elienne\).)-427(En)-253(fait)-259(une)-259(m)24(^)475(eme)]TJ 0 -1.1988 Td [(partie)-301(p)-24(eut)-271(^)469(etre)-295(obten)24(ue)-295(de)-295(plusieurs)-301(fa)-6(\273)439(cons)-301(\265)500(a)-289(partir)-301(de)-289(segmen)18(ts)-295(gr)0(^)500(ace)-295(aux)-301(op)-6(\266)475(erations)-295(\(A\))]TJ 0 -1.1988 Td [(et)-246(\(B\))-253(et)-253(le)-253(probl)24(\265)469(eme)-246(est)-253(de)-246(mon)24(trer)-253(que)-253(le)-253(r)30(\266)469(esultat)-253(obten)24(u)-246(p)-30(our)-253(la)-246(mesure)-253(de)-246(cette)-253(partie)-253(ne)]TJ 0 -1.1988 Td [(d)30(\266)469(ep)-30(end)-271(pas)-277(du)-271(pro)-30(c)18(\266)475(ed)24(\266)475(e)-277(de)-271(construction)-271(consid)24(\266)475(er)18(\266)475(e.)-433(Borel)-271(donne)-277(des)-277(indications)-271(de)-271(la)-271(preuv)30(e)]TJ 0 -1.2048 Td [(de)-379(cette)-379(ind)24(\266)475(ep)-30(endance)-379(en)-379(s'appuy)30(an)24(t)-379(sur)-385(une)-379(v)54(arian)24(te)-379(de)-379(ce)-385(qu'on)-379(app)-30(elle)-379(main)18(tenan)24(t)-379(le)]TJ 0 -1.1988 Td [(th)24(\266)469(eor)24(\265)469(eme)-277(de)-283(Borel-Leb)-30(esgue)-283(mais)-277(la)-283(preuv)24(e)-283(con)30(v)48(aincan)24(te)-283(sera)-277(app)-36(ort)24(\266)475(ee)-283(par)-283(la)-283(construction)]TJ 0 -1.1988 Td [(de)-331(Leb)-36(esgue)-331(en)-337(1901.)]TJ 0 -2.1024 Td [(Ce)-331(pro)-24(c)18(\266)475(ed)24(\266)475(e)-331(s')24(\266)469(etend)-331(naturellemen)24(t)-325(\265)500(a)-325(tout)-331(segmen)24(t)-331(de)]TJ /F3 1 Tf 23.102 0 Td [(R)]TJ /F2 1 Tf 1.0482 0 Td [(et,)-325(comme)-331(toute)-325(partie)-331(de)]TJ /F3 1 Tf 11.633 0 Td [(R)]TJ /F2 1 Tf 1.0482 0 Td [(est)-331(la)]TJ -36.831 -1.1988 Td [(r)30(\266)469(eunion)-355(d'une)-355(famille)-349(d)24(\266)475(enom)24(brable)-355(de)-349(parties)-355(disjoin)30(tes)-355(b)-30(orn)24(\266)469(ees)-349(\(donc)-355(con)24(ten)24(ues)-349(dans)-355(un)]TJ 0 -1.1988 Td [(segmen)24(t\),)-313(cela)-319(p)-30(ermet)-313(de)-307(d)24(\266)475(e\257nir)-313(la)]TJ /F1 1 Tf 15.38 0 Td [(tribu)-355(de)-361(Borel)]TJ /F2 1 Tf 7.4217 0 Td [(form)24(\266)475(ee)-313(par)-313(les)-313(parties)-313(bien)-313(d)30(\266)469(e\257nies)-313(de)]TJ /F3 1 Tf -22.801 -1.1988 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(et)-337(la)]TJ /F1 1 Tf 2.2771 0 Td [(mesure)-385(de)-391(Borel)]TJ /F2 1 Tf 8.5663 0 Td [(sur)-337(les)-307(\266)469(el)24(\266)469(emen)30(ts)-337(de)-331(cette)-337(tribu.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 13.867 Td [(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.259 16.623 Td [(3.)-289(La)-385(con)30(tribution)-385(de)-385(Leb)-36(esgue.)]TJ /F2 1 Tf -1.9277 -1.5663 Td [(A)-253(l'in)24(v)24(erse)-246(de)-253(Borel,)-271(Leb)-36(esgue)-246(consid)24(\265)475(ere)-259(toutes)-253(les)-253(parties)-253(de)-246([0;)-168(1])-253(et)-253(rec)24(herc)24(he)-246(un)-253(ma)-60(joran)24(t)]TJ /F14 1 Tf 0 -1.1988 Td [(a)-512(priori)]TJ /F2 1 Tf 3.988 0 Td [(de)-493(la)-500(mesure)-469(\266)469(ev)30(en)24(tuelle)-500(de)-493(cette)-493(partie.)-927(Il)-493(d)24(\266)469(egage)-500(ainsi)-493(la)-493(notion)-493(de)-493(mesure)]TJ -3.988 -1.2048 Td [(ext)30(\266)469(erieure)-337(de)-337(toute)-331(partie)-337(de)-337([0;)-168(1].)]TJ 0 -1.5663 Td [(Plus)-325(pr)24(\266)469(ecis)30(\266)469(emen)24(t,)-325(p)-30(our)]TJ /F8 1 Tf 10.337 0 Td [(A)]TJ /F2 1 Tf 1.0723 0 Td [(partie)-331(de)-325([0;)-168(1],)-325(Leb)-30(esgue)-325(consid)24(\265)475(ere)-331(tous)-325(les)-325(recouvremen)24(ts)-325(de)]TJ /F8 1 Tf 27.006 0 Td [(A)]TJ /F2 1 Tf -38.416 -1.1988 Td [(par)-361(une)-367(famille)-367(d)30(\266)469(enom)24(brable)-367(d'in)24(terv)54(alles.)-536(La)-367(somme)-361(des)-367(longueurs)-367(de)-361(ces)-367(in)24(terv)60(alles)-367(est)]TJ 0 -1.1988 Td [(alors)-349(un)-349(ma)-60(joran)30(t)-355(de)-343(la)-349(mesure)-355(de)]TJ /F8 1 Tf 15.452 0 Td [(A)]TJ /F2 1 Tf 1.0964 0 Td [(\(si)-349(on)-349(p)-30(eut)-349(la)-349(d)24(\266)475(e\257nir\))-349(et)-349(Leb)-36(esgue)-349(d)30(\266)469(e\257nit)-349(la)-343(mesure)]TJ -16.548 -1.1988 Td [(ext)30(\266)469(erieure)-355(de)]TJ /F8 1 Tf 6.012 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-355(not)24(\266)469(ee)]TJ /F8 1 Tf 3.3133 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.302 11.306 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.747 7.4906 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\),)-355(comme)-349(la)-349(b)-30(orne)-349(inf)24(\266)469(erieure)-349(de)-355(ces)-349(ma)-54(joran)24(ts)-349(;)-355(il)-349(s'agit)-355(bien)]TJ -12.554 -1.2048 Td [(d'un)-337(ma)-60(joran)30(t)]TJ /F14 1 Tf 6.6145 0 Td [(a)-361(priori)]TJ /F2 1 Tf 3.6867 0 Td [(de)-331(la)-337(mesure)-307(\266)469(ev)24(en)30(tuelle)-337(de)]TJ /F8 1 Tf 11.819 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)]TJ /F5 1 Tf -22.867 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.5 0 Td [(6)]TJ ET endstream endobj 49 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 48 0 R /ProcSet 2 0 R >> /Contents 47 0 R >> endobj 48 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F14 43 0 R >> endobj 50 0 obj << /Length 12355 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-18024(I.)-385(In)30(tro)-36(duction)]TJ /F2 1 Tf 0 -2.4036 Td [(Si)-391(on)-391(p)-30(eut)-391(d)24(\266)475(e\257nir)-397(la)-391(mesure)-391(de)]TJ /F8 1 Tf 14.217 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)-403(utilisan)30(t)-391(l'op)-6(\266)469(eration)-391(\(B\))-397(on)-391(v)24(oit)-391(que)-397(l'on)-391(p)-30(eut)-391(d)24(\266)469(e\257nir)]TJ -14.97 -1.1988 Td [(la)-439(mesure)-445(du)-439(compl)24(\266)469(emen)30(taire)]TJ /F8 1 Tf 13.578 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.129 87.909 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.506 61.021 Td [(de)]TJ /F8 1 Tf 1.4398 0 Td [(A)]TJ /F2 1 Tf 1.1867 0 Td [(dans)-439([0;)-168(1])-439(par)]TJ /F8 1 Tf 6.7771 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.69 87.909 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.331 61.021 Td [(\))-457(=)-457(1)]TJ /F6 1 Tf 2.8675 0 Td [(\241)]TJ /F8 1 Tf 1.0723 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)-765(On)-439(a)-439(alors)]TJ -32.958 -1.1988 Td [(1)]TJ /F6 1 Tf 0.72289 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.707 86.193 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.133 59.822 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.991 86.193 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.681 59.822 Td [(\))-337(et)-331(donc)]TJ /F8 1 Tf 4.2771 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(1)]TJ /F6 1 Tf 0.72289 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.776 86.193 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.759 59.822 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.069 86.193 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.307 59.822 Td [(\).)]TJ -20.976 -1.6566 Td [(P)30(our)-265(toute)-259(partie)]TJ /F8 1 Tf 7.7169 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-277(la)-259(quan)24(tit)24(\266)475(e)]TJ /F8 1 Tf 5.4639 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.819 83.021 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.596 58.165 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(=)-277(1)]TJ /F6 1 Tf 2.3012 0 Td [(\241)]TJ /F8 1 Tf 0.8494 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.871 83.823 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.223 58.165 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.155 83.823 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.771 58.165 Td [(\))-265(est)-265(par)-259(d)24(\266)475(e\257nition)-265(la)-259(mesure)-265(in)24(t)24(\266)475(erieure)]TJ -22.44 -1.2048 Td [(de)]TJ /F8 1 Tf 1.4277 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)-451(c'est)-433(un)-433(minoran)30(t)]TJ /F14 1 Tf 9.0542 0 Td [(a)-451(priori)]TJ /F2 1 Tf 3.8675 0 Td [(de)-433(la)-427(mesure)-403(\266)469(ev)24(en)30(tuelle)-433(de)]TJ /F8 1 Tf 12.301 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)-734(Leb)-30(esgue)-433(app)-30(elle)-433(alors)]TJ -28.151 -1.1988 Td [(mesurables)-403(les)-397(parties)]TJ /F8 1 Tf 10.054 0 Td [(A)]TJ /F2 1 Tf 1.1506 0 Td [(don)24(t)-403(la)-397(mesure)-403(in)24(t)24(\266)475(erieure)-403(est)-373(\266)469(egale)-403(\265)500(a)-397(la)-397(mesure)-403(ext)24(\266)475(erieure,)-415(leur)]TJ -11.205 -1.1988 Td [(v)54(aleur)-337(comm)24(une)-337(est)-337(par)-337(d)30(\266)469(e\257nition)-337(la)-331(mesure)-337(de)]TJ /F8 1 Tf 21.084 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)]TJ -21.831 -1.6566 Td [(Leb)-30(esgue)-373(prouv)24(e)-373(que)-373(l'ensem)24(ble)]TJ /F6 1 Tf 14.434 0 Td [(L)]TJ /F2 1 Tf 1.0602 0 Td [(des)-373(parties)-373(mesurables)-373(\(au)-373(sens)-373(de)-367(Leb)-36(esgue\))-367(p)-30(oss)24(\265)475(ede)]TJ -15.494 -1.2048 Td [(les)-331(segmen)24(ts)-331(et)-331(est)-331(stable)-331(par)-331(les)-331(op)0(\266)469(erations)-331(\(A\))-331(et)-331(\(B\))-331(;)-331(il)-331(con)30(tien)18(t)-331(donc)-331(l'ensem)30(ble)]TJ /F6 1 Tf 36.753 0 Td [(B)]TJ /F2 1 Tf 1.0181 0 Td [(des)]TJ -37.771 -1.1988 Td [(parties)-403(b)-30(or)30(\266)469(eliennes.)-650(Cet)-403(ensem)24(ble)]TJ /F6 1 Tf 15.434 0 Td [(L)]TJ /F2 1 Tf 1.0904 0 Td [(est)-379(\266)475(egalemen)24(t)-403(stable)-403(par)-403(in)24(tersection,)-421(c'est)-403(la)-403(tribu)]TJ -16.524 -1.1988 Td [(de)-349(Leb)-30(esgue.)-500(La)-355(mesure)-355(d)24(\266)475(e\257nie)-355(sur)]TJ /F6 1 Tf 15.886 0 Td [(L)]TJ /F2 1 Tf 1.0361 0 Td [(est)-355(e\256ectiv)24(emen)24(t)-355(d)30(\266)469(enom)24(brablemen)30(t)-355(additiv)30(e)-355(ce)-355(qui)]TJ -16.922 -1.2048 Td [(prouv)30(e)-343(en)-331(particulier)-337(la)-331(non)-337(con)24(tradiction)-337(de)-331(la)-337(construction)-337(de)-331(Borel.)]TJ 0 -1.6566 Td [(La)-240(tribu)-246(de)-240(Leb)-30(esgue)-246(est)-240(strictemen)18(t)-240(plus)-240(grande)-246(que)-240(celle)-246(de)-240(Borel)-246(\(son)-240(cardinal)-246(est)-240(stricte-)]TJ 0 -1.1988 Td [(men)24(t)-373(sup)0(\266)469(erieur\),)-385(en)-373(fait)-373(les)-349(\266)469(el)24(\266)469(emen)30(ts)-373(de)]TJ /F6 1 Tf 17.867 0 Td [(L)]TJ /F2 1 Tf 1.0602 0 Td [(son)18(t)-373(les)-373(parties)-379(que)-373(l'on)-379(obtien)30(t)-379(en)-367(prenan)18(t)-373(la)]TJ -18.928 -1.1988 Td [(r)30(\266)469(eunion)-487(d'une)-493(partie)-487(b)-30(or)30(\266)469(elienne)-487(et)-487(d'une)-487(partie)-493(n)24(\266)475(egligeable)-487(\(c'est-)-6(\265)500(a-dire)-487(de)-487(mesure)-487(ex-)]TJ 0 -1.1988 Td [(t)24(\266)475(erieure)-343(n)30(ulle\).)]TJ 0 -1.988 Td [(Plus)-337(g)24(\266)475(en)24(\266)475(eralemen)24(t)-337(la)-337(m)24(\266)469(etho)-30(de)-337(pr)30(\266)469(ec)24(\266)475(eden)24(te)-337(p)-30(ermet)-337(de)-337(d)24(\266)475(e\257nir)-337(la)-337(mesure)-343(ext)30(\266)469(erieure)-337(de)-337(n'im-)]TJ 0 -1.1988 Td [(p)-30(orte)-331(quelle)-343(partie)-337(de)]TJ /F3 1 Tf 9.7289 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-445(On)-337(p)-30(eut)-337(encore)-337(mon)24(trer)-337(\(cf.)]TJ /F1 1 Tf 13.488 0 Td [(VI)-36(I.2.)]TJ /F2 1 Tf 2.988 0 Td [(\))-451(:)]TJ -26.928 -1.6566 Td [({)-463(cette)-475(mesure)-469(ext)24(\266)475(erieure)-475(p)-30(ermet)-469(de)-469(d)24(\266)475(e\257nir)-475(la)-463(notion)-469(de)-469(partie)-475(mesurable)-469(\(au)-469(sens)-469(de)]TJ 0 -1.2048 Td [(Leb)-30(esgue\))-325(\(ces)-331(parties)-325(p)-30(ouv)48(an)24(t)-301(^)475(etre)-331(non)-325(b)-30(orn)24(\266)475(ees,)-331(la)-325(consid)24(\266)475(eration)-331(d'une)-331(mesure)-331(in)24(t)24(\266)475(erieure)]TJ 0 -1.1988 Td [(n'est)-331(plus)-337(su\261san)24(te\))-343(;)]TJ 0 -1.6566 Td [({)-373(l'ensem)30(ble)-379(de)-373(ces)-373(parties)-379(mesurables,)-385(encore)-379(not)18(\266)475(e)]TJ /F6 1 Tf 22.964 0 Td [(L)]TJ /F2 1 Tf 0.69277 0 Td [(,)-385(est)-379(stable)-373(par)-379(passage)-373(au)-373(compl)24(\266)469(e-)]TJ -23.657 -1.1988 Td [(men)24(taire)-337(et)-331(r)24(\266)475(eunion)-343(d)30(\266)469(enom)24(brable,)-337(c'est)-337(la)]TJ /F1 1 Tf 18.524 0 Td [(tribu)-379(de)-391(Leb)-36(esgue)]TJ /F2 1 Tf 9.3976 0 Td [(sur)]TJ /F3 1 Tf 1.6747 0 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(;)]TJ -30.651 -1.6566 Td [({)-283(la)-283(restriction)-289(de)-283(la)-289(mesure)-283(ext)24(\266)469(erieure)-289(\265)500(a)]TJ /F6 1 Tf 17.663 0 Td [(L)]TJ /F2 1 Tf 0.96988 0 Td [(est)-289(d)24(\266)469(enom)30(brablemen)24(t)-289(additiv)30(e,)-301(c'est)-289(la)]TJ /F1 1 Tf 16.952 0 Td [(mesure)]TJ -35.584 -1.2048 Td [(de)-385(Leb)-36(esgue)]TJ /F2 1 Tf 6.494 0 Td [(sur)]TJ /F3 1 Tf 1.6747 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -8.8916 -1.6566 Td [(La)-367(mesure)-367(de)-361(Borel)-361(est)-367(une)-367(restriction)-367(de)-361(la)-361(mesure)-367(de)-361(Leb)-36(esgue,)-373(aussi)-361(il)-361(est)-367(fr)18(\266)475(equen)24(t)-367(de)]TJ 0 -1.1988 Td [(con)30(tin)18(uer)-331(\265)500(a)-337(app)-30(eler)-337(mesure)-337(de)-337(Leb)-30(esgue)-337(cette)-337(restriction.)]TJ 0 -1.9819 Td [(T)78(out)-295(ce)-295(qui)-295(vien)24(t)-295(d')24(^)469(etre)-289(dit)-301(s')24(\266)475(etend)-301(sans)-289(di\261cult)24(\266)469(e)-295(au)-295(cas)-295(de)]TJ /F3 1 Tf 25.813 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.603 36.426 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.705 25.045 Td [(en)-295(rempla)0(\273)439(can)24(t)-295(les)-295(segmen)24(ts)]TJ -27.373 -1.2048 Td [(par)-373(les)-379(pa)30(v)48(\266)475(es)-379(\(i.e.)-566(les)-379(pro)-30(duits)-373(de)-373(segmen)18(ts\))-373(don)24(t)-379(on)-373(d)24(\266)475(e\257nit)-379(la)-373(mesure)-379(comme)-373(le)-379(pro)-30(duit)]TJ 0 -1.1988 Td [(des)-337(longueurs)-331(des)-337(pro)-60(jections.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 16.294 Td [(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.259 19.533 Td [(4.)-289(Les)-385(parado)30(xes.)]TJ /F2 1 Tf -1.9277 -1.6566 Td [(La)-319(m)24(\266)469(etho)-30(de)-319(constructiv)24(e)-325(de)-313(Borel)-325(ne)-319(p)-24(ermet)-325(pas)-319(de)-319(d)24(\266)475(e\257nir)-325(la)-319(mesure)-319(de)-319(toutes)-325(les)-319(parties)]TJ 0 -1.1988 Td [(de)-283([0;)-168(1].)-433(La)-289(m)24(\266)475(etho)-30(de)-289(de)-283(Leb)-36(esgue)-289(p)-24(ermet)-289(de)-289(d)24(\266)475(e\257nir)-289(la)-289(mesure)-289(de)-283(b)-30(eaucoup)-289(plus)-283(de)-289(parties)]TJ 0 -1.1988 Td 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[(de)-319(longueur,)-319(d'aire,)-325(de)-319(v)24(olume,)-325(etc.)-439(La)-325(consid)30(\266)469(eration)-325(d'une)-319(telle)-319(mesure)-325(conduit)-319(en)-319(e\256et)-319(\265)500(a)]TJ 0 -1.2048 Td [(des)-337(parado)24(xes)-337(que)-337(nous)-337(allons)-331(exp)-36(oser.)]TJ 0 -1.9819 Td [(Nous)-421(supp)-30(osons)-427(donc)-427(l'existence)-427(d'une)-427(application)-427(mesure)]TJ /F8 1 Tf 26.428 0 Td [(m)]TJ /F2 1 Tf 1.3072 0 Td [(:)]TJ /F8 1 Tf 0.70482 0 Td [(A)]TJ /F6 1 Tf 1.1747 0 Td [(7!)]TJ /F8 1 Tf 1.4277 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74698 0 Td [(\))-427(d)24(\266)475(e\257nie)-427(p)-30(our)]TJ -33.06 -1.1988 Td [(toutes)-397(les)-391(parties)-397(de)]TJ /F3 1 Tf 9.3133 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.991 11.306 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.91 7.4906 Td [(,)-409(\265)500(a)-391(v)48(aleurs)-391(p)-30(ositiv)24(es,)-409(additiv)30(e,)-415(in)24(v)54(arian)24(te)-391(par)-397(d)24(\266)475(eplacemen)24(t)-397(et)-391(qui)]TJ -10.578 -1.2048 Td [(v)54(aut)-337(1)-337(p)-30(our)-331(une)-337(partie)-337(unit)24(\266)469(e)]TJ /F8 1 Tf 12.789 0 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(\(segmen)24(t,)-337(carr)24(\266)469(e,)-337(cub)-30(e,)]TJ /F8 1 Tf 9.9578 0 Td [(:)-168(:)-168(:)]TJ /F2 1 Tf 1.3373 0 Td [(\))-337(\257x)24(\266)469(ee)-331(\265)500(a)-337(l'a)24(v)54(ance.)]TJ -6.0422 -2.3976 Td [(7)]TJ /F5 1 Tf 6.012 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 52 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 51 0 R /ProcSet 2 0 R >> /Contents 50 0 R >> endobj 51 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R /F14 43 0 R >> endobj 53 0 obj << /Length 20526 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-18024(I.)-385(In)30(tro)-36(duction)]TJ /F2 1 Tf 0 -2.4036 Td [(Explicitons)-337(tout)-337(d'ab)-30(ord)-337(quelques)-331(cons)24(\266)469(equences)-337(simples)-337(de)-337(l'additivit)30(\266)469(e)-337(de)]TJ /F8 1 Tf 32.452 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ -33.331 -1.6566 Td [(P)30(our)]TJ /F8 1 Tf 2.4578 0 Td [(A)]TJ /F6 1 Tf 1.0723 0 Td [(\275)]TJ /F8 1 Tf 1.1024 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.336 87.254 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.994 60.563 Td [(,)-367(si)]TJ /F8 1 Tf 1.6807 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.207 87.254 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.295 60.563 Td [(est)-367(le)-361(compl)24(\266)469(emen)24(taire)-361(de)]TJ /F8 1 Tf 11.259 0 Td [(A)]TJ /F2 1 Tf 1.1084 0 Td [(dans)]TJ /F8 1 Tf 2.3675 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.621 87.254 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.422 60.563 Td [(on)-361(a)]TJ /F8 1 Tf 2.2771 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.681 87.254 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35 60.563 Td [(\))-325(=)]TJ /F8 1 Tf 1.8133 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-240(+)]TJ /F8 1 Tf 1.6446 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.81 87.254 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.006 60.563 Td [(\))]TJ /F11 1 Tf 0.71085 0 Td [(>)]TJ /F8 1 Tf -38.386 -1.1988 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)-445(L'application)]TJ /F8 1 Tf 7.1807 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(est)-343(donc)-331(croissan)24(te.)]TJ -10.41 -1.6627 Td [(P)30(our)]TJ /F8 1 Tf 2.4096 0 Td [(A)]TJ /F2 1 Tf 1.0602 0 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(B)]TJ /F2 1 Tf 1.1205 0 Td [(parties)-313(non)-313(supp)-30(os)24(\266)469(ees)-307(disjoin)24(tes,)]TJ /F8 1 Tf 14.428 0 Td [(A)]TJ /F6 1 Tf 0.92771 0 Td [([)]TJ /F8 1 Tf 0.84337 0 Td [(B)]TJ /F2 1 Tf 1.1205 0 Td [(est)-313(la)-313(r)30(\266)469(eunion)-313(disjoin)24(te)-313(de)]TJ /F8 1 Tf 11.512 0 Td [(A)]TJ /F2 1 Tf 1.0602 0 Td [(et)-313(d'une)]TJ -35.627 -1.1988 Td [(partie)]TJ /F8 1 Tf 2.9036 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.948 81.452 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6747 56.503 Td [(de)]TJ /F8 1 Tf 1.3494 0 Td [(B)]TJ /F2 1 Tf 1.1566 0 Td [(\(l'in)24(tersection)-349(de)]TJ /F8 1 Tf 7.6807 0 Td [(B)]TJ /F2 1 Tf 1.1566 0 Td [(et)-349(du)-355(compl)24(\266)475(emen)24(taire)-355(de)]TJ /F8 1 Tf 11.205 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)-493(On)-349(a)-349(donc)]TJ /F8 1 Tf 6.0904 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F6 1 Tf 0.98193 0 Td [([)]TJ /F8 1 Tf 0.89759 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-307(=)]TJ /F8 1 Tf -37.693 -1.1988 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.793 79.728 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.319 55.304 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\).)]TJ -13.416 -1.6627 Td [(P)30(ar)-457(r)24(\266)475(ecurrence)-463(on)-463(en)-457(d)24(\266)475(eduit)-457(imm)18(\266)475(ediatemen)24(t)-463(que,)-493(p)-30(our)]TJ /F8 1 Tf 25.181 0 Td [(A)]TJ /F6 1 Tf 1.241 0 Td [(\275)]TJ /F8 1 Tf 1.2651 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.319 76.556 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.518 53.641 Td [([)]TJ /F8 1 Tf 0.97591 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 51.862 76.556 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37 53.641 Td [([)]TJ /F8 1 Tf 0.96988 0 Td [(:)-168(:)-168(:)]TJ /F6 1 Tf 1.4759 0 Td [([)]TJ /F8 1 Tf 0.96988 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.914 76.556 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.711 53.641 Td [(,)-493(on)-457(a)]TJ /F8 1 Tf -36.38 -1.1988 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.871 74.831 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.536 52.442 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 21.707 74.831 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.614 52.442 Td [(\))-228(+)]TJ /F6 1 Tf 1.6145 0 Td [(\242)-168(\242)-162(\242)]TJ /F2 1 Tf 1.3916 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.966 74.831 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.181 52.442 Td [(\).)]TJ -16.849 -1.6566 Td [(Comme)-295(toute)-295(partie)-301(b)-30(orn)30(\266)469(ee)-295(est)-301(con)24(ten)24(ue)-295(dans)-289(la)-295(r)24(\266)475(eunion)-301(d'un)-301(nom)24(bre)-295(\257ni)-295(de)-295(translat)30(\266)469(es)-295(de)]TJ /F8 1 Tf 0 -1.1988 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(on)-331(conclut)-343(que)-337(toute)-331(partie)-343(b)-30(orn)30(\266)469(ee)-337(a)-331(une)-337(mesure)-337(\257nie.)]TJ /F1 1 Tf -1.1205 -2.3193 Td [(P)30(arado)24(xe)-409(1.)]TJ /F12 1 Tf 6.9699 0 Td [(T)78(oute)-361(partie)-361(b)-30(orn)30(\266)469(ee)]TJ /F8 1 Tf 9.0783 0 Td [(A)]TJ /F12 1 Tf 1.1084 0 Td [(de)]TJ /F3 1 Tf 1.3554 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.155 68.237 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.464 47.268 Td [(p)-30(eut)-337(^)475(etre)-361(recouv)24(erte)-355(par)-361(des)-355(parties)-361(deux)-355(\265)500(a)]TJ -20.133 -1.1988 Td [(deux)-337(disjoin)24(tes)]TJ /F8 1 Tf 6.8313 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.483 65.711 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 13.717 46.069 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(p)]TJ /F12 1 Tf 0.84337 0 Td [(v)54(arian)24(t)-343(dans)]TJ /F3 1 Tf 5.747 0 Td [(N)]TJ /F12 1 Tf 0.72289 0 Td [(\))-343(don)24(t)-343(la)-343(r)30(\266)469(eunion)]TJ /F8 1 Tf 7.7831 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.871 66.512 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 30.578 46.069 Td [(est)-343(b)-30(orn)24(\266)469(ee)-343(et)-343(telles)-343(que)-349(c)24(hacune)]TJ -25.247 -1.2048 Td [(d'elles)-337(se)-337(d)30(\266)469(eduise)-337(de)-331(n'imp)-30(orte)-337(quelle)-337(autre)-337(par)-337(une)-337(translation.)]TJ /F2 1 Tf 0 -1.9819 Td [(Av)30(ec)-307(ces)-307(notations)-307(on)-301(v)18(oit)-307(que)-307(les)-307(parties)]TJ /F8 1 Tf 18.235 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.802 61.15 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.084 42.882 Td [(son)24(t)-307(b)-30(orn)24(\266)469(ees)-301(et)-307(on)24(t)-307(toutes)-307(la)-301(m)24(^)469(eme)-301(mesure)]TJ /F8 1 Tf 18.88 0 Td [(a)]TJ /F2 1 Tf -38.633 -1.2048 Td [(\(in)24(v)48(ariance)-325(par)-325(d)24(\266)469(eplacemen)30(t\).)-445(Alors,)-331(si)]TJ /F8 1 Tf 17.331 0 Td [(m)]TJ /F2 1 Tf 1.2048 0 Td [(est)-331(d)24(\266)475(enom)24(brablemen)24(t)-325(additiv)24(e,)-331(la)-331(mesure)-331(de)]TJ /F8 1 Tf 19.596 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 60.237 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 40.479 Td [(est)-295(la)-289(somme)-289(d'une)-289(in\257nit)24(\266)469(e)-295(de)-289(termes)-259(\266)469(egaux)-289(\265)500(a)]TJ /F8 1 Tf 19.982 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-427(Comme)-295(il)-289(s'agit)-289(d'une)-295(partie)-295(b)-30(orn)30(\266)469(ee)-295(cette)]TJ -20.512 -1.1988 Td [(mesure)-397(est)-397(\257nie)-391(et)-397(la)-391(seule)-391(p)-30(ossibilit)24(\266)469(e)-397(est)-397(donc)]TJ /F8 1 Tf 21.042 0 Td [(a)]TJ /F2 1 Tf 0.90361 0 Td [(=)-379(0)-391(et)]TJ /F8 1 Tf 3.2711 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.603 56.797 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.849 39.28 Td [(\))-379(=)-373(0.)-620(Mais)-397(alors)]TJ /F8 1 Tf 8.253 0 Td [(A)]TJ /F2 1 Tf 1.1145 0 Td [(\266)475(etan)18(t)]TJ -36.886 -1.2048 Td [(con)30(ten)24(u)-391(dans)]TJ /F8 1 Tf 6.1687 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.534 55.081 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.922 38.075 Td [(est)-391(aussi)-391(de)-385(mesure)-391(n)24(ulle.)-608(Ainsi)-391(toute)-385(partie)-397(b)-24(orn)24(\266)469(ee)-391(\(et,)-397(en)-391(particulier,)]TJ -7.5904 -1.1988 Td [(l'unit)24(\266)475(e)]TJ /F8 1 Tf 3.1084 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(\))-337(doit)-307(^)475(etre)-337(de)-331(mesure)-343(n)30(ulle.)]TJ -3.8976 -1.9879 Td [(L'asp)-30(ect)-367(parado)24(xal)-367(de)-361(cet)-367(exemple)-367(tien)24(t)-367(essen)24(tiellemen)24(t)-367(\265)500(a)-361(l'utilisation)-361(d'un)-373(nom)24(bre)-361(in\257ni)]TJ 0 -1.1988 Td [(\(d)24(\266)469(enom)24(brable\))-319(de)-319(parties)]TJ /F8 1 Tf 11.283 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.853 47.995 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.151 33.689 Td [(et)-319(de)-325(l'axiome)-319(d'additivit)24(\266)469(e)-325(dans)-319(sa)-325(v)24(ersion)-325(d)24(\266)475(enom)24(brable.)-445(Ef-)]TJ -12.819 -1.1988 Td [(fectiv)24(emen)24(t)-397(on)-397(p)-30(eut)-397(prouv)24(er)-403(qu'en)-391(dimension)-403(inf)24(\266)475(erieure)-403(ou)-367(\266)469(egale)-403(\265)500(a)-391(2)-397(un)-397(tel)-403(parado)24(xe)-397(ne)]TJ 0 -1.2048 Td [(p)-30(eut)-301(pas)-307(se)-301(pr)24(\266)475(esen)24(ter)-301(a)24(v)24(ec)-301(un)-307(nom)24(bre)-301(\257ni)-301(de)-301(parties)-301(;)-313(plus)-301(pr)24(\266)469(ecis)30(\266)469(emen)24(t)-301(on)-301(p)-30(eut)-307(alors)-301(mon-)]TJ 0 -1.1988 Td [(trer)-313(l'existence)-313(d'une)-319(mesure)]TJ /F8 1 Tf 12.801 0 Td [(m)]TJ /F2 1 Tf 1.1867 0 Td [(additiv)24(e)-313(\(mais)-319(pas)-313(d)24(\266)475(enom)24(brablemen)24(t)-313(additiv)24(e\),)-319(in)30(v)48(arian)24(te)]TJ -13.988 -1.1988 Td [(par)-331(d)24(\266)469(eplacemen)30(t)-337(et)-337(v)48(alan)30(t)-337(1)-337(sur)-337(une)-337(unit)30(\266)469(e)]TJ /F8 1 Tf 18.657 0 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(\257x)24(\266)469(ee)-337(\265)500(a)-331(l'a)24(v)48(ance.)]TJ /F1 1 Tf -19.777 -2.3193 Td [(P)30(arado)24(xe)-361(2.)-506(\(Hausdor\256)-108(\))]TJ /F12 1 Tf 13.337 0 Td [(Dans)-313(l'espace)-319(euclidien)-313(de)-313(dimension)-319(3,)-313(toute)-319(partie)-313(b)-30(orn)24(\266)469(ee)]TJ /F8 1 Tf -13.337 -1.1988 Td [(A)]TJ /F12 1 Tf 1.0602 0 Td [(p)-24(eut)-289(^)475(etre)-313(recouv)24(erte)-307(par)-313(des)-313(parties)-307(b)-30(orn)24(\266)475(ees)]TJ /F8 1 Tf 19.422 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.009 36.09 Td [(1)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 27.012 25.37 Td [(,)]TJ /F8 1 Tf 0.59036 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.578 36.09 Td [(2)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 29.108 25.37 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.362 36.09 Td [(3)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 31.759 25.37 Td [(telles)-313(que)]TJ /F8 1 Tf 4.3735 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.784 36.09 Td [(2)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 37.645 25.37 Td [(soit)-307(disjoin)24(te)-307(de)]TJ /F8 1 Tf -32.313 -1.1988 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 34.375 Td [(3)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8614 24.171 Td [(et)-337(c)24(hacune)-337(des)-337(parties)]TJ /F8 1 Tf 9.994 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 25.19 34.375 Td [(2)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 18.054 24.171 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.784 34.375 Td [(3)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.199 24.171 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 31.647 34.375 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.783 24.171 Td [([)]TJ /F8 1 Tf 0.89157 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 34.957 34.375 Td [(3)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.205 24.171 Td [(soit)-337(l'image)-337(de)]TJ /F8 1 Tf 6.6807 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 46.698 34.375 Td [(1)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 33.416 24.171 Td [(par)-337(un)-337(d)30(\266)469(eplacemen)24(t.)]TJ /F2 1 Tf -28.084 -1.988 Td [(Ici)-409(encore)]TJ /F8 1 Tf 4.6807 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.397 31.53 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.211 22.183 Td [(,)]TJ /F8 1 Tf 0.70482 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.121 31.53 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.114 22.183 Td [(,)]TJ /F8 1 Tf 0.70482 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 20.845 31.53 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.422 22.183 Td [(et)]TJ /F8 1 Tf 1.241 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 24.922 31.53 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.133 22.183 Td [([)]TJ /F8 1 Tf 0.93976 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.371 31.53 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.681 22.183 Td [(on)24(t)-409(la)-409(m)24(^)475(eme)-409(mesure)]TJ /F8 1 Tf 9.4458 0 Td [(a)]TJ /F2 1 Tf 0.93976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.93976 0 Td [(\257ni\).)-668(En)-409(outre,)-427(l'additivit)24(\266)475(e)]TJ -27.06 -1.1988 Td [(assure)]TJ /F8 1 Tf 3.0361 0 Td [(a)]TJ /F2 1 Tf 0.83735 0 Td [(=)]TJ /F8 1 Tf 1.0904 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.621 29.814 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.994 20.985 Td [([)]TJ /F8 1 Tf 0.90361 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 20.966 29.814 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.096 20.985 Td [(\))-319(=)]TJ /F8 1 Tf 1.7892 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.052 29.814 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.355 20.985 Td [(\))-240(+)]TJ /F8 1 Tf 1.6386 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32.931 29.814 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.458 20.985 Td [(\))-319(=)-307(2)]TJ /F8 1 Tf 2.2892 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)-361(d'o)-30(\265)530(u)]TJ /F8 1 Tf 2.8795 0 Td [(a)]TJ /F2 1 Tf 0.84337 0 Td [(=)-313(0.)-500(Comme)]TJ /F8 1 Tf 6.0602 0 Td [(A)]TJ /F2 1 Tf 1.1024 0 Td [(est)-361(con)30(ten)24(u)-355(dans)]TJ /F8 1 Tf -31.831 -1.2048 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 28.09 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7771 19.78 Td [([)]TJ /F8 1 Tf 0.91566 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 12.086 28.09 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1446 19.78 Td [([)]TJ /F8 1 Tf 0.91566 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.466 28.09 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.633 19.78 Td [(on)-373(a)]TJ /F8 1 Tf 2.3012 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F11 1 Tf 0.73494 0 Td [(6)]TJ /F2 1 Tf 1.1265 0 Td [(3)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F2 1 Tf 0.8735 0 Td [(=)-343(0.)-566(On)-379(conclut)-379(encore)-379(que)-379(toute)-373(partie)-379(b)-30(orn)30(\266)469(ee)-379(\(et)-379(donc)]TJ /F8 1 Tf -13.855 -1.1988 Td [(C)]TJ /F2 1 Tf 0.78313 0 Td [(\))-337(doit)-307(^)469(etre)-331(de)-337(mesure)-337(n)24(ulle.)]TJ -0.78313 -2.3193 Td [(En)-331(fait)-337(une)-337(cons)24(\266)469(equence)-331(du)-337(dernier)-337(parado)24(xe)-337(p)-30(eut)-307(^)469(etre)-307(\266)469(enonc)30(\266)469(ee)-337(de)-331(fa)-6(\273)439(con)-331(plus)-337(frappan)24(te)-337(:)]TJ /F1 1 Tf 0 -2.3133 Td [(P)30(arado)24(xe)-265(de)-265(Banac)24(h)-259(et)-271(T)90(arski.)]TJ /F12 1 Tf 16.404 0 Td [(Dans)-234(l'espace)-228(euclidien)-234(de)-228(dimension)-234(3,)-253(si)]TJ /F8 1 Tf 17.867 0 Td [(A)]TJ /F12 1 Tf 0.9759 0 Td [(et)]TJ /F8 1 Tf 1.0663 0 Td [(B)]TJ /F12 1 Tf 1.0361 0 Td [(son)18(t)]TJ -37.349 -1.1988 Td [(des)-373(parties)-367(b)-30(orn)24(\266)475(ees)-373(d'in)24(t)24(\266)469(erieur)-367(non)-373(vide,)-373(on)-373(p)-30(eut)-373(d)30(\266)469(ecoup)-30(er)-373(c)24(hacune)-373(d'elles)-373(en)-367(un)-373(nom)24(bre)]TJ 0 -1.2048 Td [(\257ni)-325(de)-325(parties)-331(deux)-325(\265)499(a)-325(deux)-325(disjoin)30(tes)]TJ /F8 1 Tf 16.355 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.095 16.306 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.175 11.545 Td [([)]TJ /F8 1 Tf 0.87349 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.353 16.306 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.452 11.545 Td [([)]TJ /F8 1 Tf 0.8735 0 Td [(:)-168(:)-168(:)]TJ /F6 1 Tf 1.3735 0 Td [([)]TJ /F8 1 Tf 0.87349 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.819 16.306 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 31.867 11.545 Td [(,)]TJ /F8 1 Tf 0.92771 0 Td [(B)]TJ /F2 1 Tf 1.0843 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 51.086 16.306 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 36.349 11.545 Td [([)]TJ /F8 1 Tf 0.8735 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.353 16.306 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.639 11.545 Td [([)]TJ /F8 1 Tf 0.87349 0 Td [(:)-162(:)-168(:)]TJ /F6 1 Tf 1.3675 0 Td [([)]TJ /F8 1 Tf 0.87349 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.836 16.306 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 43.386 11.545 Td [(\(le)]TJ -38.054 -1.1988 Td [(m)24(^)475(eme)-295(nom)24(bre)-295(de)-295(parties)-295(p)-30(our)]TJ /F8 1 Tf 13.265 0 Td [(A)]TJ /F12 1 Tf 1.0482 0 Td [(et)]TJ /F8 1 Tf 1.1265 0 Td [(B)]TJ /F12 1 Tf 0.80723 0 Td [(\))-295(de)-295(fa)-6(\273)439(con)-295(que,)-307(p)-30(our)-295(c)24(haque)]TJ /F8 1 Tf 12.307 0 Td [(k)]TJ /F12 1 Tf 0.8494 0 Td [(pris)-295(dans)]TJ /F6 1 Tf 4.2108 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(;)-168(:)-168(:)-168(:)-168(;)-162(n)]TJ /F6 1 Tf 2.8253 0 Td [(g)]TJ /F12 1 Tf 0.5 0 Td [(,)]TJ /F8 1 Tf -38.886 -1.1988 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7155 12.875 Td [(k)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9157 9.1473 Td [(soit)-331(l'image)-337(de)]TJ /F8 1 Tf 6.6747 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.517 12.875 Td [(k)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 15.169 9.1473 Td [(par)-331(un)-337(d)24(\266)475(eplacemen)24(t)]TJ /F8 1 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11.952 0 0 11.952 0 0 Tm 23.916 60.189 Td [(\242)]TJ /F2 1 Tf 0.73494 -0.81325 Td [(=)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.94 83.262 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.319 59.376 Td [(#)]TJ /F8 1 Tf 0.83735 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.474 84.754 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 29.651 59.376 Td [(;)]TJ /F1 1 Tf -24.434 -2.5964 Td [(\(iv\))]TJ /F12 1 Tf 2.4036 0 Td [(Le)-295(pro)-30(duit)-295(d'une)-295(famille)-295(\257nie)]TJ /F2 1 Tf 12.922 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.147 81.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.096 56.78 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.172 81.038 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.06 56.78 Td [(d'ensem)30(bles)-295(\257nis)-295(est)-295(un)-295(ensem)24(ble)-295(\257ni)-295(et)-295(l'on)-295(a)-289(:)]TJ /F2 1 Tf -4.1024 -1.6988 Td [(#)]TJ /F9 1 Tf 0.83133 0.81325 Td [(\241)]TJ 0.62651 0.13855 Td [(Y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.681 77.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.861 55.081 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.905 78.607 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 24.024 55.894 Td [(\242)]TJ /F2 1 Tf 0.73494 -0.81325 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(Y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.974 77.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.259 55.081 Td [(#)]TJ /F8 1 Tf 0.83735 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.388 78.607 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 29.59 55.081 Td [(.)]TJ /F2 1 Tf -24.259 -3.1325 Td [(Concernan)24(t)-337(les)-331(ensem)18(bles)-337(d)30(\266)469(enom)24(brables)-331(on)-337(a)-331(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 40.816 Td [(I)-30(I.A.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1928 48.93 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F1 1 Tf -5.1205 -1.6024 Td [(\(i\))]TJ /F12 1 Tf 1.8916 0 Td [(Soit)]TJ /F8 1 Tf 2.0663 0 Td [(A)]TJ /F12 1 Tf 1.0904 0 Td [(une)-337(partie)-349(de)]TJ /F3 1 Tf 6.1325 0 Td [(N)]TJ /F12 1 Tf 1.0663 0 Td [(alors,)-343(si)]TJ /F8 1 Tf 3.6928 0 Td [(A)]TJ /F12 1 Tf 1.0904 0 Td [(est)-343(ma)-60(jor)24(\266)469(ee,)]TJ /F8 1 Tf 5.6627 0 Td [(A)]TJ /F12 1 Tf 1.0904 0 Td [(est)-349(\257nie,)-343(si)]TJ /F8 1 Tf 5.0361 0 Td [(A)]TJ /F12 1 Tf 1.0904 0 Td [(n'est)-343(pas)-343(ma)-60(jor)24(\266)469(ee)]TJ /F8 1 Tf 8.0121 0 Td [(A)]TJ /F12 1 Tf -38.416 -1.1988 Td [(est)-337(in\257nie)-337(d)24(\266)469(enom)30(brable.)]TJ /F1 1 Tf 0.1747 -1.5964 Td [(\(ii\))]TJ /F12 1 Tf 2.1988 0 Td [(T)78(oute)-337(partie)-337(d'un)-343(ensem)24(ble)-331(d)24(\266)469(enom)24(brable)-337(est)-337(d)24(\266)475(enom)24(brable.)]TJ /F1 1 Tf -2.5181 -1.5964 Td [(\(iii\))]TJ /F12 1 Tf 2.5181 0 Td [(Soit)]TJ /F8 1 Tf 2.0542 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(:)]TJ /F8 1 Tf 0.55422 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(!)]TJ /F8 1 Tf 1.2771 0 Td [(Y)]TJ /F12 1 Tf 1.1386 0 Td [(une)-331(application,)]TJ -8.4639 -1.6024 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-343(injectiv)30(e)-337(et)]TJ /F8 1 Tf 6.7048 0 Td [(Y)]TJ /F12 1 Tf 1.1386 0 Td [(est)-337(d)24(\266)475(enom)24(brable,)-337(alors)]TJ /F8 1 Tf 10.157 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(est)-343(d)30(\266)469(enom)24(brable)-337(;)]TJ -21.175 -1.5964 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-343(surjectiv)24(e)-337(et)]TJ /F8 1 Tf 7.2169 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(est)-337(d)30(\266)469(enom)24(brable,)-337(alors)]TJ /F8 1 Tf 10.157 0 Td [(Y)]TJ /F12 1 Tf 1.1386 0 Td [(est)-337(d)24(\266)475(enom)24(brable.)]TJ /F1 1 Tf -22.801 -1.5964 Td [(\(iv\))]TJ /F12 1 Tf 2.6867 0 Td [(La)-433(r)24(\266)475(eunion)-439(d'une)-433(famille)-433(d)24(\266)469(enom)24(brable)-433(d'ensem)30(bles)-439(d)24(\266)475(enom)24(brables)-433(est)-439(un)-433(ensem)24(ble)]TJ -2.5723 -1.1988 Td [(d)30(\266)469(enom)24(brable.)]TJ /F1 1 Tf 0.20482 -1.6024 Td [(\(v\))]TJ /F12 1 Tf 1.9578 0 Td [(Le)-228(pro)-30(duit)-228(d'une)-228(famille)-228(\257nie)-228(d'ensem)24(bles)-228(d)24(\266)469(enom)24(brables)-222(est)-234(un)-228(ensem)24(ble)-228(d)24(\266)469(enom)30(brable.)]TJ /F1 1 Tf -2.1627 -1.5964 Td [(A)30(tten)30(tion)]TJ /F2 1 Tf 4.8494 0 Td [(,)-500(le)-475(pro)-30(duit)-469(d'une)-469(famille)-469(d)24(\266)469(enom)24(brable)-469(d'ensem)24(bles)-469(d)24(\266)475(enom)24(brables)-469(\(et)-475(m)24(^)475(eme)]TJ -4.8494 -1.1988 Td [(\257nis\))-337(n'est)-337(en)-331(g)30(\266)469(en)30(\266)469(eral)-337(pas)-337(d)30(\266)469(enom)24(brable.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 23.439 Td [(I)-30(I.A.2.)-241(Les)-381(d)25(\266)537(emonstrations.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 26.503 Td [(In)30(tro)-36(duisons,)-313(p)-30(our)-313(tout)-313(en)24(tier)]TJ /F8 1 Tf 13.301 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(,)-319(l'ensem)30(ble)]TJ /F8 1 Tf 5.3855 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.103 37.711 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.054 26.503 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(;)-168(:)-168(:)-162(:)-168(;)-168(n)]TJ /F6 1 Tf 2.8253 0 Td [(g)]TJ /F2 1 Tf 0.80723 0 Td [(des)-319(en)30(tiers)-313(compris)-313(en)30(tre)-313(1)-313(et)]TJ /F8 1 Tf -26.41 -1.1988 Td [(n)]TJ /F2 1 Tf 0.90361 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(S)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 10.353 35.995 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.9639 25.304 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(\).)-439(De)-307(m)24(^)469(eme)-301(notons)]TJ /F8 1 Tf 8.6747 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.914 35.995 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.958 25.304 Td [(l'ensem)24(ble)]TJ /F6 1 Tf 4.7892 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(;)]TJ /F2 1 Tf 0.43976 0 Td [(2)]TJ /F8 1 Tf 0.50602 0 Td [(;)-162(:)-168(:)-168(:)]TJ /F6 1 Tf 1.6084 0 Td [(g)]TJ /F2 1 Tf 0.80723 0 Td [(des)-307(en)30(tiers)-307(sup)0(\266)475(erieurs)-307(ou)-283(\266)475(egaux)-313(\265)500(a)-301(1.)]TJ -23.777 -1.5964 Td [(Une)-319(\266)475(en)24(um)24(\266)469(eration)-349(d'un)-349(ensem)24(ble)]TJ /F8 1 Tf 14.416 0 Td [(X)]TJ /F2 1 Tf 1.253 0 Td [(appara)108(^)385(\263t)-343(alors)-349(comme)-349(une)-343(bijection)-349(d'un)-349(ensem)24(ble)]TJ /F8 1 Tf 22.38 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.957 33.711 Td [(x)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 22.503 Td [(sur)]TJ /F8 1 Tf 1.759 0 Td [(X)]TJ /F2 1 Tf 1.3253 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.2771 0 Td [(x)]TJ /F6 1 Tf 0.99398 0 Td [(2)]TJ /F3 1 Tf 1.0843 0 Td [(N)]TJ /F2 1 Tf 1.1386 0 Td [(p)-30(our)-421(une)-391(\266)469(en)30(um)24(\266)469(eration)-421(\257nie)-421(et)]TJ /F8 1 Tf 13.771 0 Td [(x)]TJ /F2 1 Tf 0.98795 0 Td [(=)]TJ /F6 1 Tf 1.1988 0 Td [(1)]TJ /F2 1 Tf 1.4157 0 Td [(p)-30(our)-421(une)-391(\266)469(en)30(um)24(\266)469(eration)-421(in\257nie.)]TJ -25.952 -1.1988 Td [(Ainsi)]TJ /F8 1 Tf 2.6747 0 Td [(X)]TJ /F2 1 Tf 1.3253 0 Td [(est)-421(\257ni)-421(de)-415(cardinal)]TJ /F8 1 Tf 8.7892 0 Td [(n)]TJ /F2 1 Tf 1.0181 0 Td [(\(resp.)-698(in\257ni)-415(d)24(\266)469(enom)24(brable\))-415(si)-421(et)-421(seulemen)30(t)-421(s'il)-421(existe)-421(une)]TJ -13.807 -1.1988 Td [(bijection)-313(en)30(tre)]TJ /F8 1 Tf 6.5663 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.905 28.556 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.367 20.105 Td [(\(resp.)]TJ /F8 1 Tf 2.8916 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.147 28.556 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.717 20.105 Td [(\))-313(et)]TJ /F8 1 Tf 1.8434 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)-433(Cette)-313(propri)24(\266)469(et)24(\266)469(e)-313(est)-313(donc)-313(vraie)-313(p)-30(our)]TJ /F8 1 Tf 16.367 0 Td [(X)]TJ /F2 1 Tf 1.2169 0 Td [(si)-313(et)-313(seulemen)24(t)]TJ -32.723 -1.2048 Td [(si)-337(elle)-337(est)-337(vraie)-337(p)-30(our)-337(tout)-337(ensem)24(ble)-337(en)-331(bijection)-337(a)24(v)24(ec)]TJ /F8 1 Tf 23.229 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(.)]TJ -24.133 -1.5964 Td [(En)-367(particulier)]TJ /F8 1 Tf 6.4759 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.776 24.547 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.331 17.304 Td [(est)-373(un)-367(ensem)24(ble)-373(\257ni)-367(\265)500(a)]TJ /F8 1 Tf 9.988 0 Td [(n)]TJ /F2 1 Tf 0.93976 0 Td [(\266)475(el)18(\266)475(emen)24(ts.)-548(A)-367(l'in)24(v)24(erse,)-379(la)-367(prop)-30(osition)]TJ /F1 1 Tf 16.349 0 Td [(I)-36(I.A.1.2)]TJ /F2 1 Tf -35.277 -1.1988 Td [(mon)24(tre)-234(que)]TJ /F8 1 Tf 5.0843 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.784 22.831 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.108 16.105 Td [(n'est)-240(pas)-234(\257ni)-240(puisqu'il)-234(con)24(tien)24(t)-240(c)24(haque)]TJ /F8 1 Tf 16.518 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.845 22.831 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.018 16.105 Td [(\(#)]TJ /F8 1 Tf 1.2229 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.586 22.831 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.934 16.105 Td [(devrait)-240(donc)-210(^)475(etre)-234(sup)0(\266)469(erieur)]TJ -27.602 -1.2048 Td [(\265)499(a)-222(c)24(haque)-222(en)24(tier)]TJ /F8 1 Tf 6.6506 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(\).)-415(Ainsi)]TJ /F8 1 Tf 3.5602 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.974 21.116 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.825 14.9 Td [(et,)-246(par)-228(suite,)-246(tout)-228(ensem)24(ble)-222(admettan)24(t)-222(une)-198(\266)469(en)30(um)18(\266)475(eration)-228(in\257nie)]TJ -12.494 -1.1988 Td [(est)-337(in\257ni.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 8.9118 Td [(I)-30(I.A.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1928 10.683 Td [(1.)-289(Quelques)-391(preuv)30(es)-385(p)-36(our)-379(les)-391(exemples.)]TJ /F2 1 Tf -3.8614 -1.5964 Td [(On)-337(a)-331(une)-307(\266)469(en)30(um)24(\266)469(eration)-337(de)]TJ /F3 1 Tf 11.482 0 Td [(N)]TJ /F2 1 Tf 1.0542 0 Td [(en)-331(p)-30(osan)24(t)]TJ /F8 1 Tf 4.5602 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.914 12.788 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.741 9.087 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(p)]TJ /F6 1 Tf 0.72892 0 Td [(\241)]TJ /F2 1 Tf 1 0 Td [(1,)]TJ /F3 1 Tf 1.1145 0 Td [(N)]TJ /F2 1 Tf 1.0542 0 Td [(est)-337(donc)-337(strictemen)24(t)-337(d)24(\266)475(enom)24(brable.)]TJ -23.361 -1.5964 Td [(On)-361(a)-367(une)-337(\266)475(en)24(um)24(\266)475(eration)-367(de)]TJ /F3 1 Tf 11.627 0 Td [(Z)]TJ /F2 1 Tf 1.0301 0 Td [(en)-361(p)-30(osan)24(t)]TJ /F8 1 Tf 4.6205 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.172 10.504 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.97 7.4906 Td [(=)]TJ /F8 1 Tf 1.1024 0 Td [(p=)]TJ /F2 1 Tf 1.006 0 Td [(2)-361(si)]TJ /F8 1 Tf 1.8976 0 Td [(p)]TJ /F2 1 Tf 0.86144 0 Td [(est)-373(pair)-361(et)]TJ /F8 1 Tf 4.8795 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.069 10.504 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.078 7.4906 Td [(=)]TJ /F6 1 Tf 1.1024 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(p)]TJ /F2 1 Tf 0.74097 0 Td [(+)-246(1\))]TJ /F8 1 Tf 1.9096 0 Td [(=)]TJ /F2 1 Tf 0.5 0 Td [(2)-367(si)]TJ /F8 1 Tf 1.9036 0 Td [(p)]TJ /F2 1 Tf 0.86145 0 Td [(est)]TJ -37.934 -1.2048 Td [(impair,)]TJ /F3 1 Tf 3.4458 0 Td [(Z)]TJ /F2 1 Tf 1 0 Td [(est)-343(donc)-331(strictemen)18(t)-337(d)30(\266)469(enom)24(brable.)]TJ /F5 1 Tf -4.4458 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(14)]TJ ET endstream endobj 74 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 73 0 R /ProcSet 2 0 R >> /Contents 72 0 R >> endobj 73 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 75 0 obj << /Length 19361 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10162(I)-36(I.A.)-385(Ensem)24(bles)-379(d)18(\266)548(enom)30(brables)]TJ /F2 1 Tf 0 -2.4036 Td [(Les)-331(propri)24(\266)469(et)24(\266)475(es)-337(d'arithm)24(\266)469(etique)-307(\266)475(el)18(\266)475(emen)24(taire)-337(mon)24(tren)24(t)-337(que)-337(l'on)-337(a)-331(une)-307(\266)469(en)30(um)24(\266)469(eration)-337(de)]TJ /F3 1 Tf 36.5 0 Td [(N)]TJ /F6 1 Tf 0.93976 0 Td [(\243)]TJ /F3 1 Tf 1 0 Td [(N)]TJ /F2 1 Tf -38.44 -1.1988 Td [(en)-331(notan)24(t)]TJ /F8 1 Tf 4.5301 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.931 87.107 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.307 61.021 Td [(\(p)-36(our)]TJ /F8 1 Tf 2.759 0 Td [(n)]TJ /F6 1 Tf 0.87349 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.612 87.107 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.349 61.021 Td [(\))-337(le)-337(couple)-337(d'en)24(tiers)-331(\()]TJ /F8 1 Tf 9.3193 0 Td [(p;)-168(q)]TJ /F2 1 Tf 1.4277 0 Td [(\))-337(tel)-337(que)]TJ /F8 1 Tf 4.0301 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)-277(2)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.457 87.909 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.024 61.021 Td [(\(2)]TJ /F8 1 Tf 0.89157 0 Td [(q)]TJ /F2 1 Tf 0.70481 0 Td [(+)-222(1\).)]TJ -31.289 -1.5422 Td [(La)-337(d)24(\266)475(enom)24(brabilit)24(\266)469(e)-337(de)]TJ /F3 1 Tf 9.6566 0 Td [(Q)]TJ /F2 1 Tf 1.1145 0 Td [(sera)-337(prouv)54(\266)475(ee)-337(ult)18(\266)475(erieuremen)24(t.)]TJ -10.771 -1.5361 Td [(Mon)24(trons)-277(que)]TJ /F3 1 Tf 6.253 0 Td [(R)]TJ /F2 1 Tf 1 0 Td [(n'est)-277(pas)-277(d)24(\266)469(enom)24(brable)-277(en)-277(raisonnan)24(t)-277(par)-277(l'absurde.)-427(Consid)24(\266)475(erons)-283(donc)-277(une)]TJ -7.2831 -1.2048 Td [(\266)469(en)30(um)24(\266)469(eration)-307(\(\257nie)-301(ou)-301(non\))-301(de)]TJ /F3 1 Tf 13.241 0 Td [(R)]TJ /F2 1 Tf 0.71687 0 Td [(,)-313(le)-307(r)30(\266)469(eel)-307(d'indice)]TJ /F8 1 Tf 7.1626 0 Td [(p)]TJ /F2 1 Tf 0.77711 0 Td [(\266)469(etan)24(t)-307(not)24(\266)469(e)]TJ /F8 1 Tf 4.7711 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.569 80.978 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.006 56.738 Td [(.)-433(Notons)]TJ /F8 1 Tf 4.1024 0 Td [(X)]TJ /F2 1 Tf 1.2108 0 Td [(l'ensem)30(ble)-307(des)]TJ -32.988 -1.1988 Td [(r)30(\266)469(eels)]TJ /F8 1 Tf 2.253 0 Td [(x)]TJ /F2 1 Tf 0.87349 0 Td [(qui)-301(admetten)24(t)-307(une)-271(\266)469(ecriture)-307(d)30(\266)469(ecimale)-301(du)-301(t)24(yp)-36(e)]TJ /F8 1 Tf 19.651 0 Td [(x)]TJ /F2 1 Tf 0.8494 0 Td [(=)-277(0)]TJ /F8 1 Tf 1.5542 0 Td [(;)-168(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.06 79.262 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.934 55.539 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 46.457 79.262 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.916 55.539 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.853 79.262 Td [(3)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.06 55.539 Td [(:)-162(:)-168(:)]TJ /F2 1 Tf 1.4639 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1627 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.69 79.262 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.982 55.539 Td [(\266)469(egal)-301(\265)500(a)-301(0)-301(ou)-301(\265)500(a)]TJ -33.651 -1.1988 Td [(1)-313(\(on)-325(sait)-319(que,)-325(si)-319(une)-319(telle)-295(\266)475(ecriture)-319(existe,)-325(elle)-325(est)-319(unique\).)-439(Nous)-319(allons)-319(alors)-319(construire)-325(un)]TJ 0 -1.2048 Td [(r)30(\266)469(eel)-295(appartenan)24(t)-295(\265)500(a)]TJ /F8 1 Tf 8.241 0 Td [(X)]TJ /F2 1 Tf 1.1988 0 Td [(qui)-289(ne)-295(p)-30(eut)-295(pas)-265(^)469(etre)-289(un)]TJ /F8 1 Tf 10.313 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.716 75.831 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.41 53.135 Td [(ce)-295(qui)-289(donnera)-295(la)-295(con)30(tradiction)-301(souhait)24(\266)475(ee.)]TJ -21.078 -1.5361 Td [(P)30(our)]TJ /F8 1 Tf 2.3855 0 Td [(p)]TJ /F6 1 Tf 0.78313 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.397 73.625 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.904 51.599 Td [(,)-301(p)-30(osons)]TJ /F8 1 Tf 3.7952 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.793 73.625 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.97 51.599 Td [(=)-277(0)-289(si)]TJ /F8 1 Tf 2.8012 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.681 73.625 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.096 51.599 Td [(est)-289(d)24(\266)469(e\257ni,)-301(appartien)30(t)-295(\265)500(a)]TJ /F8 1 Tf 10.09 0 Td [(X)]TJ /F2 1 Tf 1.1928 0 Td [(et)-289(a)-289(une)]TJ /F8 1 Tf 3.753 0 Td [(p)]TJ /F2 1 Tf 0.50602 0 Td [(-i)24(\265)475(eme)-289(d)24(\266)475(ecimale)-265(\266)475(egale)]TJ -30.307 -1.2048 Td [(\265)499(a)-313(1,)]TJ /F8 1 Tf 1.9096 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.121 71.909 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.512 50.394 Td [(=)-277(1)-313(sinon.)-439(On)-319(p)-30(eut)-319(alors)-319(d)30(\266)469(e\257nir)-319(le)-319(r)30(\266)469(eel)]TJ /F8 1 Tf 17.193 0 Td [(x)]TJ /F2 1 Tf 0.8494 0 Td [(=)-277(0)]TJ /F8 1 Tf 1.5542 0 Td [(;)-168(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.621 71.909 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.536 50.394 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.017 71.909 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.512 50.394 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.422 71.909 Td [(3)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.657 50.394 Td [(:)-168(:)-162(:)]TJ /F2 1 Tf 1.1687 0 Td [(,)-319(il)-313(appartien)24(t)-319(\265)500(a)]TJ /F8 1 Tf 7.1265 0 Td [(X)]TJ /F2 1 Tf 1.2229 0 Td [(mais)-313(ne)]TJ -35.843 -1.1988 Td [(p)-30(eut)-343(^)469(etre)-343(\266)469(egal)-367(\265)500(a)-373(un)]TJ /F8 1 Tf 8.8313 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.086 70.185 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.566 49.195 Td [(car)-379(alors)]TJ /F8 1 Tf 4.1446 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.026 70.185 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.12 49.195 Td [(serait)-367(dans)]TJ /F8 1 Tf 5.1446 0 Td [(X)]TJ /F2 1 Tf 1.2771 0 Td [(a)24(v)24(ec)-373(une)]TJ /F8 1 Tf 4.1627 0 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(-i)24(\265)469(eme)-367(d)24(\266)469(ecimale)-367(di\256)24(\266)469(eren)24(te)-373(de)]TJ -26.873 -1.1988 Td [(celle)-337(de)]TJ /F8 1 Tf 3.5542 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(.)]TJ ET 512.64 572.43 m 519.84 572.43 l 519.84 580.86 l 512.64 580.86 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 46.22 Td [(Les)-331(d)24(\266)475(emonstrations)-337(suiv)54(an)24(tes)-337(s'appuieron)18(t)-337(sur)-337(la)-331(remarque)-307(\266)469(el)24(\266)469(emen)30(taire)-337(que)-337(v)24(oici)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 36.133 Td [(I)-30(I.A.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1928 43.316 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Lemme.)]TJ /F12 1 Tf 4.2349 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0663 0 Td [(X)]TJ /F12 1 Tf 1.2831 0 Td [(un)-379(ensem)24(ble)-373(et)]TJ /F8 1 Tf 6.994 0 Td [(a)]TJ /F12 1 Tf 0.5241 0 Td [(,)]TJ /F8 1 Tf 0.66265 0 Td [(b)]TJ /F12 1 Tf 0.80723 0 Td [(des)-349(\266)475(el)18(\266)475(emen)24(ts)-373(de)]TJ /F8 1 Tf 7.2711 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-566(Alors)-379(il)-373(y)-379(a)-373(une)]TJ -31.367 -1.2048 Td [(bijection)-337(de)]TJ /F8 1 Tf 5.4157 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(sur)-343(lui-m)30(^)469(eme)-331(qui)-337(en)30(v)24(oie)]TJ /F8 1 Tf 10.735 0 Td [(a)]TJ /F12 1 Tf 0.86747 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(b)]TJ /F12 1 Tf 0.42771 0 Td [(.)]TJ /F1 1 Tf -20.361 -1.7771 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Si)]TJ /F8 1 Tf 1.1627 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F2 1 Tf 0.75904 0 Td [(l'application)-337(iden)24(tique)-331(con)24(vien)24(t.)-451(Sinon)-331(il)-337(est)-331(facile)-337(de)-331(v)54(\266)469(eri\257er)]TJ -12.602 -1.1988 Td [(que)]TJ /F8 1 Tf 1.8614 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(b)]TJ /F6 1 Tf 0.70482 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(a)]TJ /F2 1 Tf 0.86747 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2831 0 Td [(x)]TJ /F2 1 Tf 0.90361 0 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(x)-391(=)]TJ /F6 1 Tf 0.8494 0 Td [(2)-283(f)]TJ /F8 1 Tf 1.4458 0 Td [(a;)-168(b)]TJ /F6 1 Tf 1.4036 0 Td [(g)]TJ /F2 1 Tf 0.83132 0 Td [(d)24(\266)475(e\257nit)-337(une)-331(bijection)-343(con)30(v)24(enable.)]TJ ET 512.64 466.52 m 519.84 466.52 l 519.84 474.94 l 512.64 474.94 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 37.358 Td [(Une)-487(premi)24(\265)475(ere)-487(application)-487(de)-481(ce)-487(lemme)-487(est)-487(l')24(\266)475(enonc)24(\266)469(e)-487(suiv)54(an)24(t)-487(qui)-481(justi\257e)-481(en)24(tre)-487(autres)-481(la)]TJ 0 -1.2048 Td [(d)30(\266)469(e\257nition)-337(du)-337(nom)24(bre)-331(d')24(\266)469(el)24(\266)469(emen)30(ts)-337(p)-30(our)-331(un)-337(ensem)18(ble)-331(\257ni.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 27.736 Td [(I)-30(I.A.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1928 33.25 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F1 1 Tf -5.1205 -1.5361 Td [(\(i\))]TJ /F12 1 Tf 1.9096 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0422 0 Td [(n)]TJ /F12 1 Tf 0.95181 0 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(m)]TJ /F12 1 Tf 1.2289 0 Td [(des)-349(en)30(tiers)-349(et)]TJ /F8 1 Tf 6.1446 0 Td [(f)]TJ /F12 1 Tf 0.94578 0 Td [(une)-349(application)-349(injectiv)24(e)-349(de)]TJ /F8 1 Tf 12.422 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.034 45.168 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 35.157 31.714 Td [(dans)]TJ /F8 1 Tf 2.3554 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.56 45.168 Td [(m)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.886 31.714 Td [(.)-493(Alors)]TJ /F8 1 Tf 3.4337 0 Td [(n)]TJ /F12 1 Tf 0.94579 0 Td [(est)]TJ -37.934 -1.1988 Td [(inf)30(\266)469(erieur)-331(ou)-307(\266)469(egal)-331(\265)500(a)]TJ /F8 1 Tf 8.259 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(.)]TJ /F1 1 Tf -8.9639 -1.5422 Td [(\(ii\))]TJ /F12 1 Tf 2.1988 0 Td [(S'il)-337(y)-337(a)-331(une)-337(bijection)-337(en)24(tre)]TJ /F8 1 Tf 11.928 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.966 41.245 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 21.12 28.973 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.776 41.245 Td [(m)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 23.994 28.973 Td [(alors)]TJ /F8 1 Tf 2.3976 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F12 1 Tf 0.87349 0 Td [(.)]TJ /F1 1 Tf -24.018 -1.5422 Td [(\(iii\))]TJ /F12 1 Tf 2.5602 0 Td [(Si)-355(un)-361(ensem)24(ble)-355(admet)-355(une)-331(\266)475(en)24(um)24(\266)475(eration)-361(index)24(\266)475(ee)-361(par)]TJ /F8 1 Tf 23.108 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.034 39.038 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.367 27.43 Td [(et)-355(une)-331(\266)475(en)24(um)24(\266)469(eration)-355(index)24(\266)469(ee)]TJ -27.036 -1.1988 Td [(par)]TJ /F8 1 Tf 1.7771 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.052 37.323 Td [(m)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8193 26.232 Td [(alors)]TJ /F8 1 Tf 2.3976 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(.)]TJ /F1 1 Tf -8.6988 -1.7771 Td [(D)24(\266)548(emonstration.)]TJ 0.49398 -1.5422 Td [(\(i\))]TJ /F2 1 Tf 1.9337 0 Td [(Raisonnons)-367(par)-361(r)30(\266)469(ecurrence)-367(sur)]TJ /F8 1 Tf 13.717 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(.)-530(C'est)-337(\266)469(eviden)24(t)-367(p)-30(our)]TJ /F8 1 Tf 9.3193 0 Td [(n)]TJ /F2 1 Tf 0.92771 0 Td [(=)-325(0.)-530(Si)]TJ /F8 1 Tf 3.6024 0 Td [(n)]TJ /F11 1 Tf 0.92169 0 Td [(>)]TJ /F2 1 Tf 1.1084 0 Td [(1,)]TJ /F8 1 Tf 1.1446 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.836 32.573 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.62 22.912 Td [(n'est)-361(pas)]TJ -35.289 -1.1988 Td [(vide)-361(donc)]TJ /F8 1 Tf 4.5723 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.052 30.857 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.633 21.714 Td [(non)-361(plus,)-361(par)-361(suite)]TJ /F8 1 Tf 8.6145 0 Td [(m)]TJ /F11 1 Tf 1.1928 0 Td [(>)]TJ /F2 1 Tf 1.0964 0 Td [(1.)-512(Cela)-361(prouv)24(e)-361(le)-361(r)30(\266)469(esultat)-361(p)-30(our)]TJ /F8 1 Tf 14.06 0 Td [(n)]TJ /F2 1 Tf 0.91566 0 Td [(=)-319(1.)-512(Supp)-30(osons)]TJ -32.181 -1.2048 Td [(main)24(tenan)24(t)]TJ /F8 1 Tf 5.2771 0 Td [(n)]TJ /F11 1 Tf 0.87952 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(2.)]TJ -7.2108 -1.5361 Td [(P)30(ar)-361(restriction)]TJ /F8 1 Tf 6.7169 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(fournit)-355(une)-361(injection)-361(de)]TJ /F8 1 Tf 10.717 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.819 26.935 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.259 18.973 Td [(dans)]TJ /F8 1 Tf 2.3675 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.845 26.935 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.355 18.973 Td [(qui)-361(ne)-355(prend)-361(pas)-361(la)-361(v)54(aleur)]TJ /F8 1 Tf 11.886 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(n)]TJ /F2 1 Tf 0.59638 0 Td [(\).)]TJ -38.494 -1.1988 Td [(P)30(ar)-439(ailleurs,)-469(le)-445(lemme)-439(pr)24(\266)469(ec)24(\266)475(eden)24(t)-439(fournit)-439(une)-445(bijection)-439(de)]TJ /F8 1 Tf 25.964 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.664 25.219 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.108 17.774 Td [(dans)-439(lui-m)24(^)469(eme)-439(qui)-439(en)30(v)24(oie)]TJ /F8 1 Tf -27.777 -1.2048 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(\))-343(sur)]TJ /F8 1 Tf 2.4096 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-463(P)24(ar)-337(comp)-30(osition)-343(on)-343(obtien)24(t)-343(une)-337(injection)-343(de)]TJ /F8 1 Tf 20.373 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.638 23.495 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.102 16.569 Td [(dans)]TJ /F8 1 Tf 2.3434 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.603 23.495 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.157 16.569 Td [(qui)-343(ne)-337(prend)-343(pas)]TJ -31.825 -1.1988 Td [(la)-367(v)54(aleur)]TJ /F8 1 Tf 4.1566 0 Td [(m)]TJ /F2 1 Tf 1.247 0 Td [(et)-367(donc)-373(une)-367(injection)-373(de)]TJ /F8 1 Tf 11.042 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.034 21.78 Td [(n)]TJ /F10 1 Tf 0.71552 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.325 15.37 Td [(dans)]TJ /F8 1 Tf 2.3735 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.086 21.78 Td [(m)]TJ /F10 1 Tf 1.0172 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.096 15.37 Td [(.)-554(L'h)30(yp)-36(oth)24(\265)469(ese)-367(de)-367(r)24(\266)475(ecurrence)-373(assure)]TJ -23.765 -1.1988 Td [(alors)]TJ /F8 1 Tf 2.3976 0 Td [(n)]TJ /F6 1 Tf 0.81928 0 Td [(\241)]TJ /F2 1 Tf 1.006 0 Td [(1)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F6 1 Tf 1.1024 0 Td [(\241)]TJ /F2 1 Tf 1 0 Td [(1)-331(d'o)-30(\265)524(u)-331(l'on)-337(tire)]TJ /F8 1 Tf 6.8373 0 Td [(n)]TJ /F11 1 Tf 0.87952 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ /F1 1 Tf -17.633 -1.5422 Td [(\(ii\))]TJ /F2 1 Tf 2.3072 0 Td [(Lorsqu'il)-391(y)-391(a)-385(une)-385(bijection)-391(en)24(tre)]TJ /F8 1 Tf 14.687 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.069 17.857 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.042 12.629 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.026 17.857 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.639 12.629 Td [(,)-397(le)-391(r)24(\266)475(esultat)-397(pr)30(\266)469(ec)24(\266)469(eden)30(t)-391(s'applique)-385(\265)500(a)-391(cette)]TJ -21.307 -1.1988 Td [(bijection)-337(et)-337(\265)500(a)-331(la)-337(bijection)-337(r)30(\266)469(ecipro)-30(que)-337(ce)-337(qui)-331(donne)]TJ /F8 1 Tf 22.042 0 Td [(n)]TJ /F11 1 Tf 0.87952 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(m)]TJ /F11 1 Tf 1.1566 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)-337(d'o)-30(\265)530(u)]TJ /F8 1 Tf 2.8373 0 Td [(n)]TJ /F2 1 Tf 0.8735 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ /F1 1 Tf -34.964 -1.5422 Td [(\(iii\))]TJ /F2 1 Tf 2.3916 0 Td [(Si)-271(un)-277(ensem)24(ble)]TJ /F8 1 Tf 6.6807 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(admet)-271(une)-240(\266)469(en)30(um)24(\266)469(eration)-277(index)30(\266)469(ee)-277(par)]TJ /F8 1 Tf 15.753 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.517 13.935 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.62 9.8882 Td [(et)-271(une)-246(\266)475(en)24(um)24(\266)469(eration)-271(index)24(\266)469(ee)]TJ -27.289 -1.1988 Td [(par)]TJ /F8 1 Tf 1.8133 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.103 12.219 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5181 8.6894 Td [(,)-379(celles-ci)-373(fournissen)24(t)-367(une)-373(bijection)-373(de)]TJ /F8 1 Tf 16.886 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.233 12.219 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.928 8.6894 Td [(dans)]TJ /F8 1 Tf 2.3735 0 Td [(X)]TJ /F2 1 Tf 1.2771 0 Td [(et)-373(une)-367(bijection)-373(de)]TJ /F8 1 Tf 8.6145 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.966 12.219 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.934 8.6894 Td [(dans)]TJ /F8 1 Tf 2.3735 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)]TJ -38.886 -1.1988 Td [(d'o)-30(\265)530(u)-343(par)-349(passage)-343(\265)500(a)-343(la)-343(bijection)-349(r)30(\266)469(ecipro)-30(que)-343(et)-349(comp)-30(osition)-343(une)-349(bijection)-343(en)24(tre)]TJ /F8 1 Tf 33.934 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.069 10.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.765 7.4906 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.905 10.504 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 7.4906 Td [(et)]TJ /F8 1 Tf -38.331 -1.2048 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ ET 512.64 73.976 m 519.84 73.976 l 519.84 82.4 l 512.64 82.4 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.247 3.8882 Td [(15)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 77 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 76 0 R /ProcSet 2 0 R >> /Contents 75 0 R >> endobj 76 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 78 0 obj << /Length 21535 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10162(I)-36(I.A.)-385(Ensem)24(bles)-379(d)18(\266)548(enom)30(brables)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I)-30(I.A.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1928 62.22 Td [(4.)-289(Preuv)30(e)-391(de)-385(la)-385(prop)-36(osition)-391(I)-30(I.A.1.2.)]TJ -3.3675 -1.5783 Td [(\(i\))]TJ /F2 1 Tf 1.8735 0 Td [(Soit)]TJ /F8 1 Tf 2.0482 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(une)-331(partie)-337(d'un)-331(ensem)24(ble)-331(\257ni)]TJ /F8 1 Tf 13.018 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(de)-325(cardinal)]TJ /F8 1 Tf 5.1626 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(,)-331(Il)-331(s'agit)-331(de)-331(prouv)24(er)-331(que)]TJ /F8 1 Tf 11.102 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(est)]TJ -37.934 -1.1988 Td [(\257ni)-307(de)-301(cardinal)-307(inf)24(\266)475(erieur)-307(ou)-277(\266)475(egal)-307(\265)500(a)]TJ /F8 1 Tf 14.94 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-433(Quitte)-307(\265)500(a)-301(appliquer)-307(une)-307(bijection)-307(con)30(v)24(enable)-307(on)-307(p)-30(eut)]TJ -15.819 -1.2048 Td [(supp)-30(oser)]TJ /F8 1 Tf 4.1506 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.431 83.125 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.94 58.238 Td [(.)]TJ -7.6084 -1.5783 Td [(Raisonnons)-331(par)-337(r)24(\266)475(ecurrence)-337(sur)]TJ /F8 1 Tf 13.602 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(,)-337(c'est)-307(\266)469(eviden)24(t)-337(p)-30(our)]TJ /F8 1 Tf 8.7651 0 Td [(m)]TJ /F2 1 Tf 1.1566 0 Td [(=)-277(0)-337(et)]TJ /F8 1 Tf 3.0542 0 Td [(m)]TJ /F2 1 Tf 1.1566 0 Td [(=)-283(1.)-445(Supp)-24(osons)]TJ /F8 1 Tf 7.2108 0 Td [(m)]TJ /F11 1 Tf 1.1566 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(2.)]TJ -38.036 -1.5783 Td [(Si)]TJ /F8 1 Tf 1.1446 0 Td [(B)]TJ /F2 1 Tf 1.0904 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.216 78.607 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.307 55.081 Td [(alors)]TJ /F8 1 Tf 2.3795 0 Td [(B)]TJ /F2 1 Tf 1.1265 0 Td [(est)-319(\257ni)-319(de)-313(cardinal)]TJ /F8 1 Tf 8.3795 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-439(Sinon,)-319(notons)]TJ /F8 1 Tf 6.9699 0 Td [(a)]TJ /F2 1 Tf 0.84338 0 Td [(un)-289(\266)469(el)24(\266)469(emen)30(t)-319(de)]TJ /F8 1 Tf 6.4157 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.259 78.607 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.994 55.081 Td [(qui)-313(n'est)-313(pas)]TJ -33.663 -1.1988 Td [(dans)]TJ /F8 1 Tf 2.3012 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(.)-433(Gr)-6(^)499(ace)-295(au)-301(lemme)-295(pr)24(\266)469(ec)24(\266)475(eden)24(t,)-307(on)-295(p)-30(eut)-301(trouv)24(er)-301(une)-301(bijection)-301(de)]TJ /F8 1 Tf 27.127 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.776 76.892 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.235 53.882 Td [(sur)-301(lui)-301(m)30(^)469(eme)-295(qui)]TJ -31.904 -1.2048 Td [(en)30(v)24(oie)]TJ /F8 1 Tf 3.0422 0 Td [(a)]TJ /F2 1 Tf 0.87952 0 Td [(sur)]TJ /F8 1 Tf 1.6928 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(.)-493(En)-349(comp)-30(osan)18(t)-349(a)30(v)24(ec)-355(cette)-355(bijection)-349(on)-349(se)-355(ram)24(\265)475(ene)-355(au)-349(cas)-349(o)-30(\265)530(u)]TJ /F8 1 Tf 26.765 0 Td [(m)-415(=)]TJ /F6 1 Tf 1.1807 0 Td [(2)]TJ /F8 1 Tf 0.96988 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(.)-493(On)-349(a)]TJ -36.211 -1.1988 Td [(alors)]TJ /F8 1 Tf 2.4337 0 Td [(B)]TJ /F6 1 Tf 1.1506 0 Td [(\275)]TJ /F8 1 Tf 1.1205 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.241 73.452 Td [(m)]TJ /F10 1 Tf 1.0172 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.807 51.479 Td [(et,)-385(d'apr)24(\265)475(es)-373(l'h)24(yp)-36(oth)24(\265)469(ese)-373(de)-367(r)24(\266)469(ecurrence,)]TJ /F8 1 Tf 16.825 0 Td [(B)]TJ /F2 1 Tf 1.1807 0 Td [(est)-373(\257ni)-379(de)-367(cardinal)-379(inf)30(\266)469(erieur)-373(ou)]TJ -25.512 -1.1988 Td [(\266)469(egal)-331(\265)500(a)]TJ /F8 1 Tf 2.9217 0 Td [(m)]TJ /F6 1 Tf 1.0964 0 Td [(\241)]TJ /F2 1 Tf 1 0 Td [(1)-337(donc)-331(\265)500(a)]TJ /F8 1 Tf 4.0602 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ /F1 1 Tf -9.753 -1.5843 Td [(\(ii\))]TJ /F2 1 Tf 2.1988 0 Td [(Soit)]TJ /F8 1 Tf 2.0542 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(une)-337(application)-337(de)]TJ /F8 1 Tf 8.3916 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(dans)]TJ /F8 1 Tf 2.3434 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(.)]TJ -18.139 -1.5783 Td [(Lorsque)]TJ /F8 1 Tf 3.8012 0 Td [(Y)]TJ /F2 1 Tf 1.1627 0 Td [(est)-361(\257ni)-367(le)-361(r)24(\266)475(esultat)-367(pr)24(\266)475(ec)24(\266)469(eden)24(t)-361(a\261rme)-361(que)]TJ /F8 1 Tf 17.934 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-367(est)-361(\257ni)-367(de)-355(cardinal)-367(inf)30(\266)469(erieur)-361(ou)]TJ -24.819 -1.1988 Td [(\266)469(egal)-361(\265)500(a)-367(celui)-361(de)]TJ /F8 1 Tf 6.7108 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(.)-536(Lorsque)]TJ /F8 1 Tf 4.6205 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(est)-373(injectiv)30(e,)]TJ /F8 1 Tf 5.8855 0 Td [(f)]TJ /F2 1 Tf 0.96385 0 Td [(induit)-367(une)-361(bijection)-373(de)]TJ /F8 1 Tf 10.373 0 Td [(X)]TJ /F2 1 Tf 1.2711 0 Td [(sur)]TJ /F8 1 Tf 1.7108 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90963 0 Td [(\))-367(et)-367(donc)]TJ /F8 1 Tf -35.157 -1.1988 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(est)-307(\266)469(egalemen)24(t)-337(\257ni)-337(a)30(v)24(ec)-337(#)]TJ /F8 1 Tf 11.012 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(=)-283(#)]TJ /F8 1 Tf 1.8916 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(#)]TJ /F8 1 Tf 0.83735 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(.)]TJ -20.578 -1.5843 Td [(Lorsque)]TJ /F8 1 Tf 3.753 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(est)-325(surjectif,)-325(on)-313(p)-30(eut)-319(d)24(\266)469(e\257nir)-319(une)-319(application)]TJ /F8 1 Tf 19.44 0 Td [(g)]TJ /F2 1 Tf 0.83132 0 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(Y)]TJ /F2 1 Tf 1.1205 0 Td [(dans)]TJ /F8 1 Tf 2.3253 0 Td [(X)]TJ /F2 1 Tf 1.2229 0 Td [(en)-319(asso)-24(cian)24(t)-319(\265)500(a)-319(tout)]TJ -30.952 -1.1988 Td [(\266)469(el)24(\266)469(emen)30(t)]TJ /F8 1 Tf 3.8072 0 Td [(y)]TJ /F2 1 Tf 0.93976 0 Td [(de)]TJ /F8 1 Tf 1.4157 0 Td [(Y)]TJ /F2 1 Tf 1.2169 0 Td [(un)-421(an)24(t)24(\266)475(ec)24(\266)469(eden)30(t)]TJ /F8 1 Tf 6.6145 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))-421(de)]TJ /F8 1 Tf 2.2229 0 Td [(y)]TJ /F2 1 Tf 0.93976 0 Td [(par)]TJ /F8 1 Tf 1.8614 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)-692(On)-415(a)-415(alors)]TJ /F8 1 Tf 6.1145 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.92771 0 Td [(=)-415(Id)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.362 59.797 Td [(Y)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.283 41.936 Td [(,)-439(par)-415(suite)]TJ /F8 1 Tf 5.0542 0 Td [(g)]TJ /F2 1 Tf 0.92771 0 Td [(est)]TJ -37.934 -1.1988 Td [(injectiv)30(e)-349(de)]TJ /F8 1 Tf 5.3193 0 Td [(Y)]TJ /F2 1 Tf 1.1446 0 Td [(dans)]TJ /F8 1 Tf 2.3494 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(et)-343(le)-343(r)24(\266)475(esultat)-349(pr)24(\266)475(ec)24(\266)469(eden)30(t)-343(assure)-343(que,)-349(si)]TJ /F8 1 Tf 16.645 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(est)-349(\257ni,)]TJ /F8 1 Tf 3.5783 0 Td [(Y)]TJ /F2 1 Tf 1.1446 0 Td [(l'est)-313(\266)469(egalemen)24(t)]TJ -32.675 -1.1988 Td [(a)30(v)24(ec)-337(#)]TJ /F8 1 Tf 3.0301 0 Td [(Y)]TJ /F11 1 Tf 1.0783 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(#)]TJ /F8 1 Tf 0.83735 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(.)]TJ /F1 1 Tf -7.0482 -1.5843 Td [(\(iii\))]TJ /F2 1 Tf 2.5181 0 Td [(P)24(our)-337(mon)24(trer)-337(l')24(\266)475(enonc)24(\266)469(e)-337(sur)-337(la)-337(r)24(\266)475(eunion)-343(d'une)-337(famille)-337(\257nie)-337(d'ensem)24(bles)-337(\257nis)-337(on)-337(p)-30(eut)-337(se)]TJ -2.3735 -1.1988 Td [(ramener)-361(par)-361(r)24(\266)475(ecurrence)-367(au)-361(cas)-361(de)-361(la)-361(r)30(\266)469(eunion)-361(de)-361(deux)-361(ensem)24(bles)-367(\257nis)]TJ /F8 1 Tf 30.488 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(et)]TJ /F8 1 Tf 1.1928 0 Td [(Y)]TJ /F2 1 Tf 0.80723 0 Td [(.)-524(On)-361(a)-361(donc)]TJ -33.753 -1.1988 Td [(des)-240(\266)469(en)30(um)24(\266)469(erations)-265(de)]TJ /F8 1 Tf 9.012 0 Td [(X)]TJ /F2 1 Tf 1.1687 0 Td [(et)-271(de)]TJ /F8 1 Tf 2.3675 0 Td [(Y)]TJ /F2 1 Tf 1.0723 0 Td [(sous)-265(la)-271(forme)]TJ /F8 1 Tf 5.8976 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.302 50.668 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.608 35.557 Td [(;)-168(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 42.397 50.668 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.078 35.557 Td [(;)-162(:)-168(:)-168(:)-168(;)-168(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.043 50.668 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.416 35.557 Td [(g)]TJ /F2 1 Tf 0.76506 0 Td [(et)]TJ /F8 1 Tf 1.1024 0 Td [(Y)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(y)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.966 50.668 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.861 35.557 Td [(;)-162(y)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.948 50.668 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.241 35.557 Td [(;)-168(:)-168(:)-168(:)-168(;)-162(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.474 50.668 Td [(m)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 43.717 35.557 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)]TJ -38.886 -1.2048 Td [(On)-295(p)-30(eut)-295(alors)-301(d)30(\266)469(e\257nir)-295(une)-295(application)]TJ /F8 1 Tf 16.301 0 Td [(f)]TJ /F2 1 Tf 0.89157 0 Td [(de)]TJ /F8 1 Tf 1.2952 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.966 48.952 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(+)]TJ /F7 1 Tf 0.87931 0 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.596 34.352 Td [(dans)]TJ /F8 1 Tf 2.2952 0 Td [(X)]TJ /F6 1 Tf 1.0542 0 Td [([)]TJ /F8 1 Tf 0.80723 0 Td [(Y)]TJ /F2 1 Tf 1.0964 0 Td [(par)]TJ /F8 1 Tf 1.741 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.483 48.952 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.133 34.352 Td [(si)-295(1)]TJ /F11 1 Tf 1.741 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(p)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.89759 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.1988 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.043 47.228 Td [(p)]TJ /F10 1 Tf 0.58621 0 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.024 33.153 Td [(si)]TJ /F8 1 Tf 0.99398 0 Td [(n)]TJ /F2 1 Tf 0.80723 0 Td [(+)-210(1)]TJ /F11 1 Tf 1.7651 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(p)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.80723 0 Td [(+)]TJ /F8 1 Tf 0.98795 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-439(Il)-325(est)-331(clair)-331(que)]TJ /F8 1 Tf 7.3072 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-331(surjectiv)24(e)-331(et)-325(le)-331(r)30(\266)469(esultat)-331(pr)24(\266)475(ec)24(\266)469(eden)24(t)]TJ -23.06 -1.1988 Td [(assure)-307(que)]TJ /F8 1 Tf 4.8253 0 Td [(X)]TJ /F6 1 Tf 1.0843 0 Td [([)]TJ /F8 1 Tf 0.83735 0 Td [(Y)]TJ /F2 1 Tf 1.1145 0 Td [(est)-313(\257ni)-313(\(de)-313(cardinal)-313(inf)24(\266)475(erieur)-313(ou)-283(\266)475(egal)-313(\265)500(a)]TJ /F8 1 Tf 16.91 0 Td [(n)]TJ /F2 1 Tf 0.77711 0 Td [(+)]TJ /F8 1 Tf 0.95181 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\).)-445(En)-307(outre,)-319(si)]TJ /F8 1 Tf 6.506 0 Td [(X)]TJ /F2 1 Tf 1.2169 0 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(Y)]TJ /F2 1 Tf 1.1084 0 Td [(son)18(t)]TJ -37.349 -1.1988 Td [(disjoin)30(ts,)]TJ /F8 1 Tf 4.2349 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-343(bijectiv)30(e)-337(et)-337(#\()]TJ /F8 1 Tf 7.9277 0 Td [(X)]TJ /F6 1 Tf 1.1325 0 Td [([)]TJ /F8 1 Tf 0.88554 0 Td [(Y)]TJ /F2 1 Tf 0.80723 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(n)]TJ /F2 1 Tf 0.81927 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(m)]TJ /F2 1 Tf 1.1566 0 Td [(=)-283(#)]TJ /F8 1 Tf 1.8916 0 Td [(X)]TJ /F2 1 Tf 1.1265 0 Td [(+)-228(#)]TJ /F8 1 Tf 1.8373 0 Td [(Y)]TJ /F2 1 Tf 0.80121 0 Td [(.)]TJ /F1 1 Tf -26.386 -1.5843 Td [(\(iv\))]TJ /F2 1 Tf 2.5 0 Td [(P)24(our)-343(mon)24(trer)-343(l')30(\266)469(enonc)24(\266)475(e)-349(sur)-343(le)-343(pro)-30(duit)-343(d'une)-343(famille)-343(\257nie)-343(d'ensem)24(bles)-343(\257nis)-343(on)-343(p)-30(eut)-343(se)]TJ -2.3855 -1.1988 Td [(ramener)-355(par)-349(r)24(\266)475(ecurrence)-361(au)-349(cas)-355(du)-355(pro)-30(duit)-355(de)-349(deux)-355(ensem)24(bles)-355(\257nis)]TJ /F8 1 Tf 29.343 0 Td [(X)]TJ /F2 1 Tf 1.259 0 Td [(et)]TJ /F8 1 Tf 1.1867 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(.)-500(P)24(our)-355(prouv)30(er)]TJ -32.59 -1.1988 Td [(ce)-337(cas)-331(on)-337(p)-30(eut)-337(supp)-30(oser)]TJ /F8 1 Tf 10.747 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.095 38.099 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.807 26.774 Td [(et)-337(raisonner)-337(par)-331(r)24(\266)475(ecurrence)-343(sur)]TJ /F8 1 Tf 13.819 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(.)]TJ -28.892 -1.5843 Td [(C'est)-319(\266)469(eviden)24(t)-349(p)-30(our)]TJ /F8 1 Tf 8.4578 0 Td [(n)]TJ /F11 1 Tf 0.89759 0 Td [(6)]TJ /F2 1 Tf 1.0723 0 Td [(1)-343(car)]TJ /F8 1 Tf 2.5241 0 Td [(S)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.043 35.84 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.578 25.189 Td [(\243)]TJ /F8 1 Tf 1.006 0 Td [(Y)]TJ /F2 1 Tf 1.0964 0 Td [(=)]TJ /F6 1 Tf 1.0783 0 Td [(;)-228(\243)]TJ /F8 1 Tf 1.7349 0 Td [(Y)]TJ /F2 1 Tf 1.0964 0 Td [(=)]TJ /F6 1 Tf 1.0783 0 Td [(;)]TJ /F2 1 Tf 0.84337 0 Td [(et)]TJ /F8 1 Tf 1.1747 0 Td [(S)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.931 35.84 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.982 25.189 Td [(\243)]TJ /F8 1 Tf 1.006 0 Td [(Y)]TJ /F2 1 Tf 1.0964 0 Td [(=)]TJ /F6 1 Tf 1.0783 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(1)]TJ /F6 1 Tf 0.5 0 Td [(g)-228(\243)]TJ /F8 1 Tf 1.7349 0 Td [(Y)]TJ /F2 1 Tf 1.1506 0 Td [(est)-349(clairemen)30(t)-349(en)]TJ -31.717 -1.1988 Td [(bijection)-337(a)30(v)24(ec)]TJ /F8 1 Tf 6.2771 0 Td [(Y)]TJ /F2 1 Tf 0.80723 0 Td [(.)]TJ -7.0843 -1.5783 Td [(Supp)-24(osons)]TJ /F8 1 Tf 5.012 0 Td [(n)]TJ /F11 1 Tf 1.0241 0 Td [(>)]TJ /F2 1 Tf 1.1988 0 Td [(2.)-704(On)-421(a)-415(alors)]TJ /F8 1 Tf 6.6386 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.362 31.857 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.639 22.412 Td [(\243)]TJ /F8 1 Tf 1.0602 0 Td [(Y)]TJ /F2 1 Tf 1.2229 0 Td [(=)-421(\()]TJ /F8 1 Tf 1.5904 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.957 31.857 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.97 22.412 Td [(\243)]TJ /F8 1 Tf 1.0602 0 Td [(Y)]TJ /F2 1 Tf 0.80121 0 Td [(\))]TJ /F6 1 Tf 0.66868 0 Td [([)]TJ /F2 1 Tf 0.94578 0 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(n)]TJ /F6 1 Tf 0.60241 0 Td [(g)-277(\243)]TJ /F8 1 Tf 1.8374 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(\).)-704(C'est)-427(la)-421(r)30(\266)469(eunion)-421(de)]TJ -29.247 -1.1988 Td [(deux)-343(ensem)18(bles)-349(disjoin)30(ts)-349(don)24(t)-349(le)-355(premier)-349(est)-349(\257ni)-355(de)-343(cardinal)-355(\()]TJ /F8 1 Tf 27.042 0 Td [(n)]TJ /F6 1 Tf 0.83133 0 Td [(\241)]TJ /F2 1 Tf 1.012 0 Td [(1\))]TJ /F6 1 Tf 1.1205 0 Td [(\243)]TJ /F2 1 Tf 1.012 0 Td [(#)]TJ /F8 1 Tf 0.83133 0 Td [(Y)]TJ /F2 1 Tf 1.1506 0 Td [(par)-349(h)24(yp)-30(oth)24(\265)469(ese)]TJ -33 -1.2048 Td [(de)-325(r)24(\266)475(ecurrence)-331(et)-331(le)-331(deuxi)24(\265)475(eme)-331(est)-331(\257ni)-331(de)-325(cardinal)-337(#)]TJ /F8 1 Tf 22.054 0 Td [(Y)]TJ /F2 1 Tf 1.1325 0 Td [(d'apr)24(\265)475(es)-331(le)-331(cas)-331(#)]TJ /F8 1 Tf 6.994 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)-277(1.)-445(L')24(\266)469(enonc)24(\266)475(e)-331(sur)]TJ -31.367 -1.1988 Td [(la)-385(r)30(\266)469(eunion)-385(de)-385(deux)-385(ensem)24(bles)-391(\257nis)-379(disjoin)24(ts)-385(prouv)30(e)-391(alors)-391(que)]TJ /F8 1 Tf 27.253 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.509 26.702 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34 18.81 Td [(\243)]TJ /F8 1 Tf 1.0301 0 Td [(Y)]TJ /F2 1 Tf 1.1928 0 Td [(est)-385(\257ni)-391(de)-379(cardinal)]TJ /F9 1 Tf -30.892 -0.39157 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.80723 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(n)]TJ /F6 1 Tf 0.8253 0 Td [(\241)]TJ /F2 1 Tf 1 0 Td [(1\))]TJ /F6 1 Tf 1.1145 0 Td [(\243)]TJ /F2 1 Tf 1 0 Td [(#)]TJ /F8 1 Tf 0.83133 0 Td [(Y)]TJ /F9 1 Tf 0.8012 0.80723 Td [(\242)]TJ /F2 1 Tf 0.68072 -0.80723 Td [(+)-228(#)]TJ /F8 1 Tf 1.8373 0 Td [(Y)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(n)]TJ /F6 1 Tf 0.81928 0 Td [(\243)]TJ /F2 1 Tf 1 0 Td [(#)]TJ /F8 1 Tf 0.83735 0 Td [(Y)]TJ /F2 1 Tf 1.1386 0 Td [(ce)-331(qui)-337(ac)24(h)24(\265)475(ev)24(e)-337(la)-337(preuv)30(e.)]TJ ET 512.64 209.26 m 519.84 209.26 l 519.84 217.69 l 512.64 217.69 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 12.203 Td [(I)-30(I.A.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1928 14.629 Td [(5.)-289(Preuv)30(e)-391(de)-385(la)-385(prop)-36(osition)-391(I)-30(I.A.1.3.)]TJ -3.3675 -1.5783 Td [(\(i\))]TJ /F2 1 Tf 1.8916 0 Td [(P)30(our)-343(tout)-337(en)24(tier)]TJ /F8 1 Tf 7.4217 0 Td [(n)]TJ /F2 1 Tf 0.93976 0 Td [(p)-30(osons)]TJ /F8 1 Tf 3.2711 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.759 18.461 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.928 13.051 Td [(=)]TJ /F6 1 Tf 1.0663 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.85542 0 Td [(2)]TJ /F8 1 Tf 0.95181 0 Td [(A)]TJ /F6 1 Tf 1.0361 0 Td [(j)]TJ /F8 1 Tf 0.56627 0 Td [(x)]TJ /F11 1 Tf 0.85542 0 Td [(6)]TJ /F8 1 Tf 1.0663 0 Td [(n)]TJ /F6 1 Tf 0.60241 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)-457(C'est)-343(une)-343(partie)-343(de)-337(la)-343(r)30(\266)469(eunion)-343(des)]TJ -23.596 -1.2048 Td [(ensem)24(bles)-349(\257nis)]TJ /F6 1 Tf 6.7952 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(0)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.8494 0 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.569 16.737 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.313 11.846 Td [(,)-355(c'est)-349(donc)-355(un)-349(ensem)24(ble)-349(\257ni)-349(et,)-355(en)-349(notan)30(t)]TJ /F8 1 Tf 18.825 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(\))-349(son)-355(cardinal,)-355(on)]TJ -31.44 -1.1988 Td [(d)30(\266)469(e\257nit)-331(une)-337(application)]TJ /F8 1 Tf 10.169 0 Td [(\256)]TJ /F2 1 Tf 0.9759 0 Td [(de)]TJ /F3 1 Tf 1.3374 0 Td [(N)]TJ /F2 1 Tf 1.0542 0 Td [(dans)-337(lui-m)30(^)469(eme.)]TJ -13.536 -1.5783 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(A)]TJ /F2 1 Tf 1.0783 0 Td [(est)-343(ma)-54(jor)18(\266)475(ee)-337(par)-337(un)-337(en)30(tier)]TJ /F8 1 Tf 11.41 0 Td [(n)]TJ /F2 1 Tf 0.93373 0 Td [(alors)]TJ /F8 1 Tf 2.3976 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(est)-313(\266)469(egale)-337(\265)499(a)-331(l'ensem)24(ble)-331(\257ni)]TJ /F8 1 Tf 11.434 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.931 12.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.133 9.0689 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(est)-337(donc)-337(\257nie.)]TJ -32.5 -1.5783 Td [(Supp)-24(osons)]TJ /F8 1 Tf 4.8795 0 Td [(A)]TJ /F2 1 Tf 1.0361 0 Td [(non)-289(ma)-60(jor)24(\266)475(ee;)-307(nous)-289(allons)-283(alors)-289(prouv)24(er)-289(que,)-301(par)-289(restriction)-289(\265)500(a)]TJ /F8 1 Tf 26.373 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)]TJ /F8 1 Tf 0.57229 0 Td [(\256)]TJ /F2 1 Tf 0.92771 0 Td [(d)24(\266)475(e\257nit)-289(une)]TJ -34.542 -1.2048 Td [(bijection)-337(de)]TJ /F8 1 Tf 5.4157 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.207 8.7797 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.301 6.2858 Td [(ce)-337(qui)-337(prouv)30(era)-343(bien)-337(que)]TJ /F8 1 Tf 11.127 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(est)-337(in\257nie)-337(d)24(\266)475(enom)24(brable.)]TJ /F5 1 Tf -22.181 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(16)]TJ ET endstream endobj 80 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 79 0 R /ProcSet 2 0 R >> /Contents 78 0 R >> endobj 79 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R >> endobj 82 0 obj << /Length 21731 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10162(I)-36(I.A.)-385(Ensem)24(bles)-379(d)18(\266)548(enom)30(brables)]TJ /F2 1 Tf 0 -2.4036 Td [(T)78(out)-391(d'ab)-36(ord,)-403(p)-30(our)]TJ /F8 1 Tf 8.9398 0 Td [(n)]TJ /F2 1 Tf 0.99398 0 Td [(dans)]TJ /F8 1 Tf 2.3916 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)]TJ /F8 1 Tf 0.68072 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.397 88.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.777 62.22 Td [(n'est)-391(pas)-391(vide)-397(\(il)-397(p)-24(oss)24(\265)475(ede)]TJ /F8 1 Tf 11.53 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(\))-391(donc)]TJ /F8 1 Tf 3.2229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.76506 0 Td [(>)]TJ /F2 1 Tf 1.1446 0 Td [(1)-391(;)]TJ /F8 1 Tf 1.5904 0 Td [(\256)]TJ ET 0.504 w 503.1 738.9 m 503.1 750.92 l S BT /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.422 88.392 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 62.22 Td [(est)]TJ -37.934 -1.1988 Td [(bien)-337(\265)500(a)-331(v)54(aleurs)-343(dans)]TJ /F8 1 Tf 8.7108 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.974 87.107 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.506 61.021 Td [(.)]TJ -10.175 -1.7169 Td [(P)30(ar)-271(ailleurs,)-283(p)-30(our)]TJ /F8 1 Tf 7.7952 0 Td [(n)]TJ /F2 1 Tf 0.86747 0 Td [(et)]TJ /F8 1 Tf 1.1084 0 Td [(m)]TJ /F2 1 Tf 1.1446 0 Td [(dans)]TJ /F8 1 Tf 2.2771 0 Td [(A)]TJ /F2 1 Tf 1.0181 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1325 0 Td [(n)-283(<)-277(m)]TJ /F2 1 Tf 2.8133 0 Td [(,)-283(on)-271(v)24(oit)-277(que)]TJ /F8 1 Tf 5.6205 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.164 84.65 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.892 59.304 Td [(est)-271(la)-271(r)24(\266)475(eunion)-277(disjoin)30(te)-271(de)]TJ /F8 1 Tf 11.307 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 84.65 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 58.105 Td [(et)-271(de)-271(l'ensem)30(ble)]TJ /F8 1 Tf 7.1205 0 Td [(B)]TJ /F2 1 Tf 1.0783 0 Td [(des)-240(\266)469(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 5.6867 0 Td [(a)]TJ /F2 1 Tf 0.8012 0 Td [(de)]TJ /F8 1 Tf 1.2711 0 Td [(A)]TJ /F2 1 Tf 1.0181 0 Td [(tels)-265(que)]TJ /F8 1 Tf 3.5723 0 Td [(n)-283(<)-277(a)]TJ /F11 1 Tf 2.741 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-421(On)-271(a)-271(donc)]TJ /F8 1 Tf 5.3976 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(\))-96(+)-90(#)]TJ /F8 1 Tf 2.1867 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(.)]TJ -38.886 -1.2048 Td [(Comme)]TJ /F8 1 Tf 3.741 0 Td [(m)]TJ /F6 1 Tf 1.2831 0 Td [(2)]TJ /F8 1 Tf 1.0723 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(,)]TJ /F8 1 Tf 0.70482 0 Td [(B)]TJ /F2 1 Tf 1.2229 0 Td [(n'est)-409(pas)-415(vide,)-427(#)]TJ /F8 1 Tf 7.6747 0 Td [(B)-457(>)]TJ /F2 1 Tf 2.3976 0 Td [(0)-403(et)]TJ /F8 1 Tf 2.1506 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\))]TJ /F8 1 Tf 0.79518 0 Td [(>)-403(\256)]TJ /F2 1 Tf 1.8253 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(\).)-674(Ainsi)]TJ /F8 1 Tf 4.006 0 Td [(\256)]TJ ET 439.09 675.32 m 439.09 687.34 l S BT /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.759 80.788 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.928 56.9 Td [(est)-415(strictemen)18(t)]TJ -32.596 -1.1988 Td [(croissan)30(te)-337(donc)-337(injectiv)24(e.)]TJ 0 -1.7169 Td [(Il)-421(reste)-421(\265)500(a)-421(prouv)24(er)-427(que)-421(l'image)]TJ /F8 1 Tf 13.711 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-421(de)]TJ /F8 1 Tf 2.2289 0 Td [(\256)]TJ ET 285.52 640.47 m 285.52 652.5 l S BT /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.371 76.616 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.09 53.985 Td [(est)]TJ /F8 1 Tf 1.6506 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.138 77.038 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.199 53.985 Td [(.)-710(P)30(our)-421(cela)-427(remarquons)-421(que)]TJ /F8 1 Tf 13.06 0 Td [(A)]TJ /F2 1 Tf 1.1747 0 Td [(n'est)]TJ -37.102 -1.1988 Td [(pas)-349(vide)-355(\(car)-349(non)-349(ma)-60(jor)24(\266)469(ee\),)-355(elle)-349(p)-30(oss)24(\265)475(ede)-355(donc)-349(un)-349(plus)-349(p)-30(etit)-319(\266)469(el)24(\266)469(emen)24(t)]TJ /F8 1 Tf 30.078 0 Td [(m)]TJ /F2 1 Tf 1.2229 0 Td [(\(rapp)-30(elons)-355(qu'une)]TJ -31.301 -1.1988 Td [(partie)-319(non)-313(vide)-319(de)]TJ /F3 1 Tf 8.2289 0 Td [(N)]TJ /F2 1 Tf 1.0361 0 Td [(p)-30(oss)30(\265)469(ede)-319(toujours)-313(un)-319(plus)-313(p)-30(etit)-289(\266)475(el)18(\266)475(emen)24(t\).)-439(Il)-313(est)-319(alors)-319(clair)-313(que)]TJ /F8 1 Tf 26.849 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.379 73.607 Td [(m)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 51.587 Td [(est)]TJ -37.934 -1.2048 Td [(r)30(\266)469(eduit)-337(au)-337(singleton)]TJ /F6 1 Tf 8.5783 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(m)]TJ /F6 1 Tf 0.87952 0 Td [(g)]TJ /F2 1 Tf 0.83735 0 Td [(et)-343(donc)]TJ /F8 1 Tf 3.5663 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\))-289(=)-289(1)-337(ce)-337(qui)-343(prouv)30(e)-343(1)]TJ /F6 1 Tf 9.5723 0 Td [(2)]TJ /F8 1 Tf 0.95181 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\).)-463(P)30(our)-343(conclure)-337(il)-337(su\261t)]TJ -28.578 -1.1988 Td [(donc,)-337(en)-331(raisonnan)24(t)-337(par)-337(r)30(\266)469(ecurrence,)-337(de)-331(prouv)24(er)-337(l'implication)]TJ /F8 1 Tf 26.62 0 Td [(n)]TJ /F6 1 Tf 0.87952 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F6 1 Tf 0.66265 0 Td [(\))]TJ /F8 1 Tf 1.2831 0 Td [(n)]TJ /F2 1 Tf 0.81928 0 Td [(+)-222(1)]TJ /F6 1 Tf 1.7771 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\).)]TJ -37.5 -1.7169 Td [(Si)]TJ /F8 1 Tf 1.2048 0 Td [(n)]TJ /F2 1 Tf 0.96988 0 Td [(est)-373(dans)-373(l'image)-373(de)]TJ /F8 1 Tf 8.8313 0 Td [(\256)]TJ /F2 1 Tf 1.012 0 Td [(on)-373(a)]TJ /F8 1 Tf 2.3012 0 Td [(n)]TJ /F2 1 Tf 0.93976 0 Td [(=)]TJ /F8 1 Tf 1.1145 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(\))-379(p)-24(our)-373(un)-373(\(unique\))]TJ /F8 1 Tf 8.7169 0 Td [(p)]TJ /F2 1 Tf 0.8735 0 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(.)-554(Comme)]TJ /F8 1 Tf 4.5361 0 Td [(A)]TJ /F2 1 Tf 1.1265 0 Td [(n'est)-367(pas)]TJ -35.283 -1.1988 Td [(ma)-54(jor)18(\266)475(ee,)-343(il)-331(y)-343(a)-331(des)-313(\266)475(el)18(\266)475(emen)24(ts)-337(de)]TJ /F8 1 Tf 13.831 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(strictemen)18(t)-337(sup)0(\266)469(erieurs)-337(\265)500(a)]TJ /F8 1 Tf 10.867 0 Td [(p)]TJ /F2 1 Tf 0.50603 0 Td [(,)-337(notons)]TJ /F8 1 Tf 3.8434 0 Td [(q)]TJ /F2 1 Tf 0.81928 0 Td [(le)-337(plus)-337(p)-30(etit)-331(de)-337(ces)]TJ -30.982 -1.1988 Td [(\266)469(el)24(\266)469(emen)30(ts.)-445(Alors)]TJ /F8 1 Tf 7.1386 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.879 64.28 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.952 45.069 Td [(est)-325(clairemen)24(t)-325(la)-319(r)24(\266)475(eunion)-325(des)-325(ensem)24(bles)-325(disjoin)24(ts)]TJ /F8 1 Tf 21.404 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.672 64.28 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.892 45.069 Td [(et)]TJ /F6 1 Tf 1.1506 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(q)]TJ /F6 1 Tf 0.48193 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)-445(Il)-319(s'ensuit)]TJ /F8 1 Tf -34.193 -1.2048 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(q)]TJ /F2 1 Tf 0.48193 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(\))-228(+)-222(1)-277(=)]TJ /F8 1 Tf 3.4458 0 Td [(n)]TJ /F2 1 Tf 0.8253 0 Td [(+)-222(1)-331(et)-337(donc)]TJ /F8 1 Tf 5.3916 0 Td [(n)]TJ /F2 1 Tf 0.8253 0 Td [(+)-222(1)]TJ /F6 1 Tf 1.7771 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)]TJ /F1 1 Tf -19.59 -1.7108 Td [(\(ii\))]TJ /F2 1 Tf 2.247 0 Td [(En)-361(particulier,)-367(l')30(\266)469(enonc)24(\266)475(e)-367(pr)30(\266)469(ec)24(\266)475(eden)24(t)-361(mon)24(tre)-361(qu'une)-361(partie)-361(de)]TJ /F8 1 Tf 26.163 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.414 60.107 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.735 42.153 Td [(est)-361(toujours)-361(d)24(\266)475(enom-)]TJ -30.404 -1.2048 Td [(brable)-409(\(au)-409(sens)-403(large\).)-668(En)-403(se)-409(ramenan)24(t)-409(\265)500(a)-403(ce)-409(cas)-409(de)-403(fa)-6(\273)439(con)-379(\266)475(eviden)18(te)-403(par)-409(une)-409(bijection)-409(on)]TJ 0 -1.1988 Td [(conclut)-337(que)-337(toute)-337(partie)-337(d'un)-337(ensem)18(ble)-331(d)24(\266)475(enom)24(brable)-337(est)-337(d)24(\266)469(enom)24(brable.)]TJ /F1 1 Tf -0.14458 -1.7169 Td [(\(iii\))]TJ /F2 1 Tf 2.7892 0 Td [(Comme)-469(p)-30(our)-475(la)-469(preuv)30(e)-475(de)-469(la)-469(prop)-30(osition)-475(pr)30(\266)469(ec)24(\266)469(eden)30(te,)-506(si)]TJ /F8 1 Tf 25.596 0 Td [(f)]TJ /F2 1 Tf 1.0663 0 Td [(est)-475(injectiv)30(e)-475(on)-469(a)-469(une)]TJ -29.307 -1.1988 Td [(bijection)-493(de)]TJ /F8 1 Tf 5.7289 0 Td [(X)]TJ /F2 1 Tf 1.3976 0 Td [(sur)-493(une)-493(partie)-493(de)]TJ /F8 1 Tf 8.4157 0 Td [(Y)]TJ /F2 1 Tf 1.2952 0 Td [(et)]TJ /F8 1 Tf 1.3253 0 Td [(X)]TJ /F2 1 Tf 1.3976 0 Td [(est)-493(d)30(\266)469(enom)24(brable)-493(d)30(\265)469(es)-493(que)]TJ /F8 1 Tf 11.699 0 Td [(Y)]TJ /F2 1 Tf 1.2892 0 Td [(l'est.)-915(Si)]TJ /F8 1 Tf 4.3012 0 Td [(f)]TJ /F2 1 Tf 1.0843 0 Td [(est)]TJ -37.934 -1.1988 Td [(surjectiv)24(e,)-361(par)-349(c)18(hoix)-349(d'an)24(t)24(\266)469(ec)24(\266)475(eden)24(ts)-349(on)-355(construit)-355(une)-349(application)-361(injectiv)30(e)-355(de)]TJ /F8 1 Tf 33.56 0 Td [(Y)]TJ /F2 1 Tf 1.1506 0 Td [(dans)]TJ /F8 1 Tf 2.3614 0 Td [(X)]TJ /F2 1 Tf 1.259 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.2048 Td [(Y)]TJ /F2 1 Tf 1.1325 0 Td [(est)-343(d)30(\266)469(enom)24(brable)-337(d)30(\265)469(es)-337(que)]TJ /F8 1 Tf 11.078 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(l'est.)]TJ /F1 1 Tf -13.566 -1.7108 Td [(\(iv\))]TJ /F2 1 Tf 2.6205 0 Td [(Consid)24(\266)469(erons)-403(une)-403(famille)-403(d)24(\266)475(enom)24(brable)-403(d'ensem)24(bles)-403(d)24(\266)475(enom)24(brables)-403(\()]TJ /F8 1 Tf 29.295 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.319 46.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.295 32.72 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.353 46.607 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.361 32.72 Td [(et)-403(notons)]TJ /F8 1 Tf -35.03 -1.2048 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(la)-331(r)24(\266)475(eunion)-337(des)]TJ /F8 1 Tf 6.4518 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.828 44.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.187 31.515 Td [(,)-337(il)-331(s'agit)-337(de)-337(prouv)30(er)-343(que)]TJ /F8 1 Tf 11.048 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(est)-337(d)24(\266)475(enom)24(brable.)]TJ -21.145 -1.7169 Td [(Comme)]TJ /F8 1 Tf 3.6687 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(est)-343(d)24(\266)475(enom)24(brable)-337(on)-337(p)-30(eut)-337(trouv)24(er)-343(une)-331(bijection)]TJ /F8 1 Tf 20.675 0 Td [(\256)]TJ /F2 1 Tf 0.9759 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(sur)-343(une)-337(partie)]TJ /F8 1 Tf 6.4639 0 Td [(S)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.336 42.435 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.645 29.798 Td [(ou)]TJ /F8 1 Tf 1.3916 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.457 42.435 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 28.599 Td [(de)]TJ /F3 1 Tf 1.3133 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)-439(P)30(our)-319(une)-313(raison)-313(analogue,)-325(p)-24(our)-319(c)24(haque)-313(indice)]TJ /F8 1 Tf 20.898 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(,)-319(on)-313(p)-30(eut)-319(trouv)24(er)-319(une)-313(bijection)]TJ /F8 1 Tf 13.675 0 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.319 40.711 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.494 28.599 Td [(de)]TJ /F8 1 Tf -38.163 -1.1988 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8103 38.995 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8253 27.4 Td [(sur)-337(une)-337(partie)-337(de)]TJ /F3 1 Tf 7.7952 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -10.012 -1.7169 Td [(In)30(tro)-36(duisons)-415(alors)-421(l'ensem)30(ble)]TJ /F8 1 Tf 13.217 0 Td [(Y)]TJ /F2 1 Tf 1.2169 0 Td [(des)-415(couples)-415(\()]TJ /F8 1 Tf 5.7831 0 Td [(i;)-168(x)]TJ /F2 1 Tf 1.3614 0 Td [(\))-415(v)48(\266)475(eri\257an)24(t)]TJ /F8 1 Tf 4.8012 0 Td [(i)]TJ /F6 1 Tf 0.75904 0 Td [(2)]TJ /F8 1 Tf 1.0783 0 Td [(I)]TJ /F2 1 Tf 0.92771 0 Td [(et)]TJ /F8 1 Tf 1.247 0 Td [(x)]TJ /F6 1 Tf 0.98193 0 Td [(2)]TJ /F8 1 Tf 1.0783 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.259 36.538 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.946 25.683 Td [(.)-686(Il)-409(est)-421(clair)]TJ -33.614 -1.1988 Td [(que)-427(\()]TJ /F8 1 Tf 2.3373 0 Td [(i;)-168(x)]TJ /F2 1 Tf 1.3614 0 Td [(\))]TJ /F6 1 Tf 0.81928 0 Td [(7!)]TJ /F9 1 Tf 1.4277 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(;)-168(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.767 34.823 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.843 24.485 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F2 1 Tf 0.87952 -0.80723 Td [(d)24(\266)475(e\257nit)-427(une)-421(injection)-427(de)]TJ /F8 1 Tf 10.783 0 Td [(Y)]TJ /F2 1 Tf 1.2229 0 Td [(dans)]TJ /F3 1 Tf 2.4337 0 Td [(N)]TJ /F6 1 Tf 1.006 0 Td [(\243)]TJ /F3 1 Tf 1.0602 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)-710(Comme)-427(on)-427(a)-421(vu)-427(que)]TJ /F3 1 Tf -28.976 -1.2048 Td [(N)]TJ /F6 1 Tf 0.9759 0 Td [(\243)]TJ /F3 1 Tf 1.0301 0 Td [(N)]TJ /F2 1 Tf 1.1024 0 Td [(est)-391(d)30(\266)469(enom)24(brable)-385(on)-379(conclut)-385(que)]TJ /F8 1 Tf 14.476 0 Td [(Y)]TJ /F2 1 Tf 1.1867 0 Td [(l'est)-355(\266)469(egalemen)30(t.)-596(Main)24(tenan)24(t)-385(\()]TJ /F8 1 Tf 13.193 0 Td [(i;)-168(x)]TJ /F2 1 Tf 1.3614 0 Td [(\))]TJ /F6 1 Tf 0.74699 0 Td [(7!)]TJ /F8 1 Tf 1.3614 0 Td [(x)]TJ /F2 1 Tf 0.95181 0 Td [(d)24(\266)475(e\257nit)]TJ -36.386 -1.1988 Td [(une)-331(surjection)-337(de)]TJ /F8 1 Tf 7.8133 0 Td [(Y)]TJ /F2 1 Tf 1.1386 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-331(il)-337(s'ensuit)-337(que)]TJ /F8 1 Tf 6.988 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(est)-337(encore)-337(d)24(\266)475(enom)24(brable.)]TJ /F1 1 Tf -19.56 -1.7169 Td [(\(v\))]TJ /F2 1 Tf 2.0602 0 Td [(P)24(our)-283(mon)24(trer)-283(que)-283(le)-283(pro)-30(duit)-283(d'une)-283(famille)-283(\257nie)-283(d'ensem)24(bles)-283(d)24(\266)475(enom)24(brables)-283(est)-283(d)24(\266)475(enom-)]TJ -2.2651 -1.1988 Td [(brable)-385(on)-385(se)-385(ram)24(\265)475(ene)-391(par)-385(r)24(\266)475(ecurrence)-391(au)-385(cas)-385(du)-385(pro)-30(duit)-385(de)-385(deux)-379(ensem)24(bles)-391(d)30(\266)469(enom)24(brables.)]TJ 0 -1.1988 Td [(Quitte)-295(\265)500(a)-295(faire)-301(des)-301(bijections)-295(con)30(v)24(enables)-301(on)-295(p)-30(eut)-301(toujours)-295(supp)-30(oser)-301(que)-295(ces)-301(ensem)24(bles)-295(son)18(t)]TJ 0 -1.2048 Td [(des)-373(parties)]TJ /F3 1 Tf 5.0904 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)-560(Il)-373(su\261t)-373(alors)-373(d'utiliser)-373(le)-379(fait)-373(que)]TJ /F3 1 Tf 15.88 0 Td [(N)]TJ /F6 1 Tf 0.96988 0 Td [(\243)]TJ /F3 1 Tf 1.0301 0 Td [(N)]TJ /F2 1 Tf 1.0904 0 Td [(est)-379(d)30(\266)469(enom)24(brable)-373(et)-373(la)-373(partie)]TJ /F1 1 Tf 12.849 0 Td [(\(ii\))]TJ /F2 1 Tf -37.633 -1.1988 Td [(de)-331(la)-337(prop)-30(osition.)]TJ 0 -2.0723 Td [(T)78(erminons)-307(cette)-313(preuv)24(e)-313(en)-307(mon)24(tran)24(t)-313(que)-313(cet)-277(\266)469(enonc)24(\266)475(e)-313(est)-313(en)-307(d)24(\266)469(efaut)-307(si)-313(l'on)-313(prend)-307(un)-313(pro)-30(duit)]TJ 0 -1.1988 Td [(d'une)-283(famille)-283(d)24(\266)475(enom)24(brable)-283(\(et)-289(non)-283(plus)-283(\257nie\))-283(d'ensem)30(bles)-289(d)30(\266)469(enom)24(brables.)-427(Plus)-283(pr)24(\266)469(ecis)30(\266)469(emen)24(t)]TJ 0 -1.2048 Td [(nous)-319(allons)-319(mon)24(trer)-319(que)]TJ /F8 1 Tf 10.795 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(0)]TJ /F8 1 Tf 0.5 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(1)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F16 1 Tf 8.352 0 0 8.352 0 0 Tm 29.784 16.461 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.687 11.087 Td [(n'est)-325(pas)-319(d)24(\266)469(enom)24(brable.)-439(En)-325(e\256et,)-319(si)]TJ /F8 1 Tf 15.47 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.914 15.659 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.916 11.087 Td [(;)-168(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 60.009 15.659 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.386 11.087 Td [(;)-162(:)-168(:)-168(:)]TJ /F6 1 Tf 1.6084 0 Td [(g)]TJ /F2 1 Tf -38.663 -1.1988 Td [(est)-361(un)-337(\266)475(en)24(um)24(\266)469(eration)-361(de)]TJ /F8 1 Tf 10.199 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(on)-361(p)-30(eut)-361(d)24(\266)475(e\257nir)-361(un)-337(\266)469(el)24(\266)475(emen)24(t)]TJ /F8 1 Tf 12.078 0 Td [(a)]TJ /F2 1 Tf 0.8494 0 Td [(=)-325(\()]TJ /F8 1 Tf 1.488 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.422 13.935 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.193 9.8882 Td [(;)-162(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.457 13.935 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.614 9.8882 Td [(;)-162(:)-168(:)-168(:)]TJ /F2 1 Tf 1.6084 0 Td [(\))-361(de)]TJ /F8 1 Tf 2.1084 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(par)]TJ /F8 1 Tf 1.8072 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 58.578 13.935 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.705 9.8882 Td [(=)-319(0)-361(et)]TJ -36.373 -1.1988 Td [(p)-30(our)]TJ /F8 1 Tf 2.3133 0 Td [(p)]TJ /F11 1 Tf 0.77711 0 Td [(>)]TJ /F2 1 Tf 1.0602 0 Td [(1,)]TJ /F8 1 Tf 1.0663 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.853 12.219 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.819 8.6894 Td [(=)-277(1)-283(si)-283(la)]TJ /F8 1 Tf 3.8494 0 Td [(p)]TJ /F2 1 Tf 0.50602 0 Td [(-i)24(\265)475(eme)-283(comp)-30(osan)24(te)-289(de)]TJ /F8 1 Tf 9.2952 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.267 12.219 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.789 8.6894 Td [(est)-283(0,)]TJ /F8 1 Tf 2.5783 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.784 12.219 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.639 8.6894 Td [(=)-277(0)-283(dans)-277(le)-289(cas)-283(con)24(traire.)-427(Ainsi,)]TJ -25.307 -1.1988 Td [(p)-30(our)-289(tout)]TJ /F8 1 Tf 4.4398 0 Td [(p)]TJ /F11 1 Tf 0.78313 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(1)-289(la)]TJ /F8 1 Tf 1.8554 0 Td [(p)]TJ /F2 1 Tf 0.50602 0 Td [(-i)30(\265)469(eme)-289(comp)-30(osan)24(te)-295(de)]TJ /F8 1 Tf 9.3133 0 Td [(a)]TJ /F2 1 Tf 0.81928 0 Td [(est)-295(di\256)24(\266)475(eren)24(te)-295(de)-289(celle)-289(de)]TJ /F8 1 Tf 10.62 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.509 10.504 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.759 7.4906 Td [(,)]TJ /F8 1 Tf 0.57229 0 Td [(a)]TJ /F2 1 Tf 0.81927 0 Td [(ne)-289(p)-30(eut)-289(donc)-289(pas)]TJ -31.849 -1.2048 Td [(^)469(etre)-259(l'un)-271(des)]TJ /F8 1 Tf 5.5482 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.345 8.7797 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.88 6.2858 Td [(.)-421(P)24(ar)-259(suite)]TJ /F6 1 Tf 4.8313 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 25.448 8.7797 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.235 6.2858 Td [(;)-168(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.543 8.7797 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.699 6.2858 Td [(;)-168(:)-168(:)-162(:)]TJ /F6 1 Tf 1.6145 0 Td [(g)]TJ /F2 1 Tf 0.75904 0 Td [(n'est)-265(pas)-265(une)-234(\266)469(en)30(um)24(\266)469(eration)-265(de)-265(tout)]TJ /F8 1 Tf 14.898 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-277(con)30(trairemen)24(t)]TJ -13.633 -2.3976 Td [(17)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream 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[(En)-228(application)-240(de)-228(la)-234(prop)-30(osition)-234(que)-234(l'on)-234(vien)18(t)-234(de)-228(prouv)24(er)-234(mon)24(trons)-234(que)]TJ /F3 1 Tf 30.825 0 Td [(Q)]TJ /F2 1 Tf 1.012 0 Td [(est)-234(d)24(\266)469(enom)30(brable.)]TJ -31.837 -1.1988 Td [(P)30(our)-463(cela)-469(il)-463(su\261t)-463(de)-463(remarquer)-463(que)-463(\()]TJ /F8 1 Tf 17.133 0 Td [(p;)-168(q)]TJ /F2 1 Tf 1.4277 0 Td [(\))]TJ /F6 1 Tf 0.88554 0 Td [(7!)]TJ /F8 1 Tf 1.488 0 Td [(p=q)]TJ /F2 1 Tf 1.9518 0 Td [(d)30(\266)469(e\257nit)-463(une)-463(application)-463(surjectiv)24(e)-469(du)]TJ -22.886 -1.1988 Td [(pro)-30(duit)-331(d'ensem)24(bles)-337(d)24(\266)475(enom)24(brables)]TJ /F3 1 Tf 15.392 0 Td [(Z)]TJ /F6 1 Tf 0.88554 0 Td [(\243)]TJ /F8 1 Tf 1 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.233 80.521 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.404 56.418 Td [(sur)]TJ /F3 1 Tf 1.6747 0 Td [(Q)]TJ /F2 1 Tf 0.78313 0 Td [(.)]TJ /F5 1 Tf -21.53 -52.53 Td 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Td [(;)]TJ /F2 1 Tf 0.81326 0 Td [(\(c'est)-307(le)-319(seul)-313(cas)]TJ -32.163 -1.1988 Td [(o)-24(\265)524(u)-331(la)-337(b)-30(orne)-331(inf)24(\266)469(erieure)-337(est)-337(plus)-337(grande)-331(que)-343(la)-331(b)-30(orne)-337(sup)0(\266)475(erieure\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 12.449 Td [(I)-30(I.B.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1325 14.924 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.9096 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4458 0 Td [(a)]TJ /F12 1 Tf 0.87952 0 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(b)]TJ /F12 1 Tf 0.77109 0 Td [(dans)]TJ ET 303.91 188.35 m 312.55 188.35 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 25.428 14.924 Td [(R)]TJ /F12 1 Tf 1.0723 0 Td [(v)54(\266)469(eri\257an)24(t)]TJ /F8 1 Tf 3.9337 0 Td [(a)-301(<)-301(b)]TJ /F12 1 Tf 2.3373 0 Td [(,)-349(on)-349(p)-30(eut)-349(trouv)24(er)]TJ /F8 1 Tf 7.8675 0 Td [(c)]TJ /F12 1 Tf 0.78313 0 Td [(dans)]TJ /F3 1 Tf 2.3494 0 Td [(R)]TJ /F12 1 Tf -38.44 -1.1988 Td [(et)-337(m)30(^)469(eme)-331(dans)]TJ /F3 1 Tf 6.3976 0 Td [(Q)]TJ /F12 1 Tf 1.1084 0 Td [(tel)-337(que)]TJ /F8 1 Tf 3.3072 0 Td [(a)-283(<)-277(c)-283(<)-283(b)]TJ /F12 1 Tf 4.0602 0 Td [(.)]TJ /F1 1 Tf -14.873 -1.9277 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Ce)-325(r)30(\266)469(esultat)-331(est)-325(bien)-325(conn)24(u)-325(p)-30(our)]TJ /F8 1 Tf 14.12 0 Td [(a)]TJ /F2 1 Tf 0.85542 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(b)]TJ /F2 1 Tf 0.75301 0 Td [(\257nis.)-439(Si)]TJ /F8 1 Tf 3.6566 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(\2411)]TJ /F2 1 Tf 2.1024 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(b)]TJ /F2 1 Tf 0.75301 0 Td [(\257ni)-325(on)-325(se)]TJ -35.235 -1.2048 Td [(ram)24(\265)475(ene)-343(au)-337(cas)-337(pr)24(\266)475(ec)24(\266)469(eden)30(t)-343(en)-337(rempla)0(\273)439(can)24(t)]TJ /F8 1 Tf 17.645 0 Td [(a)]TJ /F2 1 Tf 0.86747 0 Td [(par)]TJ /F8 1 Tf 1.7831 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F2 1 Tf 1 0 Td [(1.)-451(De)-343(m)24(^)475(eme,)-337(si)]TJ /F8 1 Tf 6.9518 0 Td [(a)]TJ /F2 1 Tf 0.86747 0 Td [(est)-343(\257ni)-337(et)]TJ /F8 1 Tf 4.4578 0 Td [(b)]TJ /F2 1 Tf 0.71084 0 Td [(=)-283(+)]TJ /F6 1 Tf 1.8373 0 Td [(1)]TJ /F2 1 Tf 1.3374 0 Td [(on)]TJ -38.108 -1.1988 Td [(remplace)]TJ /F8 1 Tf 4.2229 0 Td [(b)]TJ /F2 1 Tf 0.76506 0 Td [(par)]TJ /F8 1 Tf 1.7831 0 Td [(a)]TJ /F2 1 Tf 0.74699 0 Td [(+)-228(1.)-445(En\257n)-337(c'est)-307(\266)469(eviden)24(t)-337(p)-30(our)]TJ /F8 1 Tf 13.06 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(b)]TJ /F2 1 Tf 0.70482 0 Td [(=)-283(+)]TJ /F6 1 Tf 1.8373 0 Td [(1)]TJ /F2 1 Tf 1.3313 0 Td [(\(prendre)]TJ /F8 1 Tf 4.0663 0 Td [(c)]TJ /F2 1 Tf 0.71084 0 Td [(=)-277(0\).)]TJ ET 512.64 111.06 m 519.84 111.06 l 519.84 119.48 l 512.64 119.48 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.247 3.8882 Td [(19)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 90 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 89 0 R /ProcSet 2 0 R >> /Contents 88 0 R >> endobj 89 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 91 0 obj << /Length 15487 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.424 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-13710(I)-36(I.B.)-385(G)24(\266)548(en)18(\266)548(eralit)18(\266)554(es)-391(sur)]TJ ET 519.7 781.3 m 531.86 781.3 l 531.86 782.53 l 519.7 782.53 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 43.482 64.424 Td [(I)283(R)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I)-30(I.B.2.)-246(Ordre)-376(et)-376(top)-35(ologie.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 60.611 Td [(Soit)]TJ /F8 1 Tf 2.0422 0 Td [(X)]TJ /F2 1 Tf 1.2349 0 Td [(un)-325(ensem)24(ble)-325(m)24(uni)-325(d'une)-331(relation)-319(d'ordre)-331(total)]TJ /F11 1 Tf 20.398 0 Td [(6)]TJ /F2 1 Tf 0.77711 0 Td [(.)-439(On)-325(app)-36(elle)-325(alors)]TJ /F1 1 Tf 8.1747 0 Td [(in)30(terv)66(alles)]TJ /F2 1 Tf 5.4217 0 Td [(les)]TJ -38.048 -1.1988 Td [(ensem)24(bles)-337(suiv)54(an)24(ts)-337(:)]TJ 0.018072 -1.6145 Td [(\(I\))]TJ /F8 1 Tf 2.0422 -0.02409 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(et)]TJ /F6 1 Tf 1.1687 0 Td [(;)]TJ /F2 1 Tf 0.83133 0 Td [(;)]TJ -5.6687 -3.3855 Td [(\(I)-30(I\))]TJ /F9 1 Tf 2.0241 2.3494 Td [(8)]TJ 0 -0.90361 Td [(>)]TJ 0 -0.29518 Td [(<)]TJ 0 -1.8012 Td [(>)]TJ 0 -0.3012 Td [(:)]TJ /F2 1 Tf 0.89157 2.5482 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(\303)]TJ /F2 1 Tf 1 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [([)-168(=)]TJ /F6 1 Tf 1.5 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.85542 0 Td [(2)]TJ /F8 1 Tf 0.93976 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(x)-283(<)-283(a)]TJ /F6 1 Tf 2.4398 0 Td [(g)]TJ /F2 1 Tf 1.5 0 Td [(section)-331(commen)-6(\273)439(can)30(te)-337(ouv)18(erte)-331(d'extr)24(\266)469(emit)24(\266)469(e)]TJ /F8 1 Tf 18.464 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(;)]TJ -30.994 -1.5964 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(!)]TJ /F2 1 Tf 1.006 0 Td [(\))-277(=)]TJ /F6 1 Tf 1.7229 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.93976 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(a)-277(<)-283(x)]TJ /F6 1 Tf 2.4337 0 Td [(g)]TJ /F2 1 Tf 1.5 0 Td [(section)-331(\257nissan)18(te)-331(ouv)18(erte)-331(d'extr)24(\266)469(emit)24(\266)469(e)]TJ /F8 1 Tf 16.584 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(;)]TJ -29.392 -1.6024 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [([)-168(=)-277(\()]TJ /F6 1 Tf 1.8855 0 Td [(\303)]TJ /F2 1 Tf 1.006 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [([)]TJ /F6 1 Tf 0.43976 0 Td [(\\)]TJ /F2 1 Tf 0.83735 0 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(!)]TJ /F2 1 Tf 1 0 Td [(\))-277(=)]TJ /F6 1 Tf 1.7229 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(a)-283(<)-277(x)-283(<)-277(b)]TJ /F6 1 Tf 4.1928 0 Td [(g)]TJ /F2 1 Tf 1.1205 0 Td [(in)24(terv)54(alle)-307(ouv)24(ert)-313(d'extr)24(\266)475(emit)18(\266)475(es)]TJ /F8 1 Tf 13.084 0 Td [(a)]TJ /F2 1 Tf 0.83735 0 Td [(et)]TJ /F8 1 Tf 1.1385 0 Td [(b)]TJ /F2 1 Tf 0.74097 0 Td [(;)]TJ -39.325 -3.9096 Td [(\(I)-30(I)-36(I\))]TJ /F9 1 Tf 2.4157 2.6506 Td [(8)]TJ 0 -0.90362 Td [(>)]TJ 0 -0.29518 Td [(>)]TJ 0 -0.30121 Td [(<)]TJ 0 -1.8012 Td [(>)]TJ 0 -0.30121 Td [(>)]TJ 0 -0.3012 Td [(:)]TJ /F2 1 Tf 0.89157 3.3494 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(\303)]TJ /F2 1 Tf 1 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.69277 0 Td [(])-277(=)]TJ /F6 1 Tf 1.6145 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.93976 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(x)]TJ /F11 1 Tf 0.8494 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(g)]TJ /F2 1 Tf 1.5 0 Td [(section)-331(commen)-6(\273)439(can)24(te)-331(ferm)24(\266)475(ee)-337(d'extr)24(\266)469(emit)24(\266)469(e)]TJ /F8 1 Tf 18.102 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(;)]TJ -31.078 -1.6024 Td [([)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(!)]TJ /F2 1 Tf 1.006 0 Td [(\))-277(=)]TJ /F6 1 Tf 1.7229 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.93976 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(a)]TJ /F11 1 Tf 0.80121 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(x)]TJ /F6 1 Tf 0.57229 0 Td [(g)]TJ /F2 1 Tf 1.5 0 Td [(section)-331(\257nissan)18(te)-331(ferm)24(\266)475(ee)-337(d'extr)24(\266)469(emit)24(\266)469(e)]TJ /F8 1 Tf 16.223 0 Td [(a)]TJ /F2 1 Tf 0.86144 0 Td [(;)]TJ -29.199 -2.0482 Td [([)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.59639 0 Td [(])-283(=)-277(\()]TJ /F6 1 Tf 2.006 0 Td [(\303)]TJ /F2 1 Tf 1 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.59639 0 Td [(])]TJ /F6 1 Tf 0.5 0 Td [(\\)]TJ /F2 1 Tf 0.89157 0 Td [([)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(!)]TJ /F2 1 Tf 1.006 0 Td [(\))-277(=)]TJ /F6 1 Tf 1.7229 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(a)]TJ /F11 1 Tf 0.8012 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(x)]TJ /F11 1 Tf 0.8494 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F6 1 Tf 0.42771 0 Td [(g)]TJ /F2 1 Tf 1.1687 0.60241 Td [(in)24(terv)54(alle)-487(ferm)24(\266)475(e)-493(ou)]TJ /F1 1 Tf 8.9518 0 Td [(segmen)30(t)]TJ /F2 1 Tf -8.9518 -1.1988 Td [(d'extr)24(\266)469(emit)24(\266)469(es)]TJ /F8 1 Tf 5.7048 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(b)]TJ /F2 1 Tf 0.76506 0 Td [(;)]TJ -32.765 -2.5602 Td [(\(IV\))]TJ /F9 1 Tf 2.3855 1.4096 Td [(\275)]TJ /F2 1 Tf 1.0843 -0.60843 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.59639 0 Td [(])-277(=)-283(\()]TJ /F6 1 Tf 2 0 Td [(\303)]TJ /F2 1 Tf 1 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.59638 0 Td [(])]TJ /F6 1 Tf 0.44578 0 Td [(\\)]TJ /F2 1 Tf 0.83133 0 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(!)]TJ /F2 1 Tf 1 0 Td [(\))-283(=)]TJ /F6 1 Tf 1.7229 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(a)-283(<)-277(x)]TJ /F11 1 Tf 2.7108 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(b)]TJ /F6 1 Tf 0.42771 0 Td [(g)]TJ /F2 1 Tf -19.38 -1.6024 Td [([)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [([)-168(=)-277(\()]TJ /F6 1 Tf 1.8916 0 Td [(\303)]TJ /F2 1 Tf 1 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [([)]TJ /F6 1 Tf 0.44578 0 Td [(\\)]TJ /F2 1 Tf 0.66265 0 Td [([)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(!)]TJ /F2 1 Tf 1 0 Td [(\))-283(=)]TJ /F6 1 Tf 1.7229 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(a)]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(x)-283(<)-277(b)]TJ /F6 1 Tf 2.3373 0 Td [(g)]TJ /F2 1 Tf 2.5843 1.3012 Td [(in)24(terv)60(alles)-391(semi-ouv)24(erts)]TJ 0 -1.1988 Td [(d'extr)24(\266)475(emit)18(\266)475(es)]TJ /F8 1 Tf 5.7048 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(b)]TJ /F2 1 Tf 0.75904 0 Td [(.)]TJ -32.922 -1.9157 Td [(On)-349(app)-36(elle)]TJ /F1 1 Tf 5.1205 0 Td [(in)30(terv)66(alles)-409(ouv)30(erts)]TJ /F2 1 Tf 9.5422 0 Td [(les)-355(in)24(terv)60(alles)-355(du)-355(t)24(yp)-36(e)-355(\(I\))-349(et)-355(ceux)-349(du)-355(t)24(yp)-36(e)-355(\(I)-30(I\))-355(et)]TJ /F1 1 Tf 21.741 0 Td [(in)30(ter-)]TJ -36.404 -1.1988 Td [(v)60(alles)-385(ferm)18(\266)548(es)]TJ /F2 1 Tf 6.7289 0 Td [(les)-337(in)24(terv)54(alles)-337(du)-337(t)24(yp)-30(e)-337(\(I\))-337(et)-337(ceux)-337(du)-331(t)24(yp)-36(e)-337(\(I)-30(I)-36(I\).)]TJ /F1 1 Tf -6.7289 -1.9157 Td [(Remarques.)]TJ 0 -1.6084 Td [(1.)]TJ /F2 1 Tf 1.8916 0 Td [(Dans)-379(le)-379(cas)]TJ /F8 1 Tf 5.3976 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(=)]TJ ET 0.504 w 179.14 450.07 m 187.78 450.07 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 14.988 36.828 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-385([)]TJ /F6 1 Tf 1.1024 0 Td [(\2411)]TJ /F2 1 Tf 1.7831 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(a)]TJ /F2 1 Tf 0.6988 0 Td [([=)-343(\()]TJ /F6 1 Tf 1.7892 0 Td [(\303)]TJ /F2 1 Tf 1 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(a)]TJ /F2 1 Tf 0.6988 0 Td [([)-373(et)-373(cet)-373(in)24(terv)54(alle)-373(est)-373(donc)-373(un)-379(in)30(terv)54(alle)-373(ouv)18(ert.)]TJ -18.331 -1.2048 Td [(De)-337(m)24(^)475(eme)-500(])]TJ /F8 1 Tf 4.8735 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1.1687 0 Td [(])-277(=)-168(])]TJ /F8 1 Tf 1.7771 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(!)]TJ /F2 1 Tf 1 0 Td [(\))-337(est)-337(un)-337(in)24(terv)54(alle)-331(ouv)24(ert.)]TJ /F1 1 Tf -11.548 -1.6084 Td [(2.)]TJ /F2 1 Tf 1.8916 0 Td [(P)24(our)]TJ /F8 1 Tf 2.4578 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(quelconque,)-367(un)-361(in)24(terv)54(alle)-355(p)-30(eut-)30(^)469(etre)-361(de)-355(plusieurs)-361(t)24(yp)-36(es.)-518(Ainsi,)-361(si)]TJ /F8 1 Tf 28.235 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(=)]TJ /F3 1 Tf 1.0904 0 Td [(N)]TJ /F2 1 Tf 1.0843 0 Td [(on)-355(a)]TJ -37.253 -1.1988 Td [([)-168(0;)-162(1)-168(])-277(=)-283([)-168(0;)-162(2[)-168(=)-283(\()]TJ /F6 1 Tf 7.4458 0 Td [(\303)]TJ /F2 1 Tf 1 0 Td [(;)-168(1)-168(].)]TJ /F1 1 Tf -8.4458 -1.6084 Td [(3.)]TJ /F2 1 Tf 1.8916 0 Td [(L'in)24(tersection)-301(de)-301(deux)-301(in)24(terv)54(alles)-301(ouv)24(erts)-307(\(resp.)-433(ferm)24(\266)475(es\))-307(est)-307(un)-301(in)24(terv)60(alle)-301(ouv)18(ert)-301(\(resp.)]TJ -1.8916 -1.2048 Td [(ferm)30(\266)469(e\).)]TJ 0 -2.2169 Td [(Une)-337(propri)24(\266)469(et)30(\266)469(e)-337(imp)-36(ortan)24(te)-337(des)-337(in)30(terv)54(alles)-337(est)-343(la)-331(suiv)54(an)24(te)-337(dite)]TJ /F1 1 Tf 26.572 0 Td [(propri)24(\266)548(et)18(\266)548(e)-385(de)-385(con)24(v)30(exit)18(\266)548(e)]TJ /F2 1 Tf 11.59 0 Td [(:)]TJ /F15 1 Tf -38.163 -1.9157 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.9458 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4578 0 Td [(a)]TJ /F12 1 Tf 0.88554 0 Td [(et)]TJ /F8 1 Tf 1.1867 0 Td [(b)]TJ /F12 1 Tf 0.78916 0 Td [(dans)-355(un)-361(in)24(terv)54(alle)]TJ /F8 1 Tf 8.1928 0 Td [(I)]TJ /F12 1 Tf 0.87952 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.2169 0 Td [(a)]TJ /F11 1 Tf 0.84337 0 Td [(6)]TJ /F8 1 Tf 1.0964 0 Td [(b)]TJ /F12 1 Tf 0.43373 0 Td [(,)-361(le)-361(segmen)24(t)]TJ /F2 1 Tf 5.6084 0 Td [([)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.59639 0 Td [(])]TJ /F12 1 Tf 0.63253 0 Td [(est)-361(con)24(ten)24(u)]TJ -34.187 -1.1988 Td [(dans)]TJ /F8 1 Tf 2.3373 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(.)]TJ /F1 1 Tf -2.8554 -1.9157 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Il)-415(su\261t)-427(de)-421(consid)30(\266)469(erer)-421(les)-421(di\256)24(\266)469(eren)24(ts)-421(cas)-421(d'in)24(terv)54(alles)-391(\266)469(en)24(um)24(\266)475(er)24(\266)469(es)-421(pr)24(\266)475(ec)24(\266)469(e-)]TJ -8.8193 -1.1988 Td [(demmen)24(t.)]TJ ET 512.64 256.42 m 519.84 256.42 l 519.84 264.85 l 512.64 264.85 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 19.34 Td [(On)-343(d)24(\266)475(e\257nit)-343(alors)-343(sur)]TJ /F8 1 Tf 8.8855 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(une)-343(top)-36(ologie)-343(dite)]TJ /F1 1 Tf 8.2229 0 Td [(top)-36(ologie)-397(de)-397(l'ordre)]TJ /F2 1 Tf 10.211 0 Td [(en)-343(prenan)24(t)-343(p)-30(our)-343(ouv)24(erts)]TJ -28.566 -1.2048 Td [(les)-409(r)30(\266)469(eunions)-409(d'in)24(terv)54(alles)-409(ouv)24(erts.)-668(Cette)-409(top)-30(ologie)-409(est)-409(donc)-409(engendr)24(\266)469(ee)-409(par)-409(les)-409(in)24(terv)54(alles)]TJ 0 -1.1988 Td [(ouv)24(erts)-337(ou)-337(encore)-337(par)-337(les)-337(sections)-337(ouv)24(ertes)-337(commen)-6(\273)439(can)24(tes)-337(et)-337(\257nissan)24(tes.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 11.595 Td [(I)-30(I.B.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1325 13.9 Td [(1.)-289(Une)-385(remarque)-385(imp)-36(ortan)30(te.)]TJ /F2 1 Tf -3.8012 -1.6084 Td [(Si)]TJ /F8 1 Tf 1.0783 0 Td [(Y)]TJ /F2 1 Tf 1.0482 0 Td [(est)-246(une)-246(partie)-253(de)]TJ /F8 1 Tf 7.3253 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-265(on)-246(p)-30(eut)-246(d)24(\266)469(e\257nir)-246(sur)]TJ /F8 1 Tf 8.6747 0 Td [(X)]TJ /F2 1 Tf 1.1566 0 Td [(une)-246(relation)-246(d'ordre)-246(induite)-246(\(par)-253(restriction\))]TJ -20.187 -1.2048 Td [(et)-265(une)-265(top)-30(ologie)-265(induite)-271(\(en)-265(prenan)24(t)-265(p)-30(our)-265(ouv)18(erts)-271(les)-265(traces)-265(des)-271(ouv)24(erts)-271(de)]TJ /F8 1 Tf 32.066 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\).)-427(On)-265(v)54(\266)469(eri\257era)]TJ -32.97 -1.1988 Td [(que,)-379(si)]TJ /F14 1 Tf 3.1867 0 Td [(Y)]TJ /F2 1 Tf 1.0783 0 Td [(est)-337(un)-337(in)24(terv)54(alle)-331(de)]TJ /F14 1 Tf 8.6807 0 Td [(X)]TJ ET 89.784 116.56 m 227.38 116.56 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.018 9.8882 Td [(,)-379(les)-373(in)30(terv)54(alles)-373(de)]TJ /F8 1 Tf 8.2771 0 Td [(Y)]TJ /F2 1 Tf 1.1747 0 Td [(son)24(t)-373(exactemen)30(t)-373(les)-373(in)24(terv)54(alles)-367(de)]TJ /F8 1 Tf 15.12 0 Td [(X)]TJ /F2 1 Tf -38.259 -1.1988 Td [(qui)-337(son)24(t)-343(con)24(ten)24(us)-343(dans)]TJ /F8 1 Tf 10.325 0 Td [(Y)]TJ /F2 1 Tf 1.1386 0 Td [(et)-343(les)-343(in)24(terv)54(alles)-343(ouv)24(erts)-343(de)]TJ /F8 1 Tf 12.229 0 Td [(Y)]TJ /F2 1 Tf 1.1446 0 Td [(son)24(t)-343(exactemen)30(t)-343(les)-343(in)24(tersections)]TJ -24.837 -1.1988 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.3434 0 Td [(Y)]TJ /F2 1 Tf 1.2831 0 Td [(des)-487(in)24(terv)60(alles)-487(ouv)24(erts)-487(de)]TJ /F8 1 Tf 11.904 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(;)-560(il)-481(s'ensuit)-481(que)-487(la)-481(top)-36(ologie)-481(induite)-481(sur)]TJ /F8 1 Tf 18.735 0 Td [(Y)]TJ /F2 1 Tf 1.2892 0 Td [(par)-481(la)]TJ -36.458 -1.2048 Td [(top)-30(ologie)-337(de)-331(l'ordre)-337(de)]TJ /F8 1 Tf 10.151 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(est)-337(la)-331(top)-36(ologie)-331(de)-337(l'ordre)-337(de)]TJ /F8 1 Tf 12.819 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(.)]TJ /F5 1 Tf -25.012 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(20)]TJ ET endstream endobj 93 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 92 0 R /ProcSet 2 0 R >> /Contents 91 0 R >> endobj 92 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F14 43 0 R /F15 68 0 R >> endobj 94 0 obj << /Length 14521 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.424 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-13710(I)-36(I.B.)-385(G)24(\266)548(en)18(\266)548(eralit)18(\266)554(es)-391(sur)]TJ ET 519.7 781.3 m 531.86 781.3 l 531.86 782.53 l 519.7 782.53 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 43.482 64.424 Td [(I)283(R)]TJ /F2 1 Tf -38.151 -2.2048 Td [(Le)-421(r)30(\266)469(esultat)-421(pr)30(\266)469(ec)24(\266)469(eden)30(t)-421(est)-421(en)-415(d)24(\266)469(efaut)-415(si)]TJ /F8 1 Tf 17.181 0 Td [(Y)]TJ /F2 1 Tf 1.2169 0 Td [(n'est)-421(pas)-421(un)-421(in)30(terv)54(alle.)-698(Ainsi,)-439(p)-30(our)]TJ /F8 1 Tf 16.277 0 Td [(X)]TJ /F2 1 Tf 1.3193 0 Td [(=)]TJ ET 0.504 w 508.25 753.55 m 516.89 753.55 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 42.524 62.22 Td [(R)]TJ /F2 1 Tf 1.1386 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.1988 Td [(Y)]TJ /F2 1 Tf 1.2711 0 Td [(=)-475([)]TJ /F6 1 Tf 1.6988 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1)-168(])]TJ /F6 1 Tf 0.94578 0 Td [([)]TJ /F2 1 Tf 0.83132 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2.006 0 Td [(1)]TJ /F2 1 Tf 1.1627 0 Td [(])-451(on)-451(p)-30(ourra)-451(v)48(\266)475(eri\257er)-451(que)-457(la)-445(suite)-457(des)-451(1)]TJ /F8 1 Tf 17.133 0 Td [(=n)]TJ /F2 1 Tf 1.5482 0 Td [(tend)-457(v)24(ers)]TJ /F6 1 Tf 4.5783 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1)-451(p)-30(our)-451(la)]TJ -34.952 -1.1988 Td [(top)-30(ologie)-331(de)-331(l'ordre)-331(de)]TJ /F8 1 Tf 10.133 0 Td [(Y)]TJ /F2 1 Tf 1.1325 0 Td [(mais)-331(n'a)-331(pas)-331(de)-331(limite)-331(\(dans)]TJ /F8 1 Tf 12.663 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(\))-337(p)-30(our)-331(la)-325(top)-36(ologie)-331(induite)-331(par)-331(la)]TJ -24.729 -1.2048 Td [(top)-30(ologie)-337(de)-331(l'ordre)-337(de)]TJ ET 185.04 710.56 m 193.68 710.56 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 15.482 58.617 Td [(R)]TJ /F2 1 Tf 0.71687 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 45.037 Td [(I)-30(I.B.3.)-246(Le)-371(cas)-376(de)]TJ ET 183.53 655.2 m 192.17 655.2 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 15.355 53.991 Td [(R)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 13.412 45.037 Td [(.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 52.268 Td [(Comme)-349(tout)-355(espace)-349(totalemen)24(t)-355(ordonn)24(\266)469(e,)]TJ ET 281.09 634.6 m 289.73 634.6 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 23.518 52.268 Td [(R)]TJ /F2 1 Tf 1.0723 0 Td [(est)-355(donc)-349(naturellemen)24(t)-355(m)24(uni)-349(d'une)-355(top)-36(ologie.)]TJ -19.259 -1.1988 Td [(Alors)-343(tout)-343(in)24(terv)54(alle)-343(de)]TJ ET 189.5 620.28 m 198.14 620.28 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 15.855 51.069 Td [(R)]TJ /F2 1 Tf 1.0663 0 Td [(et,)-349(en)-343(particulier,)]TJ /F3 1 Tf 7.9277 0 Td [(R)]TJ /F2 1 Tf 1.0181 0 Td [(=)-168(])]TJ /F6 1 Tf 1.4518 0 Td [(\241)-228(1)]TJ /F2 1 Tf 2.006 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)-512(est)-349(m)30(uni)-349(d'une)-343(top)-36(ologie)-343(de)]TJ -26.217 -1.2048 Td [(l'ordre)-337(qui)-331(est)-337(aussi)-337(la)-337(top)-30(ologie)-337(induite)-337(par)-331(celle)-337(de)]TJ ET 339.05 605.95 m 347.69 605.95 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 28.367 49.864 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-445(On)-331(en)-337(d)24(\266)475(eduit)-337(:)]TJ /F15 1 Tf -23.759 -2.0783 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.1145 0 Td [(L'espace)-415(top)-30(ologique)]TJ ET 263.16 581.04 m 271.8 581.04 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 22.018 47.786 Td [(R)]TJ /F12 1 Tf 1.1325 0 Td [(est)-415(hom)24(\266)469(eomorphe)-415(au)-415(segmen)24(t)]TJ /F2 1 Tf 13.584 0 Td [([)]TJ /F6 1 Tf 0.44578 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1;)-168(1)-162(])]TJ /F12 1 Tf 1.8855 0 Td [(.)-680(Il)-415(est)-415(en)]TJ -34.512 -1.2048 Td [(particulier)-379(compact)-379(et)-379(p)-30(eut)-355(^)469(etre)-379(consid)30(\266)469(er)24(\266)469(e)-379(comme)-379(un)-379(espace)-379(m)24(\266)469(etrique)-379(p)-30(our)-379(une)-379(distance)]TJ 0 -1.1988 Td [(con)30(v)24(enable.)]TJ /F1 1 Tf 0 -2.0843 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(D)24(\266)469(e\257nissons)-427(une)-433(application)]TJ /F8 1 Tf 12.518 0 Td [(f)]TJ /F2 1 Tf 1.0241 0 Td [(de)]TJ ET 348.12 527.4 m 356.76 527.4 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 29.127 43.298 Td [(R)]TJ /F2 1 Tf 1.1506 0 Td [(dans)-427([)]TJ /F6 1 Tf 2.8795 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1;)-168(1)-168(])-427(par)]TJ /F8 1 Tf 4.1988 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F6 1 Tf 0.39156 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(\))-439(=)]TJ /F6 1 Tf 2.0422 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1,)]TJ /F8 1 Tf -38.386 -1.4578 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))-469(=)-463(1)-445(et)]TJ /F8 1 Tf 4.3253 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-469(=)]TJ /F8 1 Tf 3.7229 0.68073 Td [(x)]TJ ET 193.54 503.06 m 236.3 503.06 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.193 41.87 Td [(p)]TJ ET 203.54 500.68 m 236.38 500.68 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.03 41.039 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 25.19 59.142 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.271 41.039 Td [(+)-222(1)]TJ 2.0663 0.8012 Td [(si)]TJ /F8 1 Tf 1.1205 0 Td [(x)]TJ /F2 1 Tf 1.0181 0 Td [(est)-451(\257ni.)-783(Il)-451(est)-451(facile)-445(de)-445(v)48(\266)475(eri\257er)-451(qu'il)-445(s'agit)-445(d'une)]TJ -17.145 -1.9217 Td [(bijection)-439(croissan)30(te.)-753(On)-433(en)-433(d)24(\266)469(eduit)-433(que)]TJ /F8 1 Tf 17.855 0 Td [(f)]TJ /F2 1 Tf 1.0301 0 Td [(transforme)-433(les)-439(in)24(terv)60(alles)-439(ouv)24(erts)-439(de)]TJ ET 487.51 487.08 m 496.15 487.08 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 40.789 39.918 Td [(R)]TJ /F2 1 Tf 1.1566 0 Td [(en)-433(les)]TJ -36.614 -1.1988 Td [(in)24(terv)60(alles)-313(ouv)24(erts)-307(de)-307([)]TJ /F6 1 Tf 9.9337 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1;)-168(1)-168(])-301(donc)-307(les)-307(ouv)18(erts)-307(de)]TJ ET 320.4 472.75 m 329.04 472.75 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 26.807 38.72 Td [(R)]TJ /F2 1 Tf 1.0241 0 Td [(en)-307(les)-307(ouv)24(erts)-313(de)-301([)]TJ /F6 1 Tf 7.9578 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1;)-168(1)-168(],)-307(il)-307(s'agit)-307(d'un)]TJ -31.235 -1.1988 Td [(hom)24(\266)475(eomorphisme.)]TJ 0 -1.7229 Td [(Cela)-337(p)-30(ermet)-337(de)-331(d)24(\266)469(e\257nir)-337(la)-331(top)-36(ologie)-331(de)]TJ ET 265.75 437.76 m 274.39 437.76 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 22.235 35.798 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(a)30(v)24(ec)-337(la)-337(distance)]TJ /F8 1 Tf 7.2048 0 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-168(b)]TJ /F2 1 Tf 1.4036 0 Td [(\))-277(=)]TJ /F6 1 Tf 1.7229 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)]TJ ET 512.64 426.63 m 519.84 426.63 l 519.84 435.06 l 512.64 435.06 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 34.075 Td [(On)-337(remarquera)-331(que)-337(la)-337(prop)-30(osition)]TJ /F1 1 Tf 15.217 0 Td [(I)-36(I.B.1.1.)]TJ /F2 1 Tf 4.488 0 Td [(en)30(tra)108(^)385(\263ne)-331(les)-313(\266)475(enonc)24(\266)475(es)-337(suiv)54(an)24(ts)-337(:)]TJ /F12 1 Tf -19.705 -1.7229 Td [({)-319(un)-325(in)24(terv)54(alle)-319(ouv)24(ert)-325(ne)-319(p)-30(oss)24(\265)475(ede)-325(pas)-331(de)-319(plus)-319(grand)-325(\(resp.)-445(plus)-319(p)-30(etit\))-295(\266)469(el)24(\266)469(emen)24(t)-325(sauf)-319(si)-325(c'est)]TJ /F2 1 Tf 0 -1.1988 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F12 1 Tf 1.3313 0 Td [(\(resp.)]TJ /F6 1 Tf 2.9036 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ -7.512 -1.7229 Td [({)-331(tout)-337(in)24(terv)54(alle)-331(ouv)24(ert)-337(non)-337(vide)-337(p)-30(oss)24(\265)475(ede)-343(un)-307(\266)469(el)24(\266)475(emen)24(t)-337(rationnel.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 21.877 Td [(I)-30(I.B.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1325 26.226 Td [(1.)-289(V)90(oisinages.)]TJ /F2 1 Tf -3.8012 -1.7229 Td [(P)30(ar)-325(d)30(\266)469(e\257nition,)-325(si)]TJ /F8 1 Tf 7.5723 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ ET 175.18 302.76 m 183.82 302.76 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 14.657 24.503 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-325(un)-319(v)24(oisinage)-331(de)]TJ /F8 1 Tf 7.6265 0 Td [(a)]TJ /F2 1 Tf 0.8494 0 Td [(est)-325(une)-325(partie)-325(qui)-319(con)24(tien)24(t)-325(un)-325(ouv)24(ert)-325(p)-30(oss)30(\266)469(edan)24(t)]TJ /F8 1 Tf -18.524 -1.1988 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-572(Plus)-385(pr)24(\266)475(ecis)24(\266)469(emen)30(t,)-391(la)-379(d)24(\266)469(e\257nition)-379(de)-373(la)-379(top)-36(ologie)-379(de)-373(l'ordre)-385(mon)24(tre)-379(qu'un)-379(v)24(oisinage)-385(de)]TJ /F8 1 Tf 38.102 0 Td [(a)]TJ /F2 1 Tf -38.633 -1.2048 Td [(est)-337(une)-337(partie)-337(qui)-331(con)24(tien)24(t)-337(un)-337(in)24(terv)54(alle)-331(ouv)24(ert)-337(qui)-337(p)-30(oss)30(\265)469(ede)]TJ /F8 1 Tf 26.12 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)]TJ -26.651 -1.7169 Td [(Remarquons)-337(alors)-337(que)-343(tout)-337(in)24(terv)54(alle)-337(ouv)24(ert)-337(p)-30(eut)-337(s')24(\266)469(ecrire)-337(comme)-343(une)-337(in)24(tersection)]TJ /F8 1 Tf 36.042 0 Td [(J)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 60 28.952 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.602 20.382 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(J)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 63.034 28.952 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 19.177 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3855 0 Td [(J)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 10.414 27.228 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.0542 19.177 Td [(\(resp.)]TJ /F8 1 Tf 2.9036 0 Td [(J)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.474 27.228 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.958 19.177 Td [(\))-337(est)-337(une)-337(section)-337(commen)0(\273)439(can)24(te)-337(\(resp.)-445(\257nissan)24(te\))-343(ouv)24(erte)-331(ou)]TJ ET 458.21 239.18 m 466.85 239.18 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 38.337 19.177 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(tout)-337(en)24(tier.)]TJ -34.06 -1.7229 Td [(On)-319(en)-319(d)24(\266)475(eduit)-319(une)-319(caract)24(\266)469(erisation)-319(des)-325(limites)-319(;)-325(par)-319(exemple)-325(dire)-319(qu'une)-325(suite)-319(\()]TJ /F8 1 Tf 33.801 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.828 24.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.253 17.454 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.155 24.762 Td [(n)]TJ /F10 1 Tf 0.71552 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.548 17.454 Td [(tend)]TJ -37.217 -1.1988 Td [(v)24(ers)]TJ /F8 1 Tf 2.0663 0 Td [(`)]TJ /F2 1 Tf 0.74699 0 Td [(dans)]TJ ET 125.35 204.19 m 133.99 204.19 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 10.488 16.256 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(signi\257e)-331(:)]TJ -6.2108 -1.7229 Td [({)-289(p)-30(our)-295(toute)-289(section)-295(commen)-6(\273)439(can)30(te)-295(ou)-289(\257nissan)18(te)-295(ouv)24(erte)]TJ /F8 1 Tf 24.386 0 Td [(J)]TJ /F2 1 Tf 0.94578 0 Td [(qui)-289(p)-30(oss)24(\265)475(ede)]TJ /F8 1 Tf 5.259 0 Td [(`)]TJ /F2 1 Tf 0.42168 0 Td [(,)-295(il)-295(existe)-295(un)-295(indice)]TJ -31.012 -1.1988 Td [(\265)499(a)-331(partir)-337(duquel)]TJ /F8 1 Tf 6.9217 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.353 18.866 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.705 13.334 Td [(appartien)24(t)-337(\265)500(a)]TJ /F8 1 Tf 5.6988 0 Td [(J)]TJ /F2 1 Tf 0.65663 0 Td [(.)]TJ -14.729 -1.7229 Td [(Cela)-337(p)-30(eut)-337(s')24(\266)469(ecrire)-337(de)-331(fa)-6(\273)439(con)-331(formalis)24(\266)475(ee)-337(comme)-337(la)-337(conjonction)-331(des)-337(deux)-307(\266)475(enonc)24(\266)469(es)-331(:)]TJ 10.096 -1.1988 Td [(\(1\))]TJ /F6 1 Tf 2.6145 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(A)-283(<)-277(`)]TJ /F6 1 Tf 3.5 0 Td [(9)]TJ /F8 1 Tf 0.56024 0 Td [(N)]TJ /F6 1 Tf 1.1867 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(N)]TJ /F6 1 Tf 1.7229 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(n)-283(>)-283(N)-1108(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.06 14.685 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.313 10.412 Td [(>)-277(A)]TJ /F2 1 Tf 2.1386 0 Td [(;)]TJ -19.084 -1.2048 Td [(\(2\))]TJ /F6 1 Tf 2.6145 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(B)-337(>)-277(`)]TJ /F6 1 Tf 3.5602 0 Td [(9)]TJ /F8 1 Tf 0.56024 0 Td [(N)]TJ /F6 1 Tf 1.1867 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(N)]TJ /F6 1 Tf 1.7229 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(n)-283(>)-283(N)-1108(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.06 12.961 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.313 9.2075 Td [(<)-277(B)]TJ /F2 1 Tf 2.1988 0 Td [(.)]TJ -29.181 -1.7169 Td [(En)-343(fait,)-343(p)-30(our)]TJ /F8 1 Tf 6.0422 0 Td [(`)]TJ /F2 1 Tf 0.70482 0 Td [(=)]TJ /F6 1 Tf 1.0663 0 Td [(\2411)]TJ /F2 1 Tf 2.1205 0 Td [(\(resp.)]TJ /F8 1 Tf 2.9157 0 Td [(`)]TJ /F2 1 Tf 0.71084 0 Td [(=)-289(+)]TJ /F6 1 Tf 1.8434 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))-343(le)-343(premier)-343(\(resp.)-469(second\))-313(\266)469(enonc)30(\266)469(e)-343(est)-349(sans)-337(ob)-60(jet)-343(et,)]TJ -16.404 -1.2048 Td [(dans)-331(l'autre)-337(on)-331(p)-30(eut)-337(se)-337(limiter)-331(au)-337(cas)]TJ /F8 1 Tf 16.669 0 Td [(B)]TJ /F2 1 Tf 1.1446 0 Td [(r)30(\266)469(eel)-337(n)30(\266)469(egatif)-337(\(resp.)]TJ /F8 1 Tf 8.0964 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(r)30(\266)469(eel)-337(p)-30(ositif)-78(\).)]TJ -8.0783 -2.3976 Td [(21)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 96 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 95 0 R /ProcSet 2 0 R >> /Contents 94 0 R >> endobj 95 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R /F16 81 0 R >> endobj 98 0 obj << /Length 17245 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.424 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-13710(I)-36(I.B.)-385(G)24(\266)548(en)18(\266)548(eralit)18(\266)554(es)-391(sur)]TJ ET 519.7 781.3 m 531.86 781.3 l 531.86 782.53 l 519.7 782.53 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 43.482 64.424 Td [(I)283(R)]TJ /F2 1 Tf -38.151 -2.2048 Td [(De)-427(m)24(^)475(eme,)-445(lorsque)]TJ /F8 1 Tf 8.4277 0 Td [(`)]TJ /F2 1 Tf 0.84337 0 Td [(est)-427(\257ni,)-451(on)-421(p)-30(eut)-427(se)-427(limiter)-421(\265)500(a)-427(des)]TJ /F8 1 Tf 14.958 0 Td [(A)]TJ /F2 1 Tf 1.1747 0 Td [(et)]TJ /F8 1 Tf 1.259 0 Td [(B)]TJ /F2 1 Tf 1.2349 0 Td [(\257nis,)-445(on)-427(p)-24(eut)-427(alors)-403(\266)469(ecrire)]TJ /F8 1 Tf -27.898 -1.1988 Td [(A)]TJ /F2 1 Tf 1.0602 0 Td [(=)]TJ /F8 1 Tf 1.0843 0 Td [(`)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(")]TJ /F2 1 Tf 0.81928 0 Td [(\(resp.)]TJ /F8 1 Tf 2.9518 0 Td [(B)]TJ /F2 1 Tf 1.1205 0 Td [(=)]TJ /F8 1 Tf 1.0904 0 Td [(`)]TJ /F2 1 Tf 0.6506 0 Td [(+)]TJ /F8 1 Tf 1.012 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [(\))-355(a)24(v)24(ec)]TJ /F8 1 Tf 2.9578 0 Td [(")-313(>)]TJ /F2 1 Tf 1.8614 0 Td [(0.)-506(Lorsque)-355(les)]TJ /F8 1 Tf 6.5482 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.776 87.107 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.09 61.021 Td [(son)24(t)-325(\266)469(egalemen)24(t)-355(\257nis,)-355(cela)-361(p)-30(ermet)]TJ -24.759 -1.1988 Td [(de)-331(r)24(\266)475(eunir)-337(les)-337(deux)-307(\266)475(enonc)24(\266)469(es)-331(pr)24(\266)469(ec)24(\266)475(eden)24(ts)-331(sous)-337(la)-331(forme)-337(classique)-337(:)]TJ 9.241 -1.2048 Td [(\(3\))]TJ /F6 1 Tf 2.6145 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(")-283(>)]TJ /F2 1 Tf 1.8012 0 Td [(0)]TJ /F6 1 Tf 1.5 0 Td [(9)]TJ /F8 1 Tf 0.55422 0 Td [(N)]TJ /F6 1 Tf 1.1928 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(N)]TJ /F6 1 Tf 1.7229 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(n)-283(>)-277(N)]TJ /F6 1 Tf 3.8494 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.948 83.668 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.476 58.617 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(`)]TJ /F6 1 Tf 0.42169 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(<)-277(")]TJ /F2 1 Tf 1.8554 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 45.339 Td [(I)-30(I.B.4.)-246(Liens)-376(en)30(tre)-376(limite,)-376(b)-35(orne)-376(sup)-10(\266)537(erieure)-381(et)-376(b)-35(orne)-376(inf)25(\266)537(erieure.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 52.84 Td [(Consid)30(\266)469(erons)-283(une)-289(suite)-283(d')24(\266)475(el)18(\266)475(emen)24(ts)]TJ /F8 1 Tf 14.633 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.388 75.4 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.361 52.84 Td [(de)]TJ ET 0.504 w 270.65 641.52 m 279.29 641.52 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 22.645 52.84 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-427(Si)]TJ /F8 1 Tf 1.8253 0 Td [(M)]TJ /F2 1 Tf 1.3614 0 Td [(est)-289(un)-283(ma)-60(joran)30(t)-289(des)]TJ /F8 1 Tf 8.8675 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.509 75.4 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.825 52.84 Td [(alors)-283(ces)-259(\266)475(el)18(\266)475(emen)24(ts)]TJ -31.494 -1.1988 Td [(appartiennen)24(t)-289(tous)-289(\265)500(a)-283(l'in)24(terv)54(alle)-283(ferm)24(\266)475(e)-289([)]TJ /F6 1 Tf 17.277 0 Td [(\2411)]TJ /F2 1 Tf 1.7831 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(M)]TJ /F2 1 Tf 1.247 0 Td [(].)-433(Il)-283(s'ensuit)-289(que,)-295(si)-289(cette)-289(suite)-289(a)-283(une)-289(limite)]TJ /F8 1 Tf -20.747 -1.1988 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(,)-421(celle-ci)-403(est)-409(encore)-403(dans)-403([)]TJ /F6 1 Tf 11.717 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(M)]TJ /F2 1 Tf 1.247 0 Td [(])-403(et)-403(donc)]TJ /F8 1 Tf 4.3795 0 Td [(`)]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F8 1 Tf 1.1747 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(.)-656(\(A)30(tten)24(tion,)-421(une)-403(ma)-60(joration)-409(stricte)]TJ /F8 1 Tf -23.042 -1.2048 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4483 70.245 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.741 49.238 Td [(<)-301(M)]TJ /F2 1 Tf 2.5 0 Td [(ne)-343(p)-30(ermet)-349(d'obtenir)-349(qu'une)-343(ma)-60(joration)-349(large)]TJ /F8 1 Tf 19.946 0 Td [(`)]TJ /F11 1 Tf 0.71084 0 Td [(6)]TJ /F8 1 Tf 1.0783 0 Td [(M)]TJ /F2 1 Tf 1.4217 0 Td [(p)-30(our)-349(une)-313(\266)469(ev)24(en)30(tuelle)-349(limite\).)]TJ -27.066 -1.1988 Td [(On)-367(a)-367(un)-367(r)24(\266)469(esultat)-367(analogue)-373(si)]TJ /F8 1 Tf 12.994 0 Td [(m)]TJ /F2 1 Tf 1.247 0 Td [(est)-373(un)-367(minoran)24(t)-373(des)]TJ /F8 1 Tf 9.1807 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.966 68.53 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.873 48.039 Td [(.)-542(En)-367(particulier,)-379(si)-373(la)-367(suite)-367(a)-367(une)]TJ -24.542 -1.1988 Td [(limite,)-331(on)-337(a)-337(l'encadremen)24(t)-337(:)]TJ 13.958 -1.2048 Td [(inf)]TJ /F8 1 Tf 1.3855 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.405 65.099 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 22.072 45.635 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(lim)]TJ /F8 1 Tf 1.5542 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.138 65.099 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 26.078 45.635 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(sup)]TJ /F8 1 Tf 1.6747 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.043 65.099 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.259 45.635 Td [(.)]TJ -24.928 -1.506 Td [(Il)-265(est)-265(\265)500(a)-265(noter)-265(que)-271(dans)-259(l'encadremen)18(t)-265(pr)24(\266)475(ec)24(\266)469(eden)30(t)-271(les)-265(termes)-265(extr)30(^)469(emes)-265(existen)24(t)-265(toujours)-271(\(dans)]TJ ET 63.72 523 m 72.36 523 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 42.924 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))-421(tandis)-427(que)-427(le)-427(terme)-421(cen)18(tral)-427(p)-24(eut)-427(ne)-421(pas)-427(exister.)-710(On)-427(a)-421(toutefois)-427(l'imp)-24(ortan)18(t)-421(r)24(\266)475(esultat)]TJ -0.72289 -1.1988 Td [(suiv)54(an)24(t)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 32.434 Td [(I)-30(I.B.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1325 38.882 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.0663 0 Td [(Soit)]TJ /F2 1 Tf 2.1145 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.095 55.426 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.578 38.882 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.431 55.426 Td [(n)]TJ /F10 1 Tf 0.70689 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 23.952 38.882 Td [(une)-397(suite)-397(croissan)24(te)-397(\()]TJ /F2 1 Tf 9.4578 0 Td [(resp.)]TJ /F12 1 Tf 2.6988 0 Td [(d)24(\266)475(ecroissan)24(te\))-403(d')30(\266)469(el)24(\266)469(e-)]TJ -30.777 -1.1988 Td [(men)24(ts)-331(de)]TJ ET 114.62 460.29 m 123.26 460.29 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 9.5904 37.683 Td [(R)]TJ /F12 1 Tf 0.71687 0 Td [(.)-445(Alors)-337(cette)-337(suite)-337(admet)-331(p)-30(our)-337(limite)-331(sa)-337(b)-30(orne)-337(sup)0(\266)475(erieure)-343(\()]TJ /F2 1 Tf 25.717 0 Td [(resp.)]TJ /F12 1 Tf 2.512 0 Td [(inf)30(\266)469(erieure\).)]TJ /F1 1 Tf -33.205 -1.7229 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Explicitons)-313(le)-319(cas)-313(de)-313(la)-313(suite)-313(croissan)24(te,)-319(l'autre)-313(se)-313(traite)-313(de)-313(fa)-6(\273)439(con)-307(ana-)]TJ -8.8193 -1.1988 Td [(logue.)-451(P)30(osons)]TJ /F8 1 Tf 6.3373 0 Td [(M)]TJ /F2 1 Tf 1.3614 0 Td [(=)-283(sup)]TJ /F8 1 Tf 2.7289 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.379 49.53 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.217 34.762 Td [(et)-337(v)54(\266)469(eri\257ons)-331(les)-313(\266)475(enonc)24(\266)469(es)-337(\(1\))-331(et)-337(\(2\))-337(de)-337(la)-337(section)-337(pr)30(\266)469(ec)24(\266)469(eden)30(te)-337(a)24(v)24(ec)]TJ /F8 1 Tf -11.886 -1.1988 Td [(`)]TJ /F2 1 Tf 0.69277 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F2 1 Tf 1.0843 0 Td [(.)]TJ -2.8313 -1.5121 Td [(L')24(\266)469(enonc)30(\266)469(e)-397(\(2\))-391(est)-397(imm)24(\266)469(ediat)-391(car)-397(on)-391(a,)-409(p)-30(our)-391(tout)]TJ /F8 1 Tf 21.042 0 Td [(n)]TJ /F2 1 Tf 0.59638 0 Td [(,)]TJ /F8 1 Tf 0.68675 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.397 45.65 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 29.145 32.051 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(M)]TJ /F2 1 Tf 1.4518 0 Td [(=)]TJ /F8 1 Tf 1.1506 0 Td [(`)-379(<)-373(B)]TJ /F2 1 Tf 2.753 0 Td [(.)-620(P)30(our)-397(l')30(\266)469(enonc)30(\266)469(e)-397(\(1\),)]TJ -30.325 -1.1988 Td [(prenons)]TJ /F8 1 Tf 3.8614 0 Td [(A)]TJ /F2 1 Tf 1.2108 0 Td [(strictemen)18(t)-463(inf)24(\266)469(erieur)-463(\265)500(a)]TJ /F8 1 Tf 10.47 0 Td [(`)]TJ /F2 1 Tf 0.90964 0 Td [(=)]TJ /F8 1 Tf 1.2711 0 Td [(M)]TJ /F2 1 Tf 1.0843 0 Td [(.)-831(Alors)]TJ /F8 1 Tf 3.8916 0 Td [(A)]TJ /F2 1 Tf 1.2108 0 Td [(n'est)-469(pas)-463(un)-469(ma)-60(joran)30(t)-469(de)-463(la)-463(suite)]TJ -23.91 -1.2048 Td [(\(puisque)]TJ /F8 1 Tf 4.1145 0 Td [(M)]TJ /F2 1 Tf 1.488 0 Td [(est)-421(le)-415(plus)-415(p)-30(etit)-409(d'en)24(tre)-415(eux\).)-686(Ainsi)-415(il)-409(existe)-415(un)-415(indice)]TJ /F8 1 Tf 24.813 0 Td [(N)]TJ /F2 1 Tf 1.3253 0 Td [(tel)-415(que)]TJ /F8 1 Tf 3.4639 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.828 42.211 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.277 29.647 Td [(>)-409(A)]TJ /F2 1 Tf 1.9398 0 Td [(.)]TJ -38.886 -1.1988 Td [(Comme)-349(la)-343(suite)-349(est)-349(croissan)24(te)-349(on)-349(a)-343(alors)]TJ /F6 1 Tf 18.404 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(n)-301(>)-301(N)-1114(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.147 40.495 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 29.596 28.448 Td [(>)]TJ /F8 1 Tf 1.0723 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.716 40.495 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.301 28.448 Td [(>)-301(A)]TJ /F2 1 Tf 1.8253 0 Td [(,)-349(ce)-355(qui)-343(prouv)24(e)-349(l')24(\266)475(enonc)24(\266)475(e)]TJ -28.795 -1.1988 Td [(\(1\))-331(et)-337(ac)24(h)24(\265)469(ev)30(e)-343(la)-331(preuv)24(e.)]TJ ET 512.64 324.46 m 519.84 324.46 l 519.84 332.89 l 512.64 332.89 l f BT /F15 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 25.527 Td [(Cor)24(ollaire.)]TJ /F12 1 Tf 6.5542 0 Td [(Dans)]TJ ET 172.51 315.07 m 181.15 315.07 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 14.434 25.527 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(,)-337(toute)-331(suite)-337(monotone)-337(a)-331(une)-337(limite.)]TJ /F2 1 Tf -9.8253 -1.7169 Td [(Ce)-397(r)30(\266)469(esultat)-397(mon)24(tre)-397(l'in)24(t)24(\266)475(er)18(^)475(et)-397(qu'il)-391(y)-397(a)-391(\265)500(a)-397(se)-391(ramener)-397(\265)500(a)-391(des)-397(suites)-397(monotones.)-620(C'est)-397(ce)-397(qui)]TJ 0 -1.2048 Td [(conduit)-439(aux)-445(notions)-445(de)-433(limite)-445(inf)30(\266)469(erieure)-439(et)-445(de)-439(limite)-439(sup)0(\266)475(erieure)-445(qui,)-469(con)30(trairemen)24(t)-439(\265)500(a)-439(la)]TJ 0 -1.1988 Td [(limite,)-331(existen)18(t)-337(p)-24(our)-337(toute)-337(suite)-331(dans)]TJ ET 261.58 265.75 m 270.22 265.75 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 21.886 21.406 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -17.277 -1.512 Td [(Consid)30(\266)469(erons)-391(une)-391(suite)-391(\()]TJ /F8 1 Tf 10.47 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.431 28.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.916 19.894 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.767 28.262 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(P)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.398 19.894 Td [(d')24(\266)469(el)24(\266)475(emen)24(ts)-391(de)]TJ ET 308.02 247.75 m 316.66 247.75 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 25.771 19.894 Td [(R)]TJ /F2 1 Tf 1.1084 0 Td [(\(o)-36(\265)530(u)]TJ /F8 1 Tf 1.8373 0 Td [(P)]TJ /F2 1 Tf 1.1747 0 Td [(est)-391(une)-391(partie)-397(non)-391(b)-30(orn)24(\266)475(ee)-397(de)]TJ /F3 1 Tf 13.217 0 Td [(N)]TJ /F2 1 Tf 0.71687 0 Td [(\).)]TJ -38.494 -1.1988 Td [(On)-337(p)-24(eut)-337(alors)-343(d)30(\266)469(e\257nir)-337(deux)-331(autre)-337(suites)-337(par)-337(:)]TJ /F8 1 Tf 9.5783 -1.1988 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.026 24.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.217 17.497 Td [(=)-433(inf)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.716 24.107 Td [(p)]TJ /F17 1 Tf 0.59483 0 Td [(>)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.97 17.497 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.966 24.823 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.584 17.497 Td [(w)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.362 24.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.127 17.497 Td [(=)-289(sup)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.052 23.831 Td [(p)]TJ /F17 1 Tf 0.59483 0 Td [(>)]TJ /F7 1 Tf 0.88793 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.88 17.497 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.302 24.823 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 15.31 Td [(Il)-355(est)-367(clair)-361(que)-361(la)-361(suite)-361(des)]TJ /F8 1 Tf 12.036 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.543 21.693 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.753 15.31 Td [(est)-367(croissan)30(te)-361(tandis)-367(que)-361(celle)-361(des)]TJ /F8 1 Tf 15.139 0 Td [(w)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.517 21.693 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.506 15.31 Td [(est)-367(d)24(\266)475(ecroissan)24(te.)-524(On)]TJ -30.175 -1.1988 Td [(p)-30(ose)-331(alors)-337(:)]TJ 4.6867 -1.1988 Td [(lim)-168(inf)]TJ /F8 1 Tf 2.9398 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.362 18.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.355 12.912 Td [(=)-277(lim)]TJ /F8 1 Tf 2.6084 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.974 18.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.271 12.912 Td [(=)-283(sup)]TJ /F8 1 Tf 2.7289 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.75 18.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.717 12.912 Td [(lim)-174(sup)]TJ /F8 1 Tf 3.2289 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.681 18.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.343 12.912 Td [(=)-277(lim)]TJ /F8 1 Tf 2.6145 0 Td [(w)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.621 18.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.494 12.912 Td [(=)-277(inf)]TJ /F8 1 Tf 2.4398 0 Td [(w)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.31 18.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.53 12.912 Td [(;)]TJ -34.199 -1.512 Td [(P)30(ar)-343(d)24(\266)475(e\257nition)-349(lim)-168(inf)]TJ /F8 1 Tf 9.2831 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.733 16.099 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.072 11.4 Td [(est)-349(la)]TJ /F1 1 Tf 2.6928 0 Td [(limite)-397(inf)24(\266)548(erieure)]TJ /F2 1 Tf 8.4277 0 Td [(de)-343(la)-343(suite)-343(des)]TJ /F8 1 Tf 6.6084 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.19 16.099 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.259 11.4 Td [(tandis)-349(que)-349(lim)-168(sup)]TJ /F8 1 Tf 8.1205 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 16.099 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 10.201 Td [(est)-337(sa)]TJ /F1 1 Tf 2.7892 0 Td [(limite)-385(sup)-12(\266)548(erieure)]TJ /F2 1 Tf 8.5301 0 Td [(.)]TJ -11.319 -1.512 Td [(On)-493(notera)-493(que)-493(ces)-487(limites)-500(inf)30(\266)469(erieures)-493(et)-487(sup)0(\266)469(erieures)-493(existen)24(t)-493(toujours.)-921(En)-487(outre)-493(on)-493(a)]TJ -0.030121 -1.1988 Td [(\266)469(evidemmen)24(t)]TJ /F8 1 Tf 5.6205 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.328 10.504 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 12.235 7.4906 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(w)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.043 10.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.861 7.4906 Td [(d'o)-30(\265)524(u)-313(l'on)-319(d)24(\266)475(eduit)-319(lim)-168(inf)]TJ /F8 1 Tf 10.169 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.638 10.504 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 26.422 7.4906 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(lim)-168(sup)]TJ /F8 1 Tf 3.2289 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.767 10.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.825 7.4906 Td [(.)-439(Le)-319(lien)-313(de)-319(cette)-319(notion)-313(a)24(v)24(ec)]TJ -26.494 -1.2048 Td [(la)-331(limite)-337(usuelle)-337(est)-337(donn)18(\266)475(e)-337(par)-337(:)]TJ /F5 1 Tf 0 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(22)]TJ ET endstream endobj 100 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 99 0 R /ProcSet 2 0 R >> /Contents 98 0 R >> endobj 99 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F16 81 0 R /F17 97 0 R >> endobj 101 0 obj << /Length 14659 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.424 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-13710(I)-36(I.B.)-385(G)24(\266)548(en)18(\266)548(eralit)18(\266)554(es)-391(sur)]TJ ET 519.7 781.3 m 531.86 781.3 l 531.86 782.53 l 519.7 782.53 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 43.482 64.424 Td [(I)283(R)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I)-30(I.B.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1325 62.22 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8494 0 Td [(P)24(our)-289(qu'une)-289(suite)-295(dans)]TJ ET 0.504 w 334.44 753.55 m 343.08 753.55 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 27.982 62.22 Td [(R)]TJ /F12 1 Tf 1.006 0 Td [(admette)-295(une)-289(limite)-289(il)-289(faut)-295(et)-289(il)-289(su\261t)]TJ -23.657 -1.1988 Td [(que)-355(sa)-355(limite)-355(inf)30(\266)469(erieure)-355(et)-355(sa)-355(limite)-355(sup)0(\266)469(erieure)-355(soien)24(t)-331(\266)469(egales,)-355(la)-355(limite)-355(de)-349(la)-355(suite)-355(est)-355(alors)]TJ 0 -1.1988 Td [(leur)-331(v)48(aleur)-331(comm)18(une.)]TJ /F1 1 Tf 0 -1.9157 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Reprenons)-331(l')24(\266)469(enonc)30(\266)469(e)-337(\(1\))-337(de)-331(la)-337(d)30(\266)469(e\257nition)-337(des)-337(limites)-337(:)]TJ /F6 1 Tf 2.4157 -1.2048 Td [(8)]TJ /F8 1 Tf 0.56024 0 Td [(A)-277(<)-283(`)]TJ /F6 1 Tf 3.5 0 Td [(9)]TJ /F8 1 Tf 0.55422 0 Td [(N)]TJ /F6 1 Tf 1.1928 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(N)]TJ /F6 1 Tf 1.7229 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(n)-283(>)-277(N)-1114(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.957 80.935 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.837 56.701 Td [(>)-283(A)]TJ /F2 1 Tf 2.1446 0 Td [(.)]TJ -27.651 -1.6084 Td [(Av)30(ec)-403(les)-397(notations)-403(pr)30(\266)469(ec)24(\266)469(eden)30(tes)-403(il)-397(implique)-403(\()]TJ /F6 1 Tf 19.44 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(A)-385(<)-385(`)]TJ /F2 1 Tf 2.7108 0 Td [(\))-403(\()]TJ /F6 1 Tf 1.1807 0 Td [(9)]TJ /F8 1 Tf 0.55422 0 Td [(N)]TJ /F6 1 Tf 1.2952 0 Td [(2)]TJ /F3 1 Tf 1.0542 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(\))-397(\()]TJ /F8 1 Tf 1.1747 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.371 78.625 Td [(N)]TJ /F4 1 Tf 1.0172 0 Td [(+1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 36.657 55.093 Td [(>)]TJ /F8 1 Tf 1.1626 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-397(et)-403(par)-397(suite)]TJ -32.91 -1.1988 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(A)-283(<)-277(`)]TJ /F2 1 Tf 2.5 0 Td [(\))-337(\()-6(lim)-168(inf)]TJ /F8 1 Tf 4.0542 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.655 76.909 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 14.554 53.894 Td [(>)]TJ /F8 1 Tf 1.0602 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\);)-439(il)-433(n'y)-427(a)-433(alors)-439(pas)-433(de)]TJ /F8 1 Tf 10.627 0 Td [(A)]TJ /F2 1 Tf 1.1807 0 Td [(v)48(\266)475(eri\257an)24(t)-439(lim)-168(inf)]TJ /F8 1 Tf 6.9639 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.095 76.909 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.693 53.894 Td [(<)-439(A)-445(<)-445(`)]TJ /F2 1 Tf 4.4819 0 Td [(et)-433(donc)]TJ -35.843 -1.1988 Td [(lim)-168(inf)]TJ /F8 1 Tf 2.9398 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.655 75.193 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6627 52.695 Td [(>)]TJ /F8 1 Tf 1.0602 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(.)]TJ -5.8072 -1.6145 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-289(si)]TJ /F8 1 Tf 8.6024 0 Td [(`)]TJ /F2 1 Tf 0.68675 0 Td [(est)-271(pris)-277(inf)30(\266)469(erieur)-271(ou)-240(\266)475(egal)-271(\265)500(a)-271(lim)-168(inf)]TJ /F8 1 Tf 14.325 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.241 72.892 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.06 51.081 Td [(,)-283(p)-30(our)]TJ /F8 1 Tf 2.8614 0 Td [(A)-283(<)-277(`)]TJ /F2 1 Tf 2.5 0 Td [(,)-283(on)-271(a)]TJ /F8 1 Tf 2.6506 0 Td [(A)-283(<)]TJ /F2 1 Tf 2.0843 0 Td [(lim)-174(inf)]TJ /F8 1 Tf 2.9458 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.5 72.892 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 51.081 Td [(,)]TJ -38.886 -1.1988 Td [(d'o)-30(\265)530(u,)-379(p)-24(our)-373(un)-367(indice)]TJ /F8 1 Tf 9.3373 0 Td [(N)]TJ /F2 1 Tf 1.2771 0 Td [(au)-367(moins,)]TJ /F8 1 Tf 4.6386 0 Td [(A)-331(<)-331(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.284 71.168 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.38 49.882 Td [(et)-373(donc)-367(\()]TJ /F6 1 Tf 4.012 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(n)-337(>)-331(N)]TJ /F2 1 Tf 2.9578 0 Td [(\))-367(\()]TJ /F8 1 Tf 1.1446 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.112 71.168 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 34.494 49.882 Td [(>)]TJ /F8 1 Tf 1.1084 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.647 71.168 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.181 49.882 Td [(>)-331(A)]TJ /F2 1 Tf 1.8614 0 Td [(\))-367(et)-367(l')24(\266)475(enonc)24(\266)475(e)]TJ -33.711 -1.1988 Td [(\(1\))-331(est)-343(v)54(\266)469(eri\257)30(\266)469(e.)]TJ 0 -1.6084 Td [(Ainsi,)-421(nous)-403(a)30(v)24(ons)-409(prouv)54(\266)475(e)-409(l')30(\266)469(equiv)54(alence)-409(de)-403(\(1\))-397(et)-403(de)-403(lim)-174(inf)]TJ /F8 1 Tf 26.096 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.793 67.15 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 32.94 47.075 Td [(>)]TJ /F8 1 Tf 1.1687 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(.)-650(On)-403(prouv)24(e)-403(de)-403(m)24(^)475(eme)]TJ -29.193 -1.2048 Td [(l')30(\266)469(equiv)54(alence)-343(de)-331(\(2\))-337(et)-331(de)-337(lim)-168(sup)]TJ /F8 1 Tf 14.512 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.216 65.426 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 21.235 45.87 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(.)]TJ -17.38 -1.6084 Td [(Alors,)-415(si)-397(la)-397(limite)-397(inf)30(\266)469(erieure)-403(des)]TJ /F8 1 Tf 14.422 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.086 63.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.265 44.262 Td [(est)-373(\266)469(egale)-403(\265)499(a)-397(leur)-397(limite)-397(sup)0(\266)475(erieure,)-415(en)-397(prenan)24(t)-397(p)-30(our)]TJ /F8 1 Tf 22.813 0 Td [(`)]TJ /F2 1 Tf -38.747 -1.1988 Td [(leur)-439(v)54(aleur)-439(comm)24(une)-439(les)-409(\266)469(enonc)24(\266)475(es)-439(\(1\))-439(et)-439(\(2\))-439(son)24(t)-439(v)48(\266)475(eri\257)24(\266)469(es)-433(et)-439(donc)]TJ /F8 1 Tf 29.259 0 Td [(`)]TJ /F2 1 Tf 0.85542 0 Td [(est)-445(la)-439(limite)-433(des)]TJ /F8 1 Tf 7.6566 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.5 61.409 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 43.063 Td [(.)]TJ -38.886 -1.1988 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-385(si)-379(la)-373(suite)-373(des)]TJ /F8 1 Tf 14.157 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.707 59.693 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.976 41.864 Td [(a)-373(une)-373(limite)]TJ /F8 1 Tf 5.6747 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(,)-385(les)-343(\266)469(enonc)30(\266)469(es)-373(\(1\))-373(et)-373(\(2\))-343(\266)469(etan)24(t)-379(v)54(\266)469(eri\257)30(\266)469(es)-373(on)]TJ -21.735 -1.2048 Td [(a)-439(lim)-174(inf)]TJ /F8 1 Tf 3.8795 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14 57.969 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 10.789 40.659 Td [(>)]TJ /F8 1 Tf 1.241 0 Td [(`)]TJ /F11 1 Tf 0.87349 0 Td [(>)]TJ /F2 1 Tf 1.2349 0 Td [(lim)-174(sup)]TJ /F8 1 Tf 3.2349 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.681 57.969 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.928 40.659 Td [(et,)-475(compte)-445(ten)24(u)-445(de)-445(lim)-168(inf)]TJ /F8 1 Tf 11.94 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.991 57.969 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 32.44 40.659 Td [(6)]TJ /F2 1 Tf 1.241 0 Td [(lim)-168(sup)]TJ /F8 1 Tf 3.2289 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.638 57.969 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.024 40.659 Td [(,)-475(on)-439(en)-445(d)24(\266)475(eduit)]TJ -32.693 -1.1988 Td [(lim)-168(inf)]TJ /F8 1 Tf 2.9398 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.655 56.254 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6627 39.461 Td [(=)-283(lim)-168(sup)]TJ /F8 1 Tf 4.2892 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.784 56.254 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.343 39.461 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(.)]TJ ET 512.64 470.41 m 519.84 470.41 l 519.84 478.83 l 512.64 478.83 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 29.193 Td [(I)-30(I.B.5.)-246(Compl)25(\266)537(emen)30(ts)-376(top)-40(ologiques.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 33.388 Td [(Rapp)-30(elons)-481(qu'un)-487(espace)-475(top)-36(ologique)-481(est)-481(dit)]TJ /F1 1 Tf 20.193 0 Td [(connexe)]TJ /F2 1 Tf 4.506 0 Td [(lorsqu'il)-475(ne)-481(con)30(tien)18(t)-481(pas)-481(d'autre)]TJ -24.699 -1.1988 Td [(partie)-445(ouv)18(erte)-445(et)-445(ferm)24(\266)475(ee)-451(que)-445(la)-445(partie)-451(vide)-451(et)-445(lui-m)24(^)475(eme.)-783(P)30(our)-445(tout)-445(espace)-451(top)-30(ologique)]TJ /F8 1 Tf 0 -1.1988 Td [(X)]TJ /F2 1 Tf 1.3012 0 Td [(et)-397(tout)-397(p)-30(oin)24(t)]TJ /F8 1 Tf 6.1325 0 Td [(a)]TJ /F2 1 Tf 0.92771 0 Td [(de)]TJ /F8 1 Tf 1.3916 0 Td [(X)]TJ /F2 1 Tf 1.3072 0 Td [(on)-391(d)24(\266)469(e\257nit)-391(alors)-403(la)]TJ /F1 1 Tf 8.259 0 Td [(comp)-36(osan)30(te)-457(connexe)]TJ /F2 1 Tf 10.765 0 Td [(de)]TJ /F8 1 Tf 1.3916 0 Td [(a)]TJ /F2 1 Tf 0.92771 0 Td [(dans)]TJ /F8 1 Tf 2.3976 0 Td [(X)]TJ /F2 1 Tf 1.3072 0 Td [(comme)]TJ -36.108 -1.2048 Td [(la)-397(r)24(\266)475(eunion)-403(des)-403(parties)-403(connexes)-397(de)]TJ /F8 1 Tf 15.669 0 Td [(X)]TJ /F2 1 Tf 1.3072 0 Td [(qui)-397(p)-30(oss)24(\265)475(eden)24(t)]TJ /F8 1 Tf 6.3976 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-638(On)-403(mon)24(tre)-403(que)-403(ces)-397(comp)-30(osan)24(tes)]TJ -23.904 -1.1988 Td [(connexes)-331(son)24(t)-337(des)-337(parties)-337(connexes,)-337(deux)-331(\265)500(a)-337(deux)-331(disjoin)30(tes)-343(qui)-331(recouvren)24(t)]TJ /F8 1 Tf 32.56 0 Td [(X)]TJ /F2 1 Tf -32.56 -1.6084 Td [(Nous)-361(a)30(v)24(ons)-361(par)-361(ailleurs)-361(observ)54(\266)475(e)-367(que,)-367(dans)-361(tout)-361(espace)-355(totalemen)24(t)-367(ordonn)24(\266)475(e,)-373(les)-361(in)24(terv)54(alles)]TJ 0 -1.1988 Td [(son)24(t)-337(des)-337(parties)-337(con)30(v)24(exes)-337(\(i.e.)-451(on)24(t)-337(la)-331(propri)24(\266)469(et)24(\266)475(e)-337(de)-337(con)30(v)24(exit)24(\266)469(e\).)]TJ 0 -1.6145 Td [(La)-277(notion)-283(de)-271(connexit)24(\266)469(e)-283(et)-277(celle)-277(de)-277(con)24(v)24(exit)30(\266)469(e)-283(ne)-277(doiv)30(en)24(t)-277(pas)-253(^)469(etre)-277(confondues,)-289(toutefois,)-289(dans)]TJ 0 -1.1988 Td [(le)-337(cas)-337(des)-337(parties)-331(de)]TJ ET 172.22 284.4 m 180.86 284.4 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 14.41 22.967 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-331(l'existence)-337(des)-337(b)-30(ornes)-337(sup)0(\266)469(erieures)-337(p)-30(ermet)-337(de)-331(mon)24(trer)-337(:)]TJ /F15 1 Tf -9.8012 -1.9157 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(P)24(our)-331(une)-337(partie)]TJ /F8 1 Tf 7.2169 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(de)]TJ ET 261 261.57 m 269.64 261.57 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 21.837 21.051 Td [(R)]TJ /F12 1 Tf 1.0542 0 Td [(les)-307(\266)469(enonc)30(\266)469(es)-337(suiv)54(an)24(ts)-331(son)24(t)-313(\266)475(equiv)48(alen)30(ts)-337(:)]TJ -17.56 -1.6084 Td [({)]TJ /F8 1 Tf 0.83133 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(est)-337(connexe)-337(;)]TJ -1.9157 -1.6084 Td [({)]TJ /F8 1 Tf 0.83133 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(est)-337(con)24(v)24(exe)-337(;)]TJ -1.9157 -1.6084 Td [({)]TJ /F8 1 Tf 0.83133 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(est)-337(un)-337(in)24(terv)54(alle.)]TJ /F2 1 Tf -1.9157 -1.9157 Td [(Ainsi)-343(toute)-337(partie)]TJ /F8 1 Tf 8.1084 0 Td [(A)]TJ /F2 1 Tf 1.0904 0 Td [(de)]TJ ET 189.65 180.93 m 198.29 180.93 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 15.867 14.31 Td [(R)]TJ /F2 1 Tf 1.0602 0 Td [(s')24(\266)469(ecrit)-343(naturellemen)30(t)-343(comme)-343(r)30(\266)469(eunion)-343(d'une)-343(famille)-337(d'in)18(terv)54(alles)]TJ -11.596 -1.1988 Td [(disjoin)30(ts)-373(qui)-379(son)24(t)-379(ses)-373(comp)-36(osan)24(tes)-379(connexes.)-566(La)-379(comp)-30(osan)18(te)-373(d'un)-379(p)-30(oin)24(t)]TJ /F8 1 Tf 32.096 0 Td [(a)]TJ /F2 1 Tf 0.90361 0 Td [(de)]TJ /F8 1 Tf 1.3735 0 Td [(A)]TJ /F2 1 Tf 1.1205 0 Td [(est)-379(alors)]TJ -35.494 -1.2048 Td [(la)-331(r)24(\266)475(eunion)-337(des)-337(in)24(terv)54(alles)-337(con)24(ten)24(us)-337(dans)]TJ /F8 1 Tf 17.645 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(qui)-337(p)-24(oss)24(\265)469(eden)30(t)]TJ /F8 1 Tf 6.2651 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)]TJ -25.524 -1.6084 Td [(Lorsque)]TJ /F8 1 Tf 3.7048 0 Td [(A)]TJ /F2 1 Tf 1.0181 0 Td [(est)-271(ouv)24(ert,)-283(p)-30(our)]TJ /F8 1 Tf 7.1265 0 Td [(a)]TJ /F2 1 Tf 0.79518 0 Td [(dans)]TJ /F8 1 Tf 2.2711 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)-277(il)-271(y)-265(a)-271(un)-265(in)24(terv)54(alle)-265(ouv)24(ert)-271(p)-30(oss)30(\266)469(edan)24(t)]TJ /F8 1 Tf 16.187 0 Td [(a)]TJ /F2 1 Tf 0.79518 0 Td [(qui)-265(est)-271(con)24(ten)24(u)]TJ -32.651 -1.1988 Td [(dans)]TJ /F8 1 Tf 2.3373 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(et)-337(donc)-337(dans)-331(la)-337(comp)-30(osan)24(te)-337(connexe)-337(de)]TJ /F8 1 Tf 17.602 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-445(Celle-ci)-331(est)-343(par)-331(cons)24(\266)469(equen)24(t)-337(ouv)18(erte.)]TJ -21.554 -1.6084 Td [(Mon)24(trons)-259(alors)-259(que)-265(les)-259(comp)-30(osan)24(tes)-265(connexes)-259(d'un)-259(ouv)24(ert)]TJ /F8 1 Tf 25.054 0 Td [(U)]TJ /F2 1 Tf 1.0482 0 Td [(son)24(t)-259(en)-259(quan)24(tit)30(\266)469(e)-259(d)24(\266)469(enom)30(brable.)]TJ -26.102 -1.2048 Td [(Nous)-451(v)24(enons)-457(de)-445(v)24(oir)-451(que)-457(ces)-451(comp)-30(osan)24(tes)-457(son)24(t)-451(des)-451(in)24(terv)54(alles)-451(ouv)18(erts)-457(non)-451(vides.)-795(Cela)]TJ 18.916 -2.3976 Td [(23)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 103 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 102 0 R /ProcSet 2 0 R >> /Contents 101 0 R >> endobj 102 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 104 0 obj << /Length 14147 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.424 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-13710(I)-36(I.B.)-385(G)24(\266)548(en)18(\266)548(eralit)18(\266)554(es)-391(sur)]TJ ET 519.7 781.3 m 531.86 781.3 l 531.86 782.53 l 519.7 782.53 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 43.482 64.424 Td [(I)283(R)]TJ /F2 1 Tf -38.151 -2.2048 Td [(p)-30(ermet)-367(donc)-361(d'asso)-36(cier)-367(\265)500(a)-361(c)24(hacune)-367(d'elles)-367(un)-343(\266)475(el)18(\266)475(emen)24(t)-367(rationnel)-367(de)-361(cette)-367(comp)-30(osan)18(te)-361(et)-367(de)]TJ 0 -1.1988 Td [(d)30(\266)469(e\257nir)-289(ainsi)-289(une)-289(application)-289(de)-289(l'ensem)24(ble)-289(des)-289(comp)-30(osan)24(tes)-295(connexes)-289(de)]TJ /F8 1 Tf 31.313 0 Td [(U)]TJ /F2 1 Tf 1.0783 0 Td [(dans)-289(l'ensem)30(ble)]TJ /F3 1 Tf -32.392 -1.1988 Td [(Q)]TJ /F2 1 Tf 1.0843 0 Td [(des)-313(rationnels.)-439(Cep)-30(endan)24(t)-307(les)-313(comp)-30(osan)24(tes)-313(connexes)-283(\266)475(etan)24(t)-313(deux)-307(\265)500(a)-307(deux)-307(disjoin)24(tes,)-313(cette)]TJ -1.0843 -1.2048 Td [(application)-259(est)-259(injectiv)30(e)-259(ce)-259(qui)-259(prouv)30(e)-259(que,)-277(comme)]TJ /F3 1 Tf 21.819 0 Td [(Q)]TJ /F2 1 Tf 0.77711 0 Td [(,)-271(l'ensem)30(ble)-259(des)-253(comp)-36(osan)24(tes)-259(connexes)]TJ -22.596 -1.1988 Td [(d'un)-337(ouv)24(ert)-337(de)]TJ ET 0.504 w 144.14 696.16 m 152.78 696.16 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 12.06 57.418 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(est)-337(d)24(\266)475(enom)24(brable.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 44.188 Td [(I)-30(I.B.6.)-246(Prolongemen)30(t)-376(des)-376(op)-10(\266)537(erations)-376(alg)20(\266)537(ebriques.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 51.37 Td [(On)-325(sait)-331(d)24(\266)475(e\257nir)-331(sur)]TJ /F3 1 Tf 8.3253 0 Td [(R)]TJ /F2 1 Tf 1.0482 0 Td [(une)-331(addition)-331(et)-331(une)-325(m)24(ultiplication.)-445(Ce)-331(son)18(t)-325(des)-331(lois)-331(de)-331(comp)-30(ositions)]TJ -9.3735 -1.1988 Td [(asso)-24(ciativ)24(es,)-301(comm)24(utativ)24(es)-289(et)-295(con)30(tin)24(ues.)-439(On)-289(p)-30(eut)-295(alors)-289(prolonger)-295(ces)-289(lois)-295(par)-289(con)24(tin)24(uit)18(\266)475(e)-295(en)]TJ 0 -1.2048 Td [(des)-337(lois)-337(non)-337(partout)-337(d)24(\266)475(e\257nies.)-451(Plus)-337(pr)24(\266)469(ecis)30(\266)469(emen)24(t)-337(on)-331(con)24(vien)24(t)-337(:)]TJ /F6 1 Tf 8.7289 -1.1988 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F2 1 Tf 0.66867 0 Td [(])]TJ /F6 1 Tf 0.5 0 Td [(\241)-222(1)]TJ /F8 1 Tf 2 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-999(\(+)]TJ /F6 1 Tf 2.4458 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6145 0 Td [(a)]TJ /F2 1 Tf 0.80121 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(a)]TJ /F2 1 Tf 0.74699 0 Td [(+)-228(\(+)]TJ /F6 1 Tf 2.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))-283(=)-277(+)]TJ /F6 1 Tf 2.506 0 Td [(1)]TJ /F2 1 Tf 1.3313 0 Td [(;)]TJ /F6 1 Tf -21.343 -1.1988 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F2 1 Tf 0.94578 0 Td [([)]TJ /F6 1 Tf 0.27711 0 Td [(\2411)]TJ /F8 1 Tf 1.7771 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)-1000(\()]TJ /F6 1 Tf 1.6687 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F2 1 Tf 0.75301 0 Td [(+)-222(\()]TJ /F6 1 Tf 1.3855 0 Td [(\2411)]TJ /F2 1 Tf 1.7831 0 Td [(\))-277(=)]TJ /F6 1 Tf 1.7229 0 Td [(\2411)]TJ /F2 1 Tf 2.1084 0 Td [(;)]TJ /F6 1 Tf -18.789 -1.2048 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(a)-283(>)]TJ /F2 1 Tf 1.8675 0 Td [(0)-1000(\(+)]TJ /F6 1 Tf 2.6687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.75301 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))-283(=)-277(+)]TJ /F6 1 Tf 2.5 0 Td [(1)]TJ /F2 1 Tf 1.3373 0 Td [(;)]TJ /F6 1 Tf -16.319 -1.1988 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(a)-283(>)]TJ /F2 1 Tf 1.8675 0 Td [(0)-1000(\()]TJ /F6 1 Tf 1.8855 0 Td [(\2411)]TJ /F2 1 Tf 1.7831 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.75301 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(\))-283(=)]TJ /F6 1 Tf 1.7229 0 Td [(\2411)]TJ /F2 1 Tf 2.1145 0 Td [(;)]TJ /F6 1 Tf -16.319 -1.1988 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(a)-283(<)]TJ /F2 1 Tf 1.8675 0 Td [(0)-1000(\(+)]TJ /F6 1 Tf 2.6687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.75301 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))-283(=)]TJ /F6 1 Tf 1.7229 0 Td [(\2411)]TJ /F2 1 Tf 2.1145 0 Td [(;)]TJ /F6 1 Tf -16.319 -1.1988 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(a)-283(<)]TJ /F2 1 Tf 1.8675 0 Td [(0)-1000(\()]TJ /F6 1 Tf 1.8855 0 Td [(\2411)]TJ /F2 1 Tf 1.7831 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.75301 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(\))-283(=)-277(+)]TJ /F6 1 Tf 2.5 0 Td [(1)]TJ /F2 1 Tf 1.3373 0 Td [(;)]TJ -27.602 -1.6024 Td [(L'asso)-24(ciativit)24(\266)475(e)-343(et)-331(la)-337(comm)24(utativit)24(\266)469(e)-337(des)-337(lois)-337(est)-313(\266)469(evidemmen)24(t)-331(conserv)54(\266)469(ee)-337(par)-337(con)30(tin)18(uit)24(\266)469(e.)]TJ 0 -1.9036 Td [(On)-337(ne)-331(p)-30(eut)-337(pas)-337(d)24(\266)475(e\257nir)-337(en)-337(conserv)54(an)30(t)-337(la)-337(con)30(tin)18(uit)24(\266)469(e)-337(les)-337(expressions)-337(:)]TJ 2.7651 -1.1988 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(\))-222(+)-222(\(+)]TJ /F6 1 Tf 2.7771 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(;)]TJ /F2 1 Tf 3.2108 0 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))-222(+)-222(\()]TJ /F6 1 Tf 2 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(;)]TJ /F2 1 Tf 3.2108 0 Td [(0)]TJ /F6 1 Tf 0.72289 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(\2471)]TJ /F2 1 Tf 1.7831 0 Td [(\))-2765(et)-2771(\()]TJ /F6 1 Tf 7.1446 0 Td [(\2471)-222(\242)]TJ /F2 1 Tf 2.5 0 Td [(0\))-337(.)]TJ -34.898 -1.6024 Td [(Ces)-415(cas)-421(son)24(t)-415(conn)24(us)-421(sous)-421(le)-421(nom)-415(de)]TJ /F1 1 Tf 16.482 0 Td [(formes)-481(ind)24(\266)548(etermin)24(\266)548(ees)]TJ /F2 1 Tf 10.819 0 Td [(.)-692(T)78(outefois,)-439(conform)30(\266)469(emen)24(t)]TJ -27.301 -1.1988 Td [(\265)499(a)-445(un)-451(usage)-451(fr)24(\266)475(equen)18(t)-445(en)-451(in)30(t)24(\266)469(egration,)-475(nous)-445(con)24(viendrons)-445(de)-445(:)-674(0)]TJ /F6 1 Tf 28.108 0 Td [(\242)]TJ /F8 1 Tf 0.57832 0 Td [(a)]TJ /F2 1 Tf 0.99397 0 Td [(=)]TJ /F8 1 Tf 1.247 0 Td [(a)]TJ /F6 1 Tf 0.8253 0 Td [(\242)]TJ /F2 1 Tf 0.57832 0 Td [(0)-463(=)-469(0)-445(m)24(^)475(eme)-451(si)]TJ /F8 1 Tf -32.331 -1.1988 Td [(a)]TJ /F2 1 Tf 0.90361 0 Td [(est)-385(in\257ni.)-572(Av)24(ec)-379(cette)-379(con)24(v)24(en)30(tion,)-391(la)-379(m)24(ultiplication)-379(est)-379(partout)-385(d)30(\266)469(e\257nie,)-391(comm)24(utativ)30(e)-385(et)]TJ -0.90361 -1.2048 Td [(asso)-24(ciativ)24(e)-361(mais)-355(n'est)-355(plus)-355(con)30(tin)18(ue)-355(\(i.e.)]TJ /F8 1 Tf 18.12 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.31 45.375 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.837 31.858 Td [(!)]TJ /F2 1 Tf 1.3133 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.3554 0 Td [(et)]TJ /F8 1 Tf 1.1867 0 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.784 45.375 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.753 31.858 Td [(!)]TJ /F2 1 Tf 1.3133 0 Td [(0)-355(ne)-355(p)-30(ermet)-355(pas)-361(de)-355(conclure)]TJ /F8 1 Tf -26.735 -1.1988 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3879 43.659 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4036 30.659 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.7759 43.659 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6566 30.659 Td [(!)]TJ /F2 1 Tf 1.2771 0 Td [(0\).)]TJ /F1 1 Tf -3.6024 -1.6024 Td [(A)30(tten)30(tion.)]TJ /F2 1 Tf 5.6145 0 Td [(P)30(our)-337(utiliser)-337(la)-331(relation)]TJ /F8 1 Tf 10.56 0 Td [(a)]TJ /F2 1 Tf 0.74699 0 Td [(+)-222(\()]TJ /F6 1 Tf 1.3916 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-283(=)-277(0)-331(on)-337(devra)-337(supp)-30(oser)-337(que)]TJ /F8 1 Tf 12.717 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(est)-337(\257ni.)]TJ -33.199 -2.1988 Td [(On)-385(prolongera)-385(de)-385(m)24(^)469(eme)-379(les)-385(notions)-391(de)-385(v)54(aleur)-385(absolue,)-397(partie)-391(p)-30(ositiv)30(e)-391(et)-385(partie)-385(n)24(\266)469(egativ)30(e.)]TJ 0 -1.1988 Td [(Plus)-337(pr)24(\266)475(ecis)24(\266)469(emen)30(t)-337(:)]TJ 0 -1.6024 Td [({)-475(la)]TJ /F1 1 Tf 1.994 0 Td [(v)60(aleur)-277(absolue)]TJ /F2 1 Tf 7.2892 0 Td [(de)]TJ /F8 1 Tf 1.241 0 Td [(a)]TJ /F2 1 Tf 0.76506 0 Td [(dans)]TJ ET 225.5 297.43 m 234.14 297.43 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 18.867 24.057 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-259(not)24(\266)469(ee)]TJ /F6 1 Tf 3.1084 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(,)-259(est)-240(sup)-6(\()]TJ /F8 1 Tf 3.8976 0 Td [(a;)]TJ /F6 1 Tf 0.9759 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\).)-415(Cest)-240(la)-240(v)48(aleur)-240(absolue)-234(ordinaire)]TJ -24.633 -1.1988 Td [(p)-30(our)]TJ /F8 1 Tf 2.3614 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(\257ni)-343(et)-331(c'est)-337(+)]TJ /F6 1 Tf 5.9578 0 Td [(1)]TJ /F2 1 Tf 1.3313 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(a)]TJ /F2 1 Tf 0.86144 0 Td [(in\257ni.)]TJ -13.741 -1.6024 Td [({)-668(la)]TJ /F1 1 Tf 2.2771 0 Td [(partie)-385(p)-36(ositiv)36(e)]TJ /F2 1 Tf 7.5723 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(dans)]TJ ET 235.66 263.95 m 244.3 263.95 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 19.717 21.256 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-337(not)24(\266)469(ee)]TJ /F8 1 Tf 3.2831 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 34.698 30.202 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.916 21.256 Td [(,)-331(est)-337(sup)0(\()]TJ /F8 1 Tf 4.0663 0 Td [(a;)]TJ /F2 1 Tf 0.9759 0 Td [(0\).)-445(C'est)]TJ /F8 1 Tf 4.1747 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(a)]TJ /F11 1 Tf 0.80723 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(0,)-331(et)-337(0)-337(sinon.)]TJ -32.53 -1.6024 Td [({)-566(la)]TJ /F1 1 Tf 2.1265 0 Td [(partie)-331(n)24(\266)548(egativ)30(e)]TJ /F2 1 Tf 7.747 0 Td [(de)]TJ /F8 1 Tf 1.2892 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(dans)]TJ ET 234.22 244.8 m 242.86 244.8 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 19.596 19.653 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-289(not)18(\266)475(ee)]TJ /F8 1 Tf 3.1867 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.397 27.909 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.711 19.653 Td [(,)-295(est)-289(sup)0(\()]TJ /F6 1 Tf 3.9819 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(a;)]TJ /F2 1 Tf 0.96988 0 Td [(0\).)-433(C'est)]TJ /F6 1 Tf 4.1084 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(a)]TJ /F2 1 Tf 0.81325 0 Td [(si)]TJ /F8 1 Tf 0.95783 0 Td [(a)]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(0,)-295(et)-289(0)-283(sinon.)]TJ /F1 1 Tf -33.627 -1.8976 Td [(A)30(tten)30(tion)]TJ /F2 1 Tf 5.1807 0 Td [(la)-337(partie)-337(n)30(\266)469(egativ)24(e)-337(de)]TJ /F8 1 Tf 9.2831 0 Td [(a)]TJ /F2 1 Tf 0.86144 0 Td [(est)-343(un)-307(\266)469(el)24(\266)469(emen)30(t)-337(p)-30(ositif.)]TJ -15.325 -2.2048 Td [(Av)30(ec)-253(ces)-246(d)24(\266)475(e\257nitions)-246(on)-253(a)-246(:)]TJ /F8 1 Tf 11.398 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.362 22.038 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.837 15.551 Td [(\241)]TJ /F8 1 Tf 0.8253 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.328 22.038 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.114 15.551 Td [(et)]TJ /F6 1 Tf 1.0783 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F2 1 Tf 0.55422 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.414 22.038 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.855 15.551 Td [(+)]TJ /F8 1 Tf 0.8253 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.371 22.038 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.886 15.551 Td [(.)-415(\(Il)-253(n'y)-246(a)-246(pas)-253(de)-246(forme)-246(ind)24(\266)475(etermin)24(\266)469(ee)]TJ -23.554 -1.1988 Td [(car)-337(l'un)-337(au)-337(moins)-337(de)]TJ /F8 1 Tf 9.2892 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 21.681 20.323 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.145 14.352 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.534 20.323 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.849 14.352 Td [(est)-343(n)30(ul.\))]TJ /F5 1 Tf -13.518 -10.464 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(24)]TJ ET endstream endobj 106 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 105 0 R /ProcSet 2 0 R >> /Contents 104 0 R >> endobj 105 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R >> endobj 108 0 obj << /Length 13642 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12831(I)-36(I.C.)-385(F)96(amilles)-385(sommables)]TJ /F0 1 Tf 20.592 0 0 20.592 0 0 Tm 7.8776 32.113 Td [(I)-31(I.C.)-1129(F)90(amilles)-381(sommables)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 50.557 Td [(La)-367(th)18(\266)475(eorie)-373(des)-373(familles)-367(sommables)-367(p)-30(ermet)-373(de)-367(donner,)-379(dans)-367(certains)-367(cas,)-379(un)-367(sens)-367(au)-367(sym-)]TJ 0 -1.3434 Td [(b)-30(ole)]TJ /F9 1 Tf 2.1687 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.879 68.719 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1084 49.214 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.853 70.211 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.373 49.214 Td [(o)-30(\265)524(u)-361(\()]TJ /F8 1 Tf 1.8072 0 Td 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Tf 8.352 0 0 8.352 0 0 Tm 49.957 10.142 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 35.241 8.0448 Td [(\242)]TJ /F2 1 Tf 0.79518 -0.80723 Td [(;)]TJ -11.789 -3.3494 Td [(25)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 110 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 109 0 R /ProcSet 2 0 R >> /Contents 108 0 R >> endobj 109 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F13 39 0 R /F18 107 0 R >> endobj 111 0 obj << /Length 18661 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12831(I)-36(I.C.)-385(F)96(amilles)-385(sommables)]TJ /F2 1 Tf 0 -2.5542 Td [(si)]TJ /F8 1 Tf 0.96385 0 Td [(I)]TJ /F2 1 Tf 0.81325 0 Td [(est)-301(la)-295(r)30(\266)469(eunion)-301(disjoin)30(te)-295(d'une)-301(famille)-295(\257nie)-301(de)-289(parties)-301(\()]TJ /F8 1 Tf 23.096 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.853 88.607 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.139 62.069 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.121 88.607 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.283 62.069 Td [(,)]TJ /F9 1 Tf 0.87349 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.026 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.765 62.069 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52 88.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.952 62.069 Td [(=)]TJ /F9 1 Tf 1.1807 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.388 87.099 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 39.873 63.183 Td [(\263)]TJ 0.8494 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.155 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 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Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.578 59.045 Td [(+)]TJ /F9 1 Tf 1 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.448 82.788 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.193 59.045 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.302 84.28 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.349 59.045 Td [(;)]TJ /F8 1 Tf -13.018 -3.0723 Td [(a)]TJ /F9 1 Tf 0.69277 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7759 78.392 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6386 55.973 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.75 79.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8193 55.973 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.284 78.392 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.488 55.973 Td [(ax)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.017 79.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.922 55.973 Td [(plus)-337(g)30(\266)469(en)30(\266)469(eralemen)30(t)]TJ /F9 1 Tf 9.2108 1.1145 Td [(\263)]TJ 0.75904 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.345 78.392 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.506 55.973 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.319 79.883 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 26.41 57.087 Td [(\264)]TJ /F6 1 Tf 0.81928 -1.1145 Td [(\242)]TJ /F9 1 Tf 0.5 1.1145 Td [(\263)]TJ 0.76506 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.793 78.392 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.108 55.973 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.784 79.883 Td [(j)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 31.018 57.087 Td [(\264)]TJ /F2 1 Tf 0.87349 -1.1145 Td [(=)]TJ /F9 1 Tf 2.1386 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.155 78.323 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F10 1 Tf 0.44828 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F10 1 Tf 0.5862 0 Td [(\243)]TJ /F7 1 Tf 0.89655 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.717 55.973 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.362 79.883 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.62 55.973 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.543 79.883 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.867 55.973 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 41.801 Td [(I)-30(I.C.2.)-241(Extension)-381(aux)-376(familles)-381(quelconques)-376(dans)]TJ ET 0.504 w 405 608.9 m 413.64 608.9 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 33.886 50.111 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.517 71.495 Td [(+)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 29.422 41.801 Td [(.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 48.442 Td [(Les)-481(d)30(\266)469(e\257nitions)-481(pr)30(\266)469(ec)24(\266)475(eden)24(tes)-487(s'appliquen)30(t)-487(en)-481(particulier)-481(lorsque)-481(les)]TJ /F8 1 Tf 29.855 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.172 69.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.572 48.442 Td [(son)24(t)-481(des)-457(\266)475(el)18(\266)475(emen)24(ts)]TJ -31.241 -1.2048 Td [(de)]TJ ET 80.712 574.56 m 89.352 574.56 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 6.753 47.238 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 10.698 67.383 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5663 47.238 Td [(\(on)-427(ne)-427(p)-30(eut)-427(pas)-427(prendre)-427(des)-397(\266)469(el)24(\266)469(emen)24(ts)-421(dans)-427(tout)]TJ ET 365.62 574.56 m 374.26 574.56 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 30.59 47.238 Td [(R)]TJ /F2 1 Tf 1.1446 0 Td [(si)-427(l'on)-427(v)24(eut)-403(\266)475(eviter)-427(les)-427(formes)]TJ -26.404 -1.1988 Td [(ind)24(\266)475(etermin)24(\266)475(ees\).)-608(Nous)-385(allons)-385(v)24(oir)-391(que,)-403(dans)-385(ce)-385(con)24(texte,)-397(on)-391(p)-24(eut)-391(g)24(\266)475(en)24(\266)475(eraliser)-391(la)-385(notion)-385(de)]TJ 0 -1.1988 Td [(somme)-331(d'une)-337(famille)-337(au)-337(cas)]TJ /F8 1 Tf 12.38 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(in\257ni.)]TJ -13.229 -1.6747 Td [(Consid)30(\266)469(erons)-295(une)-289(famille)-289(\()]TJ /F8 1 Tf 11.024 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.224 61.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.259 43.165 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.259 61.556 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.223 43.165 Td [(d')30(\266)469(el)24(\266)469(emen)24(ts)-289(de)]TJ ET 303.48 525.81 m 312.12 525.81 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 25.392 43.165 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.371 61.556 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.777 43.165 Td [(.)-433(P)30(our)-289(c)18(haque)]TJ /F8 1 Tf 6.3916 0 Td [(J)]TJ /F2 1 Tf 0.93976 0 Td [(partie)-295(\257nie)-289(de)]TJ /F8 1 Tf 6.259 0 Td [(I)]TJ /F2 1 Tf 0.80723 0 Td [(on)-295(p)-30(eut)]TJ -35.843 -1.4036 Td [(alors)-337(d)24(\266)475(e\257nir)-337(la)-337(somme)-331(partielle)]TJ /F9 1 Tf 13.855 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.543 58.056 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.801 41.762 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.586 59.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.705 41.762 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 31.334 Td [(I)-30(I.C.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1506 37.563 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(D)]TJ 0.83133 0.078312 Td [(\266)]TJ 0 -0.078312 Td [(efinition.)]TJ /F12 1 Tf 5.9096 0 Td [(Av)30(ec)-542(les)-536(notations)-542(et)-536(h)24(yp)-30(oth)18(\265)475(eses)-542(pr)24(\266)475(ec)24(\266)469(eden)30(tes)-542(on)-542(d)30(\266)469(e\257nit)-536(la)]TJ -12.307 -1.2048 Td [(somme)-331(de)-337(la)-331(famille)-337(des)]TJ /F8 1 Tf 10.765 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.853 51.814 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.337 36.358 Td [(comme)-331(l')24(\266)469(el)24(\266)469(emen)30(t)-337(de)]TJ ET 314.5 444.52 m 323.14 444.52 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 26.313 36.358 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.69 51.814 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 28.03 36.358 Td [(:)]TJ /F9 1 Tf -7.4458 -0.76506 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.612 47.857 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.199 34.641 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.586 49.357 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.38 34.641 Td [(=)-283(sup)]TJ /F9 1 Tf 2.7289 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.448 47.857 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.723 34.641 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.491 49.357 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 28.958 34.641 Td [(.)]TJ -23.627 -2.6747 Td [(\(La)-355(b)-30(orne)-355(sup)0(\266)469(erieure)-355(pr)24(\266)469(ec)24(\266)475(eden)24(te)-355(est)-361(prise)-355(p)-30(our)]TJ /F8 1 Tf 20.922 0 Td [(J)]TJ /F12 1 Tf 1.006 0 Td [(parcouran)18(t)-355(l'ensem)30(ble)-361(des)-355(parties)-355(\257nies)]TJ -21.928 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(.\))]TJ /F1 1 Tf -1.8494 -2.006 Td [(Remarques.)]TJ /F2 1 Tf 6.7048 0 Td [(Cette)-337(d)24(\266)469(e\257nition)-331(est)-343(ind)24(\266)475(ep)-30(endan)24(te)-337(de)-331(toute)-307(\266)469(en)30(um)18(\266)475(eration)-337(de)]TJ /F8 1 Tf 25.94 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)]TJ -33.163 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0 0 Tm 15.265 25.744 Td [(6)]TJ /F9 1 Tf 1.0602 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.517 35.125 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.934 25.744 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.483 36.625 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.175 25.744 Td [(.)]TJ -13.843 -2.9759 Td [(Lorsqu'aucune)-337(confusion)-331(n'est)-337(\265)500(a)-337(craindre)-331(on)-337(p)-30(ourra)-337(noter)]TJ /F9 1 Tf 25.476 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.207 32.366 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.657 22.768 Td [(au)-337(lieu)-331(de)]TJ /F9 1 Tf 4.6145 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.922 30.875 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.88 22.768 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.888 32.366 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.789 22.768 Td [(.)]TJ -35.458 -3.2831 Td [(Les)-433(propri)24(\266)475(et)24(\266)469(es)-403(\266)469(etablies)-433(p)-30(our)-433(les)-433(sommes)-439(\257nies)-433(se)-433(g)24(\266)469(en)30(\266)469(eralisen)24(t)-433(alors)-439(aux)-433(sommes)-439(in\257nies)]TJ 0 -1.2048 Td [(\(d')24(\266)469(el)24(\266)469(emen)30(ts)-337(de)]TJ ET 143.14 228.45 m 151.78 228.45 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 11.976 18.28 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.172 25.943 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.361 18.28 Td [(\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 12.635 Td [(I)-30(I.C.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1506 15.147 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)18(oposition.)]TJ /F12 1 Tf 6.8735 0 Td [(P)30(our)-337(des)-337(familles)-337(de)-331(nom)24(bres)-337(p)-30(ositifs)-337(r)24(\266)475(eels)-337(ou)-337(in\257nis)-337(on)-331(a)-337(:)]TJ /F1 1 Tf -11.627 -1.8133 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(si)]TJ /F6 1 Tf 1.006 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(i)]TJ /F6 1 Tf 0.62651 0 Td [(2)]TJ /F8 1 Tf 0.93976 0 Td [(I)-415(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.543 18.866 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 12.867 13.334 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.629 18.866 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 15.416 13.334 Td [(alors)]TJ /F9 1 Tf 2.7289 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.121 17.366 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.759 13.334 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.095 18.866 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 20.94 13.334 Td [(6)]TJ /F9 1 Tf 1.0602 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.629 17.366 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.608 13.334 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.491 18.866 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.765 13.334 Td [(;)]TJ /F1 1 Tf -18.62 -2.9759 Td [(2.)]TJ /F12 1 Tf 1.5602 0 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(est)-343(la)-331(r)24(\266)475(eunion)-337(disjoin)24(te)-337(de)]TJ /F8 1 Tf 11.633 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.069 15.418 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 22.325 10.358 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.362 15.418 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.524 10.358 Td [(,)]TJ /F9 1 Tf 0.94578 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.595 13.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.078 10.358 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.569 14.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.265 10.358 Td [(=)]TJ /F9 1 Tf 1.0542 0.81325 Td [(\241)]TJ 0.6506 0.13855 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.853 13.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 62.349 18.728 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.614 10.358 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.06 14.607 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 32.518 11.171 Td [(\242)]TJ /F2 1 Tf 0.68072 -0.81325 Td [(+)]TJ /F9 1 Tf 1 0.81325 Td [(\241)]TJ 0.76506 0.13855 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.836 13.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 72.108 18.728 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.711 10.358 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.353 14.607 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 37.62 11.171 Td [(\242)]TJ /F12 1 Tf 0.78915 -0.81325 Td [(;)]TJ /F1 1 Tf -32.265 -3.0663 Td [(3.)]TJ /F12 1 Tf 1.4578 0 Td [(si)]TJ /F8 1 Tf 0.95783 0 Td [(I)]TJ /F12 1 Tf 0.8012 0 Td [(est)-289(la)-283(r)30(\266)469(eunion)-289(disjoin)30(te)-289(d'une)-283(famille)-289(de)-283(parties)]TJ /F2 1 Tf 20.488 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.905 10.219 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.169 7.2918 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.164 10.219 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 33.313 7.2918 Td [(,)]TJ /F9 1 Tf 0.85542 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.043 8.7193 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.777 7.2918 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.017 10.219 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.964 7.2918 Td [(=)]TJ /F9 1 Tf 1.1807 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.405 8.7107 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 39.886 8.4003 Td [(\263)]TJ 0.8494 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.172 8.7193 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 83.651 11.969 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.434 7.2918 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.543 10.219 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 43.337 8.4003 Td [(\264)]TJ /F12 1 Tf 0.87951 -1.1084 Td [(,)]TJ /F5 1 Tf -38.886 -3.4036 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(26)]TJ ET endstream endobj 113 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 112 0 R /ProcSet 2 0 R >> /Contents 111 0 R >> endobj 112 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F18 107 0 R >> endobj 114 0 obj << /Length 29393 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12831(I)-36(I.C.)-385(F)96(amilles)-385(sommables)]TJ /F12 1 Tf 0 -2.5542 Td [(en)-331(particulier)-337(on)-337(a)]TJ /F9 1 Tf 9.4759 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.647 87.047 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F10 1 Tf 0.44827 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F10 1 Tf 0.58621 0 Td [(\243)]TJ /F7 1 Tf 0.89655 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.494 62.069 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 25.853 88.564 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.91 62.069 Td [(=)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.155 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 22.578 63.183 Td [(\263)]TJ 0.76506 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.422 87.116 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.958 62.069 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.534 88.564 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 27.096 63.183 Td [(\264)]TJ /F2 1 Tf 0.8735 -1.1145 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.56 87.116 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 30.639 63.183 Td [(\263)]TJ 0.76506 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.095 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.018 62.069 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.06 88.564 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 35.157 63.183 Td [(\264)]TJ /F12 1 Tf 0.92772 -1.1145 Td [(;)]TJ /F1 1 Tf -29.94 -3 Td [(4.)]TJ /F9 1 Tf 1.5602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.181 82.823 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1506 59.069 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.466 84.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.669 59.069 Td [(+)]TJ /F8 1 Tf 1 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.397 84.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.488 59.069 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.491 82.823 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.825 59.069 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.466 84.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.952 59.069 Td [(+)]TJ /F9 1 Tf 1 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.845 82.823 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.566 59.069 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.698 84.314 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.723 59.069 Td [(;)]TJ /F1 1 Tf -14.578 -2.9277 Td [(5.)]TJ /F8 1 Tf 1.3373 0 Td [(a)]TJ /F9 1 Tf 0.6988 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.853 78.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7892 56.141 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.828 80.125 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.97 56.141 Td [(=)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.371 78.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.639 56.141 Td [(ax)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.095 80.125 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 16.072 56.141 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F11 1 Tf 0.80723 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(0)]TJ /F12 1 Tf 0.5 0 Td [(\))-1006(plus)-222(g)30(\266)469(en)24(\266)475(eralemen)24(t)]TJ /F9 1 Tf 10.373 1.1084 Td [(\263)]TJ 0.76506 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.034 78.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.578 56.141 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.009 80.125 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 32.482 57.25 Td [(\264)]TJ /F6 1 Tf 0.59639 -1.1084 Td [(\242)]TJ /F9 1 Tf 0.27711 1.1084 Td [(\263)]TJ 0.76506 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.845 78.625 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.735 56.141 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.836 80.125 Td [(j)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 36.645 57.25 Td [(\264)]TJ /F2 1 Tf 0.87952 -1.1084 Td [(=)]TJ /F9 1 Tf 1.9639 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.966 78.564 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F10 1 Tf 0.44828 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F10 1 Tf 0.58621 0 Td [(\243)]TJ /F7 1 Tf 0.88793 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.181 56.141 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.172 80.125 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.084 56.141 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.353 80.125 Td [(j)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 56.141 Td [(.)]TJ /F1 1 Tf -38.886 -2.8735 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.5361 Td [(La)-265(d)24(\266)475(emonstration)-265(utilise)-265(l')24(\266)469(equiv)54(alence)-265(fondamen)18(tale)-259(en)24(tre)-602(sup)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.974 72.831 Td [(\256)]TJ /F10 1 Tf 0.75 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.56 51.732 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.345 73.814 Td [(\256)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 33.934 51.732 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 1.3976 0 Td [(et)]TJ /F6 1 Tf 1.3554 0 Td [(8)]TJ /F8 1 Tf 0.56024 0 Td [(\256)]TJ /F6 1 Tf 0.92169 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(A)-259(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.69 73.814 Td [(\256)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 42.56 51.732 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf -38.283 -1.7831 Td [(qui)-331(r)24(\266)475(esulte)-337(de)-331(la)-337(d)30(\266)469(e\257nition)-337(des)-337(b)-30(ornes)-337(sup)0(\266)475(erieures.)]TJ /F1 1 Tf 0.81325 -1.6807 Td [(1.)]TJ /F2 1 Tf 1.6145 0 Td [(Il)-355(su\261t)-367(de)-361(prouv)30(er)]TJ /F9 1 Tf 8.5904 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.474 67.366 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.958 48.268 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.517 68.857 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 19.187 48.268 Td [(6)]TJ /F9 1 Tf 1.1024 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.181 67.366 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.898 48.268 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.043 68.857 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.078 48.268 Td [(p)-30(our)]TJ /F8 1 Tf 2.3916 0 Td [(J)]TJ /F2 1 Tf 1.012 0 Td [(partie)-367(\257nie)-361(de)]TJ /F8 1 Tf 6.4759 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-524(Or)-361(on)-361(sait,)-367(p)-30(our)]TJ /F8 1 Tf 8.3434 0 Td [(J)]TJ /F2 1 Tf 1.006 0 Td [(\257ni,)]TJ /F9 1 Tf -37.494 -1.5482 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7069 63.788 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 45.768 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.75 65.288 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1265 45.768 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.216 63.788 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.795 45.768 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.147 65.288 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.614 45.768 Td [(,)-337(la)-331(relation)-337(annonc)24(\266)475(ee)-337(d)24(\266)469(ecoule)-337(alors)-337(de)-331(la)-337(remarque)-331(pr)24(\266)469(ec)24(\266)475(eden)24(te.)]TJ /F1 1 Tf -5.4699 -2.8253 Td [(2.)]TJ /F2 1 Tf 1.5602 0 Td [(Prouv)30(ons)]TJ /F9 1 Tf 4.4096 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.491 59.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.723 42.942 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.457 61.237 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 14.91 42.942 Td [(6)]TJ /F9 1 Tf 1.0843 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.845 59.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 34.386 83.885 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.633 42.942 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.052 61.237 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.759 42.942 Td [(+)]TJ /F9 1 Tf 1.1325 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.267 59.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 41.976 83.885 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.645 42.942 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.793 61.237 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.548 42.942 Td [(,)-331(il)-337(su\261t)-337(de)-331(mon)24(trer)]TJ /F9 1 Tf 9.1506 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.44 59.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.307 42.942 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.483 61.237 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 34.494 42.942 Td [(6)]TJ /F9 1 Tf 1.0843 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.871 59.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 73.554 83.885 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.217 42.942 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.078 61.237 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.337 42.942 Td [(+)]TJ /F9 1 Tf 1.1386 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.293 59.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 81.133 83.885 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.229 42.942 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.819 61.237 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.464 42.942 Td [(p)-30(our)]TJ /F8 1 Tf -37.133 -2.253 Td [(J)]TJ /F2 1 Tf 0.99398 0 Td [(partie)-355(\257nie)-349(de)]TJ /F8 1 Tf 6.4337 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-487(Or)]TJ /F8 1 Tf 2.2771 0 Td [(J)]TJ /F2 1 Tf 1 0 Td [(est)-349(alors)-349(la)-349(r)30(\266)469(eunion)-349(disjoin)24(te)-349(de)]TJ /F8 1 Tf 14.102 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.802 58.823 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.892 40.689 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(J)]TJ /F6 1 Tf 0.88554 0 Td [(\\)]TJ /F8 1 Tf 0.89759 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.466 58.823 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.892 40.689 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.983 58.823 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.53 40.689 Td [(=)]TJ /F8 1 Tf 1.0783 0 Td [(J)]TJ /F6 1 Tf 0.88554 0 Td [(\\)]TJ /F8 1 Tf 0.89759 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.974 58.823 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.422 40.689 Td [(,)-349(par)]TJ -37.09 -1.3434 Td [(suite)]TJ /F9 1 Tf 2.3916 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.138 54.59 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3373 39.346 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.181 56.09 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.518 39.346 Td [(=)]TJ /F9 1 Tf 1.1386 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.569 54.59 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 25.819 76.692 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.343 39.346 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.914 56.09 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.47 39.346 Td [(+)]TJ /F9 1 Tf 1.1928 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.138 54.59 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 33.602 76.692 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.464 39.346 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.802 56.09 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 18.645 39.346 Td [(6)]TJ /F9 1 Tf 1.0843 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.19 54.59 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 41.867 76.692 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.367 39.346 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.397 56.09 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.494 39.346 Td [(+)]TJ /F9 1 Tf 1.1386 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.621 54.59 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 49.458 76.692 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.386 39.346 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.147 56.09 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.289 39.346 Td [(.)]TJ -20.958 -2.8313 Td [(In)30(v)24(ersemen)24(t,)-463(prouv)24(ons)]TJ /F9 1 Tf 10.416 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.491 50.538 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 33.892 71.042 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.386 36.515 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.698 52.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.578 36.515 Td [(+)]TJ /F9 1 Tf 1.2108 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.121 50.538 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 41.759 71.042 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.536 36.515 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.638 52.038 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 22.886 36.515 Td [(6)]TJ /F9 1 Tf 1.2289 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.655 50.538 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.723 36.515 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.629 52.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.627 36.515 Td [(.)-753(Nous)-433(a)24(v)24(ons)-439(remarqu)30(\266)469(e)-439(que)-439(c)24(hacun)-439(des)]TJ -21.295 -2.1446 Td [(termes)-313(de)-313(la)-307(somme)-313(du)-319(premier)-313(mem)24(bre)-313(est)-319(ma)-54(jor)24(\266)469(ee)-313(par)-319(le)-313(second)-319(mem)24(bre,)-319(la)-307(relation)-313(est)]TJ 0 -1.1988 Td [(par)-313(suite)-289(\266)475(eviden)18(te)-313(si)-313(l'un)-319(de)-313(ces)-313(termes)-313(est)-319(in\257ni,)-319(supp)-30(osons)-313(donc)-313(qu'ils)-319(son)24(t)-319(\257nis.)-439(Il)-313(s'agit)]TJ 0 -1.3494 Td [(alors)-337(de)-331(prouv)24(er)]TJ /F9 1 Tf 7.4337 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.224 43.831 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 27.94 61.656 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.404 31.822 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.431 45.323 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 15.59 31.822 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.974 43.831 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.253 31.822 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.94 45.323 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.386 31.822 Td [(\241)]TJ /F9 1 Tf 1.1386 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.172 43.831 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 43.229 61.656 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.271 31.822 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.69 45.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.175 31.822 Td [(.)]TJ -17.843 -2.8253 Td [(P)30(our)-409(cela)-415(il)-409(su\261t)-415(de)-409(prouv)24(er)]TJ /F9 1 Tf 13.367 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.647 39.78 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 39.904 55.993 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.392 28.997 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30 41.28 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 21.699 28.997 Td [(6)]TJ /F9 1 Tf 1.1867 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.897 39.78 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.494 28.997 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.871 41.28 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.675 28.997 Td [(\241)]TJ /F9 1 Tf 1.1867 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.241 39.78 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 55.904 55.993 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.614 28.997 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.767 41.28 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.928 28.997 Td [(p)-30(our)]TJ /F8 1 Tf 2.4398 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.25 42.081 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.711 28.997 Td [(partie)-409(\257nie)-415(de)]TJ /F8 1 Tf 6.6205 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.457 42.081 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.127 28.997 Td [(,)-433(ce)-409(qui)]TJ -35.825 -2.5 Td [(\266)469(equiv)54(aut)-283(\265)500(a)]TJ /F9 1 Tf 4.9518 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.474 36.202 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 22.687 50.993 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.006 26.497 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.991 37.702 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 13.187 26.497 Td [(6)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.534 36.202 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.855 26.497 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.509 37.702 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.873 26.497 Td [(\241)]TJ /F9 1 Tf 0.9759 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.431 36.202 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 38.193 50.993 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.536 26.497 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.776 37.702 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.44 26.497 Td [(.)-427(Il)-283(su\261t)-283(donc)-283(de)-283(prouv)30(er)]TJ /F9 1 Tf 11.506 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.448 36.202 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 66.181 50.993 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.747 26.497 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.112 37.702 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 34.928 26.497 Td [(6)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.647 36.202 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.596 26.497 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.621 37.702 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.614 26.497 Td [(\241)]TJ /F9 1 Tf 0.97591 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.543 36.202 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 81.687 50.993 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.277 26.497 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.888 37.702 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.464 26.497 Td [(p)-30(our)]TJ /F8 1 Tf -37.133 -2.5 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.5603 34.926 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8373 23.997 Td [(partie)-343(\257nie)-349(de)]TJ /F8 1 Tf 6.4217 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.716 34.926 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.289 23.997 Td [(,)-343(ce)-349(qui)-313(\266)469(equiv)54(aut)-349(\265)499(a)]TJ /F9 1 Tf 8.5482 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.569 32.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 48.181 46.005 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.524 23.997 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.914 34.125 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.663 23.997 Td [(+)]TJ /F9 1 Tf 1.1928 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.164 32.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 56 46.005 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.657 23.997 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.828 34.125 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 29.855 23.997 Td [(6)]TJ /F9 1 Tf 1.0723 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.405 32.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.536 23.997 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.379 34.125 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.44 23.997 Td [(.)-475(Cep)-30(endan)24(t)]TJ /F8 1 Tf 5.8193 0 Td [(J)]TJ /F2 1 Tf 0.94578 0 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60 34.926 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.44 23.997 Td [([)]TJ /F8 1 Tf 0.89157 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.94 34.926 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 21.744 Td [(est)-289(une)-289(partie)-289(\257nie)-289(de)]TJ /F8 1 Tf 9.6084 0 Td [(I)]TJ /F2 1 Tf 0.8012 0 Td [(r)30(\266)469(eunion)-289(disjoin)24(te)-289(de)]TJ /F8 1 Tf 8.8133 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.069 31.711 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.771 21.744 Td [(et)]TJ /F8 1 Tf 1.1205 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.414 31.711 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.054 21.744 Td [(.)-427(On)-289(sait)-289(donc)-289(que)-289(le)-289(premier)-289(mem)24(bre)]TJ -22.723 -1.3434 Td [(de)-379(la)-385(relation)-379(pr)24(\266)475(ec)24(\266)469(eden)30(te)-385(est)-361(\266)475(egal)-385(\265)500(a)]TJ /F9 1 Tf 15.88 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.431 27.478 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.825 20.4 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.474 28.978 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.729 20.4 Td [(,)-391(c'est)-385(une)-385(somme)-379(partielle)-391(in)30(terv)24(enan)24(t)-385(dans)-379(la)]TJ -18.398 -2.5 Td [(d)30(\266)469(e\257nition)-337(de)]TJ /F9 1 Tf 5.7771 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.052 23.909 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.723 17.9 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.026 25.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.627 17.9 Td [(,)-337(la)-331(relation)-337(en)-331(d)24(\266)475(ecoule.)]TJ /F1 1 Tf -7.4819 -2.4759 Td [(3.)]TJ /F2 1 Tf 1.5542 0 Td [(Remarquons)-331(tout)-331(d'ab)-30(ord)-331(que)-337(le)-337(cas)-331(o)-24(\265)524(u)]TJ /F8 1 Tf 17.596 0 Td [(K)]TJ /F2 1 Tf 1.247 0 Td [(est)-337(un)-331(ensem)24(ble)-331(\257ni)-337(s'obtien)24(t)-331(imm)18(\266)475(ediate-)]TJ -21.211 -1.1988 Td [(men)24(t)-337(par)-331(r)24(\266)475(ecurrence)-337(\265)499(a)-337(partir)-331(du)-337(r)24(\266)475(esultat)-343(pr)30(\266)469(ec)24(\266)475(eden)24(t.)]TJ 0 -1.7771 Td [(Dans)-337(le)-343(cas)-337(g)30(\266)469(en)30(\266)469(eral,)-331(comme)-337(p)-30(our)]TJ /F1 1 Tf 14.765 0 Td [(2.)]TJ /F2 1 Tf 0.89759 0 Td [(,)-337(prouv)30(ons)]TJ /F9 1 Tf 4.9036 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.207 16.107 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.506 12.448 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.181 17.599 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 28.693 12.448 Td [(6)]TJ /F9 1 Tf 1.1867 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.578 16.09 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 31.62 13.557 Td [(\263)]TJ 0.8494 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.345 16.107 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 67.12 22.295 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.169 12.448 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.707 17.599 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 35.072 13.557 Td [(\264)]TJ /F2 1 Tf 0.59639 -1.1084 Td [(,)-337(il)-337(su\261t)-337(de)-337(mon)24(trer)]TJ /F9 1 Tf -30.337 -1.7048 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7069 12.306 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 9.7979 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.75 13.806 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1687 9.7979 Td [(6)]TJ /F9 1 Tf 1.2289 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.267 12.297 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 11.139 10.906 Td [(\263)]TJ 0.8494 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.034 12.306 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 26.157 16.993 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.687 9.7979 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.397 13.806 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 14.59 10.906 Td [(\264)]TJ /F2 1 Tf 0.95783 -1.1084 Td [(p)-30(our)]TJ /F8 1 Tf 2.3916 0 Td [(J)]TJ /F2 1 Tf 1.012 0 Td [(partie)-361(\257nie)-367(de)]TJ /F8 1 Tf 6.4699 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-530(P)30(our)-361(une)-361(telle)-367(partie)-361(il)-361(est)-367(clair)-361(que)-367(l'on)]TJ -20.608 -2.5542 Td [(p)-30(eut)-337(trouv)24(er)-337(une)-337(partie)-337(\257nie)]TJ /F8 1 Tf 12.765 0 Td [(K)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.216 10.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.633 7.2436 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(K)]TJ /F2 1 Tf 1.253 0 Td [(telle)]TJ /F8 1 Tf 2.1687 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.828 10.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.596 7.2436 Td [(=)]TJ /F9 1 Tf 1.488 0.95181 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.138 8.6418 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(K)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 56.711 12.475 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.789 7.2436 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.828 10.15 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.06 7.2436 Td [(con)30(tienne)]TJ /F8 1 Tf 4.4699 0 Td [(J)]TJ /F2 1 Tf 0.6506 0 Td [(.)]TJ -10.934 -3.3554 Td [(27)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 116 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 115 0 R /ProcSet 2 0 R >> /Contents 114 0 R >> endobj 115 0 obj << /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F18 107 0 R >> endobj 117 0 obj << /Length 22183 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12831(I)-36(I.C.)-385(F)96(amilles)-385(sommables)]TJ /F2 1 Tf 0 -2.5542 Td [(On)-421(a)-421(alors)]TJ /F9 1 Tf 5.1627 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.095 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.102 62.069 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.138 88.607 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 13.434 62.069 Td [(6)]TJ /F9 1 Tf 1.2771 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.94 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 31.928 122.15 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.404 62.069 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.293 88.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.735 62.069 Td [(=)]TJ /F9 1 Tf 1.4639 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.095 87.099 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(K)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 41.277 122.14 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 21.072 63.183 Td [(\263)]TJ 0.8494 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.25 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77587 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 46.036 121.54 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.62 62.069 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.621 88.607 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 24.524 63.183 Td [(\264)]TJ /F11 1 Tf 1.0241 -1.1145 Td [(6)]TJ /F9 1 Tf 1.3313 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.276 87.099 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(K)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 28.62 63.183 Td [(\263)]TJ 0.84337 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.043 87.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 61.12 121.54 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.163 62.069 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.414 88.607 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 32.066 63.183 Td [(\264)]TJ /F2 1 Tf 1.0181 -1.1145 Td [(,)-445(la)-421(ma)-60(joration)-427(v)24(oulue)-421(en)]TJ -27.753 -2.1928 Td [(d)30(\266)469(ecoule.)]TJ 0 -1.9036 Td [(In)30(v)24(ersemen)24(t)-385(prouv)24(ons)]TJ /F9 1 Tf 10.102 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.905 81.237 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 17.175 59.081 Td [(\263)]TJ 0.8494 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.672 81.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 38.241 113.34 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.723 57.973 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.043 82.745 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 20.627 59.081 Td [(\264)]TJ /F11 1 Tf 0.95783 -1.1084 Td [(6)]TJ /F9 1 Tf 1.1325 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.664 81.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.331 57.973 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.638 82.745 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.614 57.973 Td [(,)-391(il)-385(su\257t)-385(de)-379(mon)24(trer)]TJ /F9 1 Tf 9.1687 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.716 81.237 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(L)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 36.44 59.081 Td [(\263)]TJ 0.8494 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.241 81.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 76.759 113.34 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.982 57.973 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.603 82.745 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 39.886 59.081 Td [(\264)]TJ /F11 1 Tf 0.95783 -1.1084 Td [(6)]TJ /F9 1 Tf 1.1325 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.224 81.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.59 57.973 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 82.745 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 55.774 Td [(p)-30(our)]TJ /F8 1 Tf 2.3554 0 Td [(L)]TJ /F2 1 Tf 1.006 0 Td [(partie)-331(\257nie)-325(de)]TJ /F8 1 Tf 6.3675 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(.)-439(Mais,)-331(puisque)]TJ /F8 1 Tf 7.0482 0 Td [(L)]TJ /F2 1 Tf 1.006 0 Td [(est)-331(\257ni,)-331(on)-325(sait)-325(que)-331(le)-331(premier)-325(mem)24(bre)-325(est)-307(\266)475(egal)]TJ -18.705 -1.3434 Td [(\265)499(a)]TJ /F9 1 Tf 0.93373 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.9224 76.185 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 14.928 106.87 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.9036 54.43 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.129 77.676 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.2108 54.43 Td [(,)-427(o)-24(\265)524(u)]TJ /F8 1 Tf 2.1627 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.017 78.487 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.578 54.43 Td [(est)-409(la)-409(r)30(\266)469(eunion)-409(des)]TJ /F8 1 Tf 8.3072 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.517 77.676 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.223 54.43 Td [(p)-30(our)]TJ /F8 1 Tf 2.4337 0 Td [(k)]TJ /F2 1 Tf 0.95783 0 Td [(dans)]TJ /F8 1 Tf 2.4096 0 Td [(L)]TJ /F2 1 Tf 0.68072 0 Td [(.)-662(Comme)]TJ /F8 1 Tf 4.6807 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.517 78.487 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.59 54.43 Td [(est)-409(une)-403(partie)-415(de)]TJ /F8 1 Tf 7.9639 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-421(le)]TJ -37.741 -2.1446 Td [(r)30(\266)469(esultat)-337(en)-337(d)30(\266)469(ecoule.)]TJ 0 -1.8976 Td [(Le)-343(r)24(\266)475(esultat)-349(concernan)24(t)-343(les)-349(familles)-349(doubles)-343(s'obtien)24(t)-343(alors)-349(en)-343(remarquan)24(t)-349(que)]TJ /F8 1 Tf 34.066 0 Td [(I)]TJ /F6 1 Tf 0.74699 0 Td [(\243)]TJ /F8 1 Tf 1.006 0 Td [(J)]TJ /F2 1 Tf 0.99397 0 Td [(est)-349(la)]TJ -36.813 -1.2048 Td [(r)30(\266)469(eunion)-259(disjoin)24(tes)-259(des)]TJ /F6 1 Tf 9.512 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(i)]TJ /F6 1 Tf 0.34337 0 Td [(g)-72(\243)]TJ /F8 1 Tf 1.4157 0 Td [(J)]TJ /F2 1 Tf 0.90361 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.2892 0 Td [(i)]TJ /F2 1 Tf 0.60241 0 Td [(parcouran)24(t)]TJ /F8 1 Tf 5.012 0 Td [(I)]TJ /F2 1 Tf 0.77711 0 Td [(ainsi)-253(que)-259(des)]TJ /F8 1 Tf 5.6988 0 Td [(I)]TJ /F6 1 Tf 0.58434 0 Td [(\243)-66(f)]TJ /F8 1 Tf 1.3434 0 Td [(j)]TJ /F6 1 Tf 0.46988 0 Td [(g)]TJ /F2 1 Tf 0.75904 0 Td [(p)-24(our)]TJ /F8 1 Tf 2.2831 0 Td [(j)]TJ /F2 1 Tf 0.72891 0 Td [(parcouran)24(t)]TJ /F8 1 Tf 5.0121 0 Td [(J)]TJ /F2 1 Tf 0.6506 0 Td [(.)]TJ /F1 1 Tf -38.072 -1.5964 Td [(4.)]TJ /F2 1 Tf 1.5783 0 Td [(D)24(\266)469(e\257nissons)-343(une)-349(famille)-343(index)30(\266)469(ee)-349(par)]TJ /F8 1 Tf 15.729 0 Td [(I)]TJ /F6 1 Tf 0.74699 0 Td [(\243)-228(f)]TJ /F2 1 Tf 1.506 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(2)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.84337 0 Td [(en)-343(p)-30(osan)24(t)]TJ /F8 1 Tf 4.5783 0 Td [(z)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.284 67.84 Td [(\()]TJ /F7 1 Tf 0.44827 0 Td [(i;)]TJ /F4 1 Tf 0.74138 0 Td [(1\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.928 47.587 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.336 67.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.247 47.587 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(z)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 55.655 67.84 Td [(\()]TJ /F7 1 Tf 0.44827 0 Td [(i;)]TJ /F4 1 Tf 0.74138 0 Td [(2\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.777 47.587 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.595 67.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.675 47.587 Td [(.)-475(Le)]TJ -37.343 -1.2048 Td [(r)30(\266)469(esultat)-385(pr)30(\266)469(ec)24(\266)469(eden)30(t)-385(p)-30(ermet)-379(d')24(\266)469(ev)54(aluer)-379(la)-385(somme)-379(de)-379(cette)-385(famille)-379(de)-379(deux)-379(fa)-6(\273)439(cons)-385(di\256)30(\266)469(eren)24(tes)]TJ 0 -1.3434 Td [(et)-343(fournit)-337(l')24(\266)469(egalit)24(\266)475(e)-349(:)]TJ /F9 1 Tf 8.9578 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.603 62.745 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.735 45.039 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(z)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 23.741 64.193 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;)]TJ /F4 1 Tf 0.74138 0 Td [(1\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.41 45.039 Td [(+)]TJ /F8 1 Tf 1.006 0 Td [(z)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.448 64.193 Td [(\()]TJ /F7 1 Tf 0.44827 0 Td [(i;)]TJ /F4 1 Tf 0.75 0 Td [(2\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.476 45.039 Td [(\))-295(=)]TJ /F9 1 Tf 1.747 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.388 62.745 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.837 45.039 Td [(z)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.207 64.193 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;)]TJ /F4 1 Tf 0.74138 0 Td [(1\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.127 45.039 Td [(+)]TJ /F9 1 Tf 1 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.405 62.745 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.741 45.039 Td [(z)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.224 64.193 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;)]TJ /F4 1 Tf 0.75 0 Td [(2\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.145 45.039 Td [(.)-463(Cela)-349(se)-343(traduit)-343(bien)-343(par)-343(la)]TJ -26.813 -2.1446 Td [(relation)-307(\266)475(enonc)24(\266)475(ee.)]TJ /F1 1 Tf 0.81325 -1.5964 Td [(5.)]TJ /F2 1 Tf 1.8253 0 Td [(P)24(our)-463(toute)]TJ /F8 1 Tf 5.3133 0 Td [(J)]TJ /F2 1 Tf 1.1145 0 Td [(partie)-475(\257nie)-469(de)]TJ /F8 1 Tf 6.7952 0 Td [(I)]TJ /F2 1 Tf 0.98795 0 Td [(on)-463(sait)]TJ /F9 1 Tf 3.5482 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.897 57.383 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.343 41.298 Td [(ax)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.698 58.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.271 41.298 Td [(=)]TJ /F8 1 Tf 1.2771 0 Td [(a)]TJ /F9 1 Tf 0.69879 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.793 57.383 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.861 41.298 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.836 58.883 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 34.265 41.298 Td [(6)]TJ /F8 1 Tf 1.2771 0 Td [(a)]TJ /F9 1 Tf 0.69277 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.009 57.383 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.849 41.298 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.983 58.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.217 41.298 Td [(et)-469(par)-469(suite)]TJ /F9 1 Tf -33.886 -1.5542 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7759 53.806 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 38.798 Td [(ax)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.509 55.306 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6506 38.798 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(a)]TJ /F9 1 Tf 0.69277 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.043 53.806 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.018 38.798 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.017 55.306 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.922 38.798 Td [(.)]TJ -7.5904 -2.7169 Td [(Supp)-24(osons)-228(d'ab)-36(ord)]TJ /F8 1 Tf 8.4157 0 Td [(a)]TJ /F2 1 Tf 0.75301 0 Td [(\257ni)-228(et)-228(non)-234(n)30(ul.)-415(On)-228(a)-222(donc,)-253(en)-228(rempla)-6(\273)439(can)30(t)]TJ /F8 1 Tf 17.753 0 Td [(a)]TJ /F2 1 Tf 0.75903 0 Td [(par)]TJ /F8 1 Tf 1.6747 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.388 52.228 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.289 36.081 Td [(,)-246(p)-30(our)-228(toute)-228(famille)]TJ -30.958 -1.3434 Td [(d')24(\266)475(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 5.0241 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.526 49.495 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.614 34.738 Td [(de)]TJ ET 0.504 w 156.02 425.08 m 164.66 425.08 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 13.054 34.738 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.716 49.495 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.44 34.738 Td [(,)]TJ /F9 1 Tf 0.74096 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.871 48.004 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.789 34.738 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.784 50.306 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.392 34.738 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.026 49.495 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 19.669 34.738 Td [(6)]TJ /F8 1 Tf 1.2229 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.655 50.306 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 22.663 35.689 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.586 48.004 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.271 34.738 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.44 49.495 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.096 34.738 Td [(,)-463(d'o)-30(\265)530(u)]TJ /F8 1 Tf 3.0663 0 Td [(a)]TJ /F9 1 Tf 0.69277 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.448 48.004 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.47 34.738 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.362 50.306 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.072 34.738 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.603 49.495 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 33.343 34.738 Td [(6)]TJ /F9 1 Tf 1.2289 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.629 48.004 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.181 34.738 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.483 49.495 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.006 34.738 Td [(.)-753(En)-439(particulier,)]TJ -31.675 -2.5 Td [(lorsque)]TJ /F8 1 Tf 3.4277 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.233 45.918 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.8554 32.238 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(ax)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.19 45.918 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.681 32.238 Td [(on)-337(obtien)30(t)]TJ /F8 1 Tf 4.8072 0 Td [(a)]TJ /F9 1 Tf 0.69879 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.172 44.426 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.795 32.238 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.147 45.918 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 20.982 32.238 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.69 44.426 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.645 32.238 Td [(ax)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.414 45.918 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.416 32.238 Td [(et)-337(donc)-331(l')24(\266)469(egalit)24(\266)475(e)-343(annonc)24(\266)475(ee.)]TJ -20.084 -2.5422 Td [(L')24(\266)469(egalit)24(\266)475(e)-391(est)-361(\266)475(eviden)18(te)-385(lorsque)]TJ /F8 1 Tf 13.205 0 Td [(a)]TJ /F2 1 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Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F10 1 Tf 0.44827 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F10 1 Tf 0.58621 0 Td [(\243)]TJ /F7 1 Tf 0.89655 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.801 23.545 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.586 33.478 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.705 23.545 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.759 33.478 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.036 23.545 Td [(=)]TJ /F9 1 Tf 1.1928 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.828 31.987 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 25.843 24.659 Td [(\263)]TJ 0.75904 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.095 31.987 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.217 23.545 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.198 33.478 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 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0 0 Tm 13.217 20.78 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.612 29.521 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.127 20.78 Td [(.)-837(P)24(ar)-463(suite)]TJ /F9 1 Tf 6.7289 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.302 27.961 Td [(\()]TJ /F7 1 Tf 0.44827 0 Td [(i;j)]TJ /F4 1 Tf 1.2759 0 Td [(\))]TJ /F10 1 Tf 0.44828 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F10 1 Tf 0.58621 0 Td [(\243)]TJ /F7 1 Tf 0.88793 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.548 20.78 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.517 29.521 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.452 20.78 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.69 29.521 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.861 20.78 Td [(=)]TJ /F9 1 Tf 1.2711 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.983 28.021 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 28.747 21.888 Td [(\263)]TJ /F8 1 Tf 0.59639 -1.1084 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.81 29.521 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 30.416 21.732 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.543 28.021 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.03 20.78 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.534 29.521 Td [(j)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 32.94 21.888 Td [(\264)]TJ /F2 1 Tf 1.0602 -1.1084 Td [(et)-469(le)-469(r)30(\266)469(esultat)-469(pr)24(\266)475(ec)24(\266)469(eden)24(t)]TJ -28.669 -2.3133 Td [(donne)-337(l')30(\266)469(egalit)24(\266)469(e)-337(annonc)24(\266)469(ee.)]TJ ET 512.64 219.49 m 519.84 219.49 l 519.84 227.91 l 512.64 227.91 l f BT /F5 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 3.8882 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(28)]TJ ET endstream endobj 119 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 118 0 R /ProcSet 2 0 R >> /Contents 117 0 R >> endobj 118 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R 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Td [(une)-373(famille)-373(d')24(\266)469(el)24(\266)469(emen)30(ts)-373(de)]TJ ET 0.504 w 416.88 714.45 m 425.52 714.45 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 34.88 58.948 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 50.948 84.142 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.265 58.948 Td [(.)-566(Supp)-24(osons)-379(que)]TJ /F8 1 Tf 7.7108 0 Td [(I)]TJ /F12 1 Tf -38.645 -1.4036 Td [(soit)-313(la)-307(r)24(\266)475(eunion)-319(d'une)-313(suite)-313(croissan)24(te)-307(de)-313(parties)]TJ /F8 1 Tf 20.777 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.991 82.133 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 27.09 57.545 Td [(,)-319(alors)]TJ /F9 1 Tf 2.9699 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.172 80.642 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.675 57.545 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.147 82.133 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.892 57.545 Td [(est)-313(la)-313(limite)-307(de)-313(la)-313(suite)-313(des)]TJ -27.56 -2.5 Td [(sommes)-337(partielles)]TJ /F9 1 Tf 8.1325 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.112 77.064 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 29.06 107.5 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.187 55.045 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.552 78.564 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 16.09 55.045 Td [(.)]TJ /F1 1 Tf -10.759 -3.1386 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Remarquons)-301(tout)-307(d'ab)-30(ord)-301(que)-307(la)-301(suite)-307(des)]TJ /F8 1 Tf 18.512 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.371 74.064 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.922 51.906 Td [(\266)469(etan)24(t)-301(croissan)24(te,)-313(la)-301(suite)]TJ -28.59 -1.3434 Td [(des)-277(sommes)-283(partielles)]TJ /F9 1 Tf 9.6928 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.345 70.65 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 32.181 98.535 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.747 50.563 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.784 72.142 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.934 50.563 Td [(est)-283(croissan)24(te)-277(donc)-283(a)-277(une)-277(limite)]TJ /F8 1 Tf 13.777 0 Td [(S)]TJ /F2 1 Tf 0.94579 0 Td [(qui)-277(est)-289(sa)-277(b)-30(orne)-277(sup)0(\266)469(erieure.)]TJ -27.325 -2.5482 Td [(Chacune)-337(de)-331(ces)-331(sommes)-343(partielles)-331(est)-343(ma)-54(jor)18(\266)475(ee)-337(par)-337(la)-331(somme)-337(globale,)-337(donc)]TJ /F8 1 Tf 32.753 0 Td [(S)]TJ /F11 1 Tf 0.94579 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.517 67.004 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.699 48.015 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.491 68.495 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.602 48.015 Td [(.)]TJ -37.271 -2.7048 Td [(In)30(v)24(ersemen)24(t,)-385(si)]TJ /F8 1 Tf 6.9518 0 Td [(J)]TJ /F2 1 Tf 1.0181 0 Td [(est)-373(une)-373(partie)-373(\257nie)-367(de)]TJ /F8 1 Tf 10.018 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-379(p)-30(our)-367(c)24(haque)]TJ /F8 1 Tf 6.4277 0 Td [(j)]TJ /F6 1 Tf 0.8012 0 Td [(2)]TJ /F8 1 Tf 1 0 Td [(J)]TJ /F2 1 Tf 1.0181 0 Td [(il)-373(y)-367(a)-373(un)-367(en)24(tier)]TJ /F8 1 Tf 7.012 0 Td [(n)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.241 64.625 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.488 45.31 Td [(tel)-367(que)]TJ /F8 1 Tf -36.157 -1.2048 Td [(j)]TJ /F6 1 Tf 0.75301 0 Td [(2)]TJ /F8 1 Tf 0.95181 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.707 62.9 Td [(n)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 15.952 87.716 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3976 44.105 Td [(.)-457(En)-343(prenan)24(t)-343(p)-30(our)]TJ /F8 1 Tf 8.3855 0 Td [(n)]TJ /F2 1 Tf 0.93976 0 Td [(le)-343(maxim)24(um)-343(des)]TJ /F8 1 Tf 7.4699 0 Td [(n)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.914 62.9 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.554 44.105 Td [(\(et)]TJ /F8 1 Tf 1.5602 0 Td [(n)]TJ /F2 1 Tf 0.89157 0 Td [(=)-283(0)-343(si)]TJ /F8 1 Tf 2.9096 0 Td [(J)]TJ /F2 1 Tf 0.99397 0 Td [(est)-343(vide\))-337(on)-343(v)24(oit)-343(que)]TJ /F8 1 Tf 9.3675 0 Td [(J)]TJ /F2 1 Tf 0.98795 0 Td [(est)]TJ -37.934 -1.494 Td [(con)30(ten)24(u)-361(dans)]TJ /F8 1 Tf 6.1084 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17 60.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.777 42.611 Td [(\()-6(\265)500(a)-355(cause)-361(de)-355(la)-355(croissance)-361(des)]TJ /F8 1 Tf 12.892 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.362 60.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.651 42.611 Td [(\))-361(et)-355(donc)]TJ /F9 1 Tf 4.3494 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.44 59.262 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.608 42.611 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.483 60.762 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 33.831 42.611 Td [(6)]TJ /F9 1 Tf 1.2048 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.974 59.262 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 72.205 82.62 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.759 42.611 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.422 60.762 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 37.976 42.611 Td [(6)]TJ /F8 1 Tf 1.0964 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(.)-512(P)24(ar)-355(suite)]TJ /F9 1 Tf -34.41 -1.5964 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7759 55.616 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 40.063 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.75 57.116 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1265 40.063 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(S)]TJ /F2 1 Tf 1.006 0 Td [(et)-337(l')30(\266)469(egalit)24(\266)469(e)-337(en)-331(d)24(\266)475(ecoule.)]TJ ET 512.64 477.61 m 519.84 477.61 l 519.84 486.03 l 512.64 486.03 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 36.587 Td [(Ce)-433(r)30(\266)469(esultat)-433(est)-433(en)-433(particulier)-427(appliqu)24(\266)469(e)-433(lorsque)]TJ /F8 1 Tf 21.042 0 Td [(I)]TJ /F2 1 Tf 0.94578 0 Td [(est)-433(d)24(\266)469(enom)30(brable,)-457(on)-433(p)-30(eut)-433(alors)-433(c)24(hoisir)]TJ -21.988 -1.1988 Td [(des)]TJ /F8 1 Tf 1.8434 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.897 50.426 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6084 35.388 Td [(\257nis.)-801(Cela)-457(p)-30(ermet)-451(de)-451(consid)30(\266)469(erer)-457(la)-451(somme)-451(globale)-451(comme)-451(la)-451(limite)-451(d'une)-457(suite)]TJ -3.2771 -1.2048 Td [(de)-409(sommes)-409(\257nies.)-674(P)30(ar)-409(exemple,)-427(en)-379(\266)469(en)24(um)24(\266)475(eran)24(t)]TJ /F8 1 Tf 21.018 0 Td [(I)]TJ /F2 1 Tf 0.92771 0 Td [(sous)-409(la)-409(forme)]TJ /F8 1 Tf 6.3253 0 Td [(I)]TJ /F2 1 Tf 0.92168 0 Td [(=)]TJ /F6 1 Tf 1.1807 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(i)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.31 48.702 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37 34.183 Td [(;)-168(i)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.078 48.702 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.241 34.183 Td [(;)]TJ /F6 1 Tf 0.44579 0 Td [(\242)-162(\242)-168(\242)0(g)]TJ /F2 1 Tf 1.6627 0 Td [(,)-427(on)-415(p)-30(eut)]TJ -35.018 -2 Td [(prendre)]TJ /F8 1 Tf 3.6747 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.517 45.84 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.277 32.183 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(i)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.431 45.84 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.633 32.183 Td [(;)-162(i)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.207 45.84 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.867 32.183 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(\242)-168(\242)-168(\242)]TJ /F8 1 Tf 1.3313 0 Td [(;)-168(i)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.517 45.84 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.982 32.183 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(,)-337(on)-337(obtien)24(t)-337(que)]TJ /F9 1 Tf 7.2952 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.612 44.34 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.386 32.183 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.578 45.84 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.627 32.183 Td [(est)-343(la)-337(limite)-337(des)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.603 47.849 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 34.886 33.135 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.914 44.331 Td [(k)]TJ /F4 1 Tf 0.62931 0 Td [(=0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.5 32.183 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.052 45.84 Td [(i)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 74.711 63.849 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.843 32.183 Td [(,)-337(autremen)24(t)-337(dit)]TJ -32.512 -2.1446 Td [(c'est)-385(la)-391(somme)-385(de)-385(la)-385(s)30(\266)469(erie)-391(de)-385(termes)-385(g)30(\266)469(en)30(\266)469(eral)]TJ /F8 1 Tf 19.946 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.991 42.771 Td [(i)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 52.265 59.572 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.62 30.039 Td [(.)-596(En)-385(particulier,)-403(si)]TJ /F8 1 Tf 8.7349 0 Td [(I)]TJ /F2 1 Tf 0.88554 0 Td [(=)]TJ /F3 1 Tf 1.1386 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(,)-397(on)-385(a)-385(l')24(\266)469(egalit)24(\266)475(e)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 7.6983 41.711 Td [(+)]TJ /F10 1 Tf 0.87931 0 Td [(1)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3614 28.846 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.6293 38.254 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(=0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.006 27.894 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.845 39.711 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3976 27.894 Td [(=)]TJ /F9 1 Tf 1.1024 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.526 38.202 Td [(n)]TJ /F10 1 Tf 0.71552 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.163 27.894 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.793 39.711 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.277 27.894 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 18.449 Td [(I)-30(I.C.4.)-241(F)90(amilles)-381(sommables.)]TJ 0 -2.7337 Td [(I)-30(I.C.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1506 18.84 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(D)]TJ 0.83133 0.084337 Td [(\266)]TJ 0 -0.084337 Td [(efinition.)]TJ /F12 1 Tf 5.3494 0 Td [(On)-361(dit)-367(qu'une)-361(famille)-361(de)-361(nom)24(bres)-361(r)24(\266)475(eels)]TJ /F2 1 Tf 17.512 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.879 26.745 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.886 18.84 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.914 26.745 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 37.916 18.84 Td [(est)]TJ /F1 1 Tf 1.5843 0 Td [(sommable)]TJ /F12 1 Tf -34.169 -1.3976 Td [(lorsque)]TJ /F9 1 Tf 3.4277 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.681 23.245 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.367 17.442 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.052 24.745 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.548 17.442 Td [(j)]TJ /F12 1 Tf 0.61446 0 Td [(est)-337(\257ni.)]TJ /F2 1 Tf -6.8313 -2.7108 Td [(Lorsque)-240(l'on)-246(a)-240(une)-240(telle)-246(famille)-240(sommable)-246(on)-246(p)-30(eut)-240(l')24(\266)469(ecrire)-240(comme)-246(di\256)24(\266)475(erence)-246(de)-240(deux)-240(familles)]TJ 0 -1.1988 Td [(sommables)-421(p)-30(ositiv)30(es)]TJ /F8 1 Tf 9.3373 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.81 19.15 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16 13.533 Td [(=)]TJ /F8 1 Tf 1.1988 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.371 19.15 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.343 13.533 Td [(\241)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.379 19.15 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.163 13.533 Td [(.)-704(P)24(ar)-421(exemple)-421(en)-421(prenan)24(t)-427(p)-30(our)]TJ /F8 1 Tf 14.56 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.448 19.15 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.006 13.533 Td [(\(resp.)]TJ /F8 1 Tf 3.1566 0 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.664 19.15 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.928 13.533 Td [(\))-421(la)-421(partie)]TJ -34.596 -1.2048 Td [(p)-30(ositiv)30(e)-421(\(resp.)-686(n)24(\266)475(egativ)24(e\))-415(de)]TJ /F8 1 Tf 12.753 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.698 17.435 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.988 12.328 Td [(,)-433(on)-415(obtien)24(t)-415(des)-421(familles)-415(sommables)-415(puisque)-421(\()]TJ /F8 1 Tf 20.488 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.302 17.435 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.38 12.328 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 58.336 17.435 Td [(+)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 41.843 12.328 Td [(6)]TJ /F6 1 Tf 1.1928 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.802 17.435 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 12.328 Td [(j)]TJ /F2 1 Tf -38.886 -1.3976 Td [(et)-445(\()]TJ /F8 1 Tf 1.6627 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.828 15.426 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.9036 10.93 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.862 15.426 Td [(\241)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 9.4277 10.93 Td [(6)]TJ /F6 1 Tf 1.241 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.483 15.426 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.849 10.93 Td [(j)]TJ /F2 1 Tf 0.28313 0 Td [(.)-777(Les)-439(sommes)]TJ /F9 1 Tf 6.8072 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.259 13.926 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.548 10.93 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.164 15.426 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.855 10.93 Td [(et)]TJ /F9 1 Tf 1.2771 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.259 13.926 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.747 10.93 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.026 15.426 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.952 10.93 Td [(son)24(t)-451(alors)-445(des)-451(r)30(\266)469(eels)-445(\(\257nis\),)-469(ce)-451(qui)-445(p)-30(ermet)]TJ -20.62 -2.5 Td [(de)-409(consid)30(\266)469(erer)-409(la)-409(di\256)24(\266)469(erence)]TJ /F9 1 Tf 11.958 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.897 10.357 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.904 8.4304 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.81 11.849 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.036 8.4304 Td [(\241)]TJ /F9 1 Tf 1.0482 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.328 10.357 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.699 8.4304 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.095 11.849 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.464 8.4304 Td [(.)-668(Les)-409(propri)24(\266)469(et)24(\266)475(es)-415(des)-409(sommes)-415(de)-409(termes)-409(p)-30(ositifs)]TJ -18.133 -2.1446 Td [(vis-)0(\265)500(a-vis)-391(de)-391(l'addition)-385(prouv)24(en)30(t)-391(imm)18(\266)475(ediatemen)24(t)-391(que)-391(cette)-391(di\256)24(\266)475(erence)-391(ne)-391(d)30(\266)469(ep)-30(end)-385(pas)-397(de)-385(la)]TJ 18.916 -2.3976 Td [(29)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 122 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 121 0 R /ProcSet 2 0 R >> /Contents 120 0 R >> endobj 121 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F16 81 0 R >> endobj 123 0 obj << /Length 9861 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12831(I)-36(I.C.)-385(F)96(amilles)-385(sommables)]TJ /F2 1 Tf 0 -2.4518 Td [(d)30(\266)469(ecomp)-30(osition)]TJ /F8 1 Tf 6.6265 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.922 88.754 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.193 62.171 Td [(=)]TJ /F8 1 Tf 1.1145 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.224 88.754 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.41 62.171 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.129 88.754 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.566 62.171 Td [(que)-367(l'on)-373(a)-367(c)24(hoisie.)-542(On)-373(note)-367(alors)-373(cette)-367(di\256)24(\266)469(erence)]TJ /F9 1 Tf 21.807 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.5 87.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.988 62.171 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.474 88.754 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.892 62.171 Td [(,)-373(c'est)]TJ -36.56 -2.1446 Td [(la)-331(somme)-337(de)-331(la)-337(famille)-337(sommable)-337(des)]TJ /F8 1 Tf 16.548 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.129 85.685 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.789 60.027 Td [(.)]TJ -17.458 -2.1988 Td [(On)-331(p)-30(eut)-331(ais)24(\266)475(emen)24(t)-331(g)24(\266)475(en)24(\266)475(eraliser)-337(aux)-331(familles)-331(sommables)-331(les)-331(r)24(\266)475(esultats)-307(\266)475(enonc)24(\266)475(es)-331(dans)-331(la)-331(prop)-30(o-)]TJ 0 -1.2048 Td [(sition)]TJ /F1 1 Tf 2.7289 0 Td [(I)-36(I.C.2.2.)]TJ /F2 1 Tf 4.1627 0 Td [(.)]TJ -6.8916 -1.8976 Td [(Des)-469(remarques)-475(faites)-475(dans)-475(la)-469(section)-469(pr)24(\266)469(ec)24(\266)475(eden)24(te)-475(il)-469(r)24(\266)475(esulte)-475(imm)24(\266)475(ediatemen)24(t)-475(que,)-512(lorsque)]TJ /F8 1 Tf 0 -1.2048 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F3 1 Tf 1.0542 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(,)-259(la)-246(notion)-240(de)-240(famille)-246(sommable)-240(se)-246(confond)-240(a)24(v)24(ec)-246(celle)-240(de)-246(s)30(\266)469(erie)-246(absolumen)24(t)-246(con)30(v)24(ergen)18(te.)]TJ -2.5723 -1.1988 Td [(La)-337(somme)-331(en)-337(tan)24(t)-337(que)-337(famille)-337(sommable)-337(s'iden)30(ti\257e)-343(alors)-337(\265)500(a)-331(la)-337(somme)-331(en)-337(tan)24(t)-337(que)-337(s)24(\266)469(erie.)]TJ /F1 1 Tf 0 -1.8976 Td [(Remarques.)]TJ 0.81325 -1.6024 Td [(1.)]TJ /F2 1 Tf 1.753 0 Td [(P)24(ar)-427(com)24(binaison)-433(lin)24(\266)469(eaire,)-457(il)-433(est)-433(facile)-439(de)-427(g)24(\266)475(en)24(\266)475(eraliser)-433(aux)-439(familles)-433(de)-433(complexes)-433(ou)]TJ -2.5663 -1.1988 Td [(m)24(^)475(eme)-349(de)-349(v)24(ecteurs)-355(d'un)-349(espace)-349(de)-349(dimension)-355(\257nie)-349(ce)-355(qui)-349(vien)24(t)-349(d')24(^)469(etre)-349(fait)-349(p)-30(our)-349(les)-349(familles)]TJ 0 -1.2048 Td [(de)-331(r)24(\266)475(eels.)]TJ /F1 1 Tf 0.81325 -1.5964 Td [(2.)]TJ /F2 1 Tf 1.5 0 Td [(Comme)-307(nous)-307(l'a)30(v)18(ons)-307(signal)30(\266)469(e)-307(les)-307(notions)-307(pr)24(\266)469(ec)24(\266)475(eden)24(tes)-307(ne)-307(fon)30(t)-307(app)-30(el)-313(\265)500(a)-301(aucune)-307(structure)]TJ -2.3133 -1.2048 Td [(d'ordre)-355(sur)-355(l'ensem)30(ble)-355(d'indices.)-500(Il)-349(s'ensuit)-355(que)-355(la)-349(sommabilit)24(\266)469(e)-355(et)-355(la)-349(somme)-355(d'une)-355(famille)]TJ 0 -1.1988 Td [(son)24(t)-228(conserv)54(\266)475(ees)-228(si)-228(l'on)-222(fait)-228(un)-228(c)24(hangemen)18(t)-222(d'indices)-228(bijectif.)-409(En)-228(particulier,)-246(on)-228(en)-222(d)24(\266)475(eduit)-222(que)]TJ 0 -1.1988 Td [(les)-331(s)24(\266)469(eries)-331(absolumen)24(t)-331(con)24(v)24(ergen)24(tes)-337(son)24(t)-337(comm)24(utativ)30(emen)24(t)-337(con)30(v)24(ergen)18(tes)-331(\(sans)-337(c)24(hangemen)18(t)]TJ 0 -1.2048 Td [(de)-331(la)-337(somme\).)]TJ 0 -2.1988 Td [(La)-259(prop)-36(osition)-259(suiv)48(an)30(te)-265(mon)24(tre)-259(que,)-277(p)-30(our)-259(l')24(\266)469(etude)-259(d'une)-265(famille)-259(sommable,)-277(on)-259(p)-30(eut)-265(toujours)]TJ 0 -1.1988 Td [(se)-331(ramener)-337(au)-337(cas)-337(d'une)-337(famille)-331(d)24(\266)469(enom)24(brable.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 28.022 Td [(I)-30(I.C.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1506 33.593 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)18(oposition.)]TJ /F12 1 Tf 7.2952 0 Td [(Soit)]TJ /F2 1 Tf 2.1928 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.543 47.857 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.681 33.593 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.578 47.857 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 23.819 33.593 Td [(une)-469(famille)-469(sommable,)-506(alors)-475(l'ensem)30(ble)]TJ /F8 1 Tf 18.012 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.603 48.659 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 43.102 33.593 Td [(des)]TJ -37.771 -1.2048 Td [(indices)]TJ /F8 1 Tf 3.2831 0 Td [(i)]TJ /F12 1 Tf 0.68072 0 Td [(tels)-331(que)]TJ /F8 1 Tf 3.6988 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.414 46.133 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.175 32.388 Td [(6)]TJ /F2 1 Tf [(=)-283(0)]TJ /F12 1 Tf 1.8916 0 Td [(\(app)-30(el)24(\266)469(e)-337(supp)-30(ort)-337(de)-331(la)-337(famille\))-337(est)-343(d)30(\266)469(enom)24(brable.)]TJ /F1 1 Tf -10.735 -1.8976 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4458 0 Td [(n)]TJ /F2 1 Tf 0.93976 0 Td [(en)24(tier)-343(sup)0(\266)469(erieur)-343(ou)-319(\266)475(egal)-349(\265)500(a)-343(1,)-343(p)-30(osons)]TJ /F8 1 Tf 15.916 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.5 43.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.729 30.491 Td [(=)]TJ /F6 1 Tf 1.0723 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(i)]TJ /F6 1 Tf 0.63855 0 Td [(2)]TJ /F8 1 Tf 0.96386 0 Td [(I)]TJ /F6 1 Tf 0.81325 0 Td [(j)-295(j)]TJ /F8 1 Tf 0.8494 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.44 43.418 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 40.47 30.491 Td [(j)]TJ /F8 1 Tf 0.57229 0 Td [(>)]TJ /F2 1 Tf 1.0723 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(=n)]TJ /F6 1 Tf 1.1024 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)]TJ -38.886 -1.1988 Td [(Il)-259(est)-265(clair)-265(que)]TJ /F8 1 Tf 6.3313 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.431 42.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.723 29.292 Td [(est)-265(la)-265(r)30(\266)469(eunion)-265(de)-259(la)-265(famille)-259(\(d)18(\266)475(enom)24(brable\))-265(des)]TJ /F8 1 Tf 19.837 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.224 41.702 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.542 29.292 Td [(.)-421(Il)-259(su\261t)-271(donc)-259(de)-265(prouv)30(er)]TJ -28.211 -1.7229 Td [(que)-355(c)24(haque)]TJ /F8 1 Tf 5.2289 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.741 39.237 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.892 27.569 Td [(est)-355(\257ni.)-500(P)30(our)-355(cela)-355(on)-349(remarque)]TJ /F9 1 Tf 14.054 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.284 37.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.56 27.569 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.655 39.237 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.741 27.569 Td [(j)]TJ /F11 1 Tf 0.58434 0 Td [(>)]TJ /F9 1 Tf 1.1928 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.517 37.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 63.169 52.548 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.241 27.569 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.353 39.237 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.428 27.569 Td [(j)]TJ /F11 1 Tf 0.57831 0 Td [(>)]TJ /F9 1 Tf 1.1988 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.216 37.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 72.542 52.548 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.928 27.569 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(=n)]TJ /F2 1 Tf 1.4036 0 Td [(=)]TJ 1.2048 0.6747 Td [(#)]TJ /F8 1 Tf 0.83133 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.112 40.202 Td [(n)]TJ ET 0.504 w 478.51 332.49 m 500.26 332.49 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.645 26.882 Td [(n)]TJ /F2 1 Tf 1.3313 0.68675 Td [(,)-349(d'o)-36(\265)530(u)]TJ -36.645 -2.5482 Td [(#)]TJ /F8 1 Tf 0.83133 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.4483 35.59 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.4819 25.021 Td [(est)-337(ma)-60(jor)24(\266)469(e)-337(par)]TJ /F8 1 Tf 6.7108 0 Td [(n)]TJ /F9 1 Tf 0.76506 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.56 34.099 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.572 25.021 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.931 35.59 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.753 25.021 Td [(j)]TJ /F2 1 Tf 0.60843 0 Td [(donc)-337(\257ni.)]TJ ET 512.64 297.82 m 519.84 297.82 l 519.84 306.25 l 512.64 306.25 l f BT /F5 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 3.8882 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(30)]TJ ET endstream endobj 125 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 124 0 R /ProcSet 2 0 R >> /Contents 123 0 R >> endobj 124 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 126 0 obj << /Length 8856 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F0 1 Tf 20.592 0 0 20.592 0 0 Tm 5.4895 32.208 Td [(Chapitre)-377(I)-31(I)-31(I.)-1132(La)-377(notion)-381(de)-377(mesure.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 40.334 Td [(I)-30(I)-35(I.1.)-241(Espaces)-376(mesurables.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 46.834 Td [(Nous)-349(a)24(v)24(ons)-355(vu)-355(lors)-355(des)-349(pr)24(\266)469(eliminaires)-349(que,)-361(p)-30(our)-349(construire)-355(les)-355(th)24(\266)475(eories)-355(que)-355(nous)-349(a)30(vions)-355(en)]TJ 0 -1.1988 Td [(vue,)-313(nous)-301(devions)-301(tout)-307(d'ab)-30(ord)-307(disp)-24(oser)-307(d'une)-301(mesure)-307(p)-30(our)-301(certaines)-301(parties)-307(mais)-301(que)-307(l'on)]TJ 0 -1.1988 Td [(ne)-421(p)-30(ouv)48(ait)-427(pas)-427(exiger)-427(de)-421(d)24(\266)475(e\257nir)-427(cette)-427(mesure)-427(p)-30(our)-427(toutes)-427(les)-427(parties.)-716(Il)-421(con)24(vien)24(t)-427(donc,)]TJ 0 -1.2048 Td [(en)-331(pr)24(\266)469(ealable)-331(de)-337(pr)24(\266)475(eciser)-337(les)-337(parties)-337(que)-337(nous)-337(consid)30(\266)469(ererons)-337(comme)-337(mesurables.)-451(L'ob)-54(jet)-337(de)]TJ 0 -1.1988 Td [(cette)-301(section)-295(est)-301(de)-295(d)24(\266)475(etailler)-301(les)-301(propri)24(\266)469(et)24(\266)475(es)-301(de)-295(stabilit)24(\266)475(e)-301(que)-301(l'on)-301(demandera)-301(p)-30(our)-301(l'ensem)30(ble)]TJ 0 -1.1988 Td [(de)-331(ces)-337(parties.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 31.671 Td [(I)-30(I)-35(I.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 37.967 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83132 0.084335 Td [(\266)]TJ -0.0060234 -0.084335 Td [(efinition.)]TJ /F12 1 Tf 5.3916 0 Td [(Soit)]TJ /F8 1 Tf 2.0904 0 Td [(X)]TJ /F12 1 Tf 1.2831 0 Td [(un)-373(ensem)24(ble,)-379(une)]TJ /F1 1 Tf 7.988 0 Td [(tribu)]TJ /F12 1 Tf 2.8916 0 Td [(de)-373(parties)-373(de)]TJ /F8 1 Tf 6.0663 0 Td [(X)]TJ /F12 1 Tf 1.2771 0 Td [(ou)-373(tribu)-373(sur)]TJ /F8 1 Tf 5.6867 0 Td [(X)]TJ /F12 1 Tf -38.259 -1.1988 Td [(est)-337(un)-337(ensem)24(ble)]TJ /F6 1 Tf 7.2651 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(de)-331(parties)-337(de)]TJ /F8 1 Tf 5.9518 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(v)48(\266)475(eri\257an)24(t)-337(les)-337(axiomes)-337(:)]TJ /F1 1 Tf -14.777 -1.5181 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(la)-337(partie)]TJ /F8 1 Tf 4.006 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 5.6988 0 Td [(T)]TJ /F1 1 Tf -12.506 -1.5241 Td [(2.)]TJ /F12 1 Tf 1.5602 0 Td [(le)-337(compl)24(\266)469(emen)24(taire)-337(de)-331(toute)-337(partie)-337(appartenan)24(t)-337(\265)499(a)]TJ /F6 1 Tf 21.572 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(appartien)24(t)-313(\266)475(egalemen)24(t)]TJ /F6 1 Tf 9.5663 0 Td [(T)]TJ /F1 1 Tf -33.831 -1.5181 Td [(3.)]TJ /F12 1 Tf 1.3976 0 Td [(si)]TJ /F2 1 Tf 0.92169 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.75 45.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.9398 32.207 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.776 45.875 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 11.861 32.207 Td [(est)-259(une)-253(famille)-253(d)24(\266)475(enom)24(brable)]TJ ET 0.504 w 218.95 383.32 m 285.77 383.32 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.163 32.207 Td [(\(i.e.)]TJ /F8 1 Tf 2.0843 0 Td [(I)]TJ /F12 1 Tf 0.77109 0 Td [(est)-253(d)24(\266)469(enom)30(brable\))-253(de)-253(parties)-259(appartenan)24(t)]TJ -21.687 -1.1988 Td [(\265)499(a)]TJ /F6 1 Tf 0.83133 0 Td [(T)]TJ /F12 1 Tf 0.80121 0 Td [(,)-331(alors)-337(la)-337(r)30(\266)469(eunion)-337(des)]TJ /F8 1 Tf 9.4639 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.586 44.159 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.843 31.009 Td [(appartien)24(t)-337(encore)-337(\265)500(a)]TJ /F6 1 Tf 8.8193 0 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(.)]TJ /F2 1 Tf -22.127 -1.9578 Td [(Autremen)24(t)-253(dit)-259(une)-253(tribu)-253(est)-259(un)-253(ensem)24(ble)-253(de)-253(parties)-253(qui)-253(est)-259(stable)-253(par)-253(r)30(\266)469(eunion)-253(d)24(\266)469(enom)30(brable,)]TJ 0 -1.2048 Td [(par)-331(passage)-337(au)-337(compl)24(\266)475(emen)24(taire)-337(et)-337(qui)-331(con)24(tien)24(t)-337(la)-331(partie)-343(pleine)-331(\(i.e.)]TJ /F8 1 Tf 29.699 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(lui-m)24(^)475(eme\).)]TJ -30.94 -1.5181 Td [(Compte)-475(ten)24(u)-475(de)-475(la)-469(stabilit)24(\266)469(e)-475(par)-481(passage)-475(au)-475(compl)24(\266)475(emen)24(taire,)-512(on)-481(d)30(\266)469(eduit)-475(de)-475(l'axiome)]TJ /F1 1 Tf 38.271 0 Td [(3.)]TJ /F2 1 Tf -38.271 -1.1988 Td [(qu'une)-397(tribu)-391(est)-373(\266)469(egalemen)24(t)-397(stable)-391(par)-397(in)24(tersection)-397(d)24(\266)469(enom)30(brable.)-632(De)-403(m)30(^)469(eme)-391(on)-397(d)24(\266)469(eduit)-391(de)]TJ 0 -1.2048 Td [(l'axiome)]TJ /F1 1 Tf 3.9699 0 Td [(1.)]TJ /F2 1 Tf 1.3434 0 Td [(qu'une)-337(tribu)-331(p)-30(oss)24(\265)475(ede)-337(toujours)-337(la)-337(partie)-337(vide.)]TJ -5.3133 -1.5181 Td [(Retenons)-253(en)-246(particulier)-246(qu'une)-253(tribu)-253(est)-253(stable)-253(par)-246(r)24(\266)475(eunion)-253(et)-253(par)-253(in)30(tersection)-253(\257nie)-253(et)-253(m)24(^)475(eme)]TJ 0 -1.1988 Td [(d)30(\266)469(enom)24(brable.)]TJ 0 -1.9639 Td [(P)30(our)-427(mon)24(trer)-427(qu'un)-433(ensem)24(ble)-427(de)-427(parties)-427(est)-427(une)-427(tribu)-427(on)-427(v)48(\266)475(eri\257era)-427(normalemen)24(t)-427(les)-427(ax-)]TJ 0 -1.1988 Td [(iomes)-397(de)-397(la)-397(d)30(\266)469(e\257nition,)-415(mais)-391(il)-397(est)-397(parfois)-403(plus)-391(pratique)-403(d'utiliser)-391(un)-403(autre)-397(jeu)-397(d'axiomes)]TJ -0.030121 -1.1988 Td [(\266)469(equiv)54(alen)24(ts.)-445(En)-331(v)24(oici)-337(un)-337(exemple)-337(que)-337(nous)-337(utiliserons)-331(plusieurs)-343(fois.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 11.666 Td [(I)-30(I)-35(I.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 13.985 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8614 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0241 0 Td [(X)]TJ /F12 1 Tf 1.2289 0 Td [(un)-325(ensem)24(ble)-325(et)]TJ /F6 1 Tf 6.8374 0 Td [(T)]TJ /F12 1 Tf 1.1205 0 Td [(un)-325(ensem)24(ble)-325(de)-325(parties)-325(de)]TJ /F8 1 Tf 11.602 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-439(P)24(our)]TJ -36.343 -1.2048 Td [(que)]TJ /F6 1 Tf 1.8614 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(soit)-331(une)-337(tribu)-337(il)-331(faut)-343(et)-331(il)-337(su\261t)-337(qu'il)-337(v)54(\266)469(eri\257e)-331(:)]TJ /F1 1 Tf -2.1807 -1.5181 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(la)-337(partie)]TJ /F8 1 Tf 4.006 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 5.6988 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(;)]TJ /F1 1 Tf -13.639 -1.5181 Td [(2.)]TJ /F12 1 Tf 1.5602 0 Td [(le)-337(compl)24(\266)469(emen)24(taire)-337(de)-331(toute)-337(partie)-337(appartenan)24(t)-337(\265)499(a)]TJ /F6 1 Tf 21.572 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(appartien)24(t)-313(\266)475(egalemen)24(t)]TJ /F6 1 Tf 9.5663 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(;)]TJ /F1 1 Tf -34.964 -1.5241 Td [(3.)]TJ /F12 1 Tf 1.5602 0 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(B)]TJ /F12 1 Tf 1.1386 0 Td [(appartiennen)18(t)-331(\265)500(a)]TJ /F6 1 Tf 7.259 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(alors)]TJ /F8 1 Tf 2.3976 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(B)]TJ /F12 1 Tf 1.1446 0 Td [(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 5.6988 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(;)]TJ /F1 1 Tf -26.584 -1.5181 Td [(4.)]TJ /F12 1 Tf 1.5602 0 Td [(p)-30(our)-337(toute)-331(suite)-337(croissan)24(te)-331(d')24(\266)469(el)24(\266)469(emen)30(ts)]TJ /F8 1 Tf 16.898 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.276 9.3745 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.229 6.7015 Td [(de)]TJ /F6 1 Tf 1.3373 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(la)-331(r)24(\266)475(eunion)]TJ /F6 1 Tf 4.7229 0 Td [([)]TJ /F8 1 Tf 0.66867 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.862 9.3745 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 35.717 6.7015 Td [(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 5.7048 0 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(.)]TJ /F2 1 Tf -17.97 -2.8133 Td [(31)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 128 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 127 0 R /ProcSet 2 0 R >> /Contents 126 0 R >> endobj 127 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 129 0 obj << /Length 12233 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ 0 -2.4036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Les)-283(\266)475(enonc)24(\266)469(es)]TJ /F1 1 Tf 5.4217 0 Td [(1.)]TJ /F2 1 Tf 1.3313 0 Td [(et)]TJ /F1 1 Tf 1.1446 0 Td [(2.)]TJ /F2 1 Tf 1.3313 0 Td [(son)18(t)-313(les)-313(m)24(^)475(emes)-313(que)-319(ceux)-313(de)-307(la)-313(d)24(\266)475(e\257nition,)-319(ils)-313(son)18(t)]TJ -18.048 -1.1988 Td [(donc)-331(n)30(\266)469(ecessaires)-325(p)-30(our)-331(que)]TJ /F6 1 Tf 11.614 0 Td [(T)]TJ /F2 1 Tf 1.1265 0 Td [(soit)-331(une)-331(tribu.)-445(L')24(\266)469(enonc)24(\266)475(e)]TJ /F1 1 Tf 10.843 0 Td [(3.)]TJ /F2 1 Tf 1.3434 0 Td [(de)-325(la)-331(d)30(\266)469(e\257nition)-331(en)24(tra)108(^)391(\263ne)-331(l')24(\266)475(enonc)24(\266)475(e)]TJ /F1 1 Tf -24.928 -1.1988 Td [(3.)]TJ /F2 1 Tf 1.3795 0 Td [(de)-349(cette)-355(prop)-30(osition)-355(par)-349(passage)-349(compl)24(\266)475(emen)24(taire)-355(et)-349(l')30(\266)469(enonc)24(\266)475(e)]TJ /F1 1 Tf 26.771 0 Td [(4.)]TJ /F2 1 Tf 1.3855 0 Td [(qui)-343(n'en)-355(est)-349(qu'un)-355(cas)]TJ -29.536 -1.2048 Td [(particulier)-331(;)-337(ceux-ci)-337(son)24(t)-337(donc)-307(\266)469(egalemen)24(t)-337(n)30(\266)469(ecessaires.)]TJ 0 -1.6265 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-385(l')24(\266)469(enonc)30(\266)469(e)]TJ /F1 1 Tf 11.639 0 Td [(3.)]TJ /F2 1 Tf 1.4518 0 Td [(de)-373(la)-373(prop)-30(osition)-379(en)30(tra)102(^)391(\263ne,)-385(par)-373(passage)-373(au)-373(compl)24(\266)469(emen)30(taire,)]TJ -13.09 -1.2048 Td [(la)-295(stabilit)24(\266)475(e)-301(de)]TJ /F6 1 Tf 6.1687 0 Td [(T)]TJ /F2 1 Tf 1.0964 0 Td [(par)-295(r)24(\266)475(eunion)-301(de)-295(deux)-271(\266)475(el)18(\266)475(emen)24(ts)-295(et)-301(donc)-295(par)-295(r)24(\266)475(eunion)-301(\257nie)-301(\(r)24(\266)475(ecurrence)-301(imm)24(\266)469(e-)]TJ -7.2651 -1.1988 Td [(diate\).)-572(P)30(ar)-373(ailleurs,)-385(si)-379(l'on)-379(consid)30(\265)469(ere)-379(une)-373(famille)-379(d)30(\266)469(enom)24(brable)-379(\()]TJ /F8 1 Tf 28.096 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.914 77.9 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.723 54.587 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.25 77.9 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.078 54.587 Td [(d')24(\266)475(el)18(\266)475(emen)24(ts)-373(de)]TJ /F6 1 Tf 6.3373 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(,)]TJ -38.886 -1.4036 Td [(p)-30(our)-349(c)24(haque)-343(en)24(tier)]TJ /F8 1 Tf 8.5482 0 Td [(n)]TJ /F2 1 Tf 0.94578 0 Td [(la)-349(partie)]TJ /F8 1 Tf 4.0301 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.069 75.9 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.464 53.183 Td [(=)]TJ /F9 1 Tf 1.3012 0.95181 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.828 74.392 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.271 53.183 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.371 75.9 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.861 53.183 Td [(est)-349(encore)-349(un)-325(\266)469(el)24(\266)475(emen)24(t)-349(de)]TJ /F6 1 Tf 11.217 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)-481(Les)]TJ /F8 1 Tf 2.5723 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.543 75.9 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.102 53.183 Td [(formen)24(t)]TJ -35.771 -2.2289 Td [(alors)-325(une)-325(suite)-319(croissan)24(te)-325(don)24(t)-325(la)-325(r)30(\266)469(eunion)-325(est)-325(la)-325(m)24(^)475(eme)-325(que)-325(celle)-325(des)]TJ /F8 1 Tf 29.542 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.983 72.702 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.169 50.954 Td [(.)-439(L')18(\266)475(enonc)24(\266)469(e)]TJ /F1 1 Tf 4.8916 0 Td [(4.)]TJ /F2 1 Tf 1.3313 0 Td [(de)-325(la)]TJ -37.06 -1.1988 Td [(prop)-30(osition)-319(implique)-319(donc)-319(que)-319(la)-313(r)30(\266)469(eunion)-319(des)]TJ /F8 1 Tf 19.922 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.207 70.987 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.861 49.756 Td [(est)-319(encore)-319(un)-289(\266)469(el)24(\266)469(emen)24(t)-319(de)]TJ /F6 1 Tf 11.06 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)-433(L')18(\266)475(enonc)24(\266)469(e)]TJ /F1 1 Tf 4.8795 0 Td [(3.)]TJ /F2 1 Tf -38.271 -1.2048 Td [(de)-331(la)-337(d)30(\266)469(e\257nition)-337(est)-337(donc)-337(v)54(\266)469(eri\257)24(\266)475(e)-337(et)]TJ /F6 1 Tf 14.958 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(est)-337(bien)-337(une)-337(tribu.)]TJ ET 512.64 579.13 m 519.84 579.13 l 519.84 587.55 l 512.64 587.55 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 46.292 Td [(Exemples.)]TJ /F2 1 Tf 6.1084 0 Td [(L'ensem)24(ble)-337(de)-331(toutes)-337(les)-337(parties)-337(de)]TJ /F8 1 Tf 15.572 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(est)-337(une)-337(tribu.)]TJ -22.922 -1.6265 Td [(L'ensem)24(ble)]TJ /F6 1 Tf 5.1566 0 Td [(B)]TJ /F2 1 Tf 1.0241 0 Td [(des)-337(parties)-337(bien)-331(d)24(\266)469(e\257nies)-337(au)-337(sens)-331(de)-337(Borel)-337(\(cf.)]TJ /F1 1 Tf 20.114 0 Td [(I.4.2.)]TJ /F2 1 Tf 2.5422 0 Td [(\))-337(est)-337(une)-337(tribu.)]TJ -28.837 -1.6325 Td [(L'ensem)24(ble)]TJ /F6 1 Tf 5.1566 0 Td [(L)]TJ /F2 1 Tf 1.0241 0 Td [(des)-337(parties)-337(mesurables)-337(au)-337(sens)-331(de)-337(Leb)-30(esgue)-337(\(cf.)]TJ /F1 1 Tf 21.181 0 Td [(I.4.3.)]TJ /F2 1 Tf 2.5482 0 Td [(\))-337(est)-337(une)-337(tribu.)]TJ -29.91 -1.9458 Td [(La)-337(notion)-337(de)-331(tribu)-337(engendr)24(\266)475(ee)-337(nous)-337(fournira)-337(d'autres)-337(exemples.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 31.188 Td [(I)-30(I)-35(I.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 37.388 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83132 0.084335 Td [(\266)]TJ -0.0060234 -0.084335 Td [(efinition.)]TJ /F12 1 Tf 5.2349 0 Td [(Un)]TJ /F1 1 Tf 1.5301 0 Td [(espace)-259(mesurable)]TJ /F12 1 Tf 8.8012 0 Td [(est)-228(un)-228(couple)]TJ /F2 1 Tf 5.7892 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F12 1 Tf 0.60843 0 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.2831 0 Td [(X)]TJ /F12 1 Tf 1.1325 0 Td [(est)-228(un)-228(ensem)24(ble)]TJ -32.452 -1.1988 Td [(et)]TJ /F6 1 Tf 1.2349 0 Td [(T)]TJ /F12 1 Tf 1.1988 0 Td [(est)-409(une)-403(tribu)-403(de)-403(parties)-403(de)]TJ /F8 1 Tf 12.325 0 Td [(X)]TJ /F12 1 Tf 0.90362 0 Td [(.)-650(Les)-403(parties)-403(de)]TJ /F8 1 Tf 7.5542 0 Td [(X)]TJ /F12 1 Tf 1.3072 0 Td [(qui)-403(appartiennen)24(t)-409(\265)499(a)]TJ /F6 1 Tf 9.1626 0 Td [(T)]TJ /F12 1 Tf 1.1988 0 Td [(son)24(t)-403(alors)]TJ -34.886 -1.1988 Td [(dites)]TJ /F6 1 Tf 2.512 0 Td [(T)]TJ /F1 1 Tf 0.8012 0 Td [(-mesurables)]TJ /F12 1 Tf 5.9337 0 Td [(,)-481(ou)-457(mesurables)-457(p)-30(our)]TJ /F6 1 Tf 10 0 Td [(T)]TJ /F12 1 Tf 1.253 0 Td [(ou)-457(simplemen)24(t)-457(mesurables)-457(si)]TJ /F6 1 Tf 13.313 0 Td [(T)]TJ /F12 1 Tf 1.253 0 Td [(p)-30(eut)-427(^)469(etre)]TJ -35.066 -1.2048 Td [(sous-en)24(tendu.)]TJ /F1 1 Tf 0 -1.9458 Td [(Remarque.)]TJ /F2 1 Tf 6.2229 0 Td [(Cette)-319(terminologie)-319(p)-30(eut)-313(pr)24(^)469(eter)-313(\265)499(a)-313(confusion)-313(et)-319(on)-313(prendra)-319(garde)-313(\265)500(a)-313(distinguer)]TJ -6.2229 -1.1988 Td [(la)-331(notion)-337(d'espace)-337(mesurable)-337(de)-331(celle)-337(de)-337(partie)-337(mesurable.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 21.811 Td [(I)-30(I)-35(I.2.)-241(Op)-10(\266)537(erations)-376(sur)-376(les)-376(tribus.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 24.521 Td [(Il)-271(s'agit)-271(de)-271(d)24(\266)469(ecrire)-271(quelques)-271(op)-6(\266)475(erations)-277(classiques)-271(qui)-271(p)-30(ermetten)24(t)-271(de)-277(construire)-271(de)-271(nouv)24(elles)]TJ 0 -1.2048 Td [(tribus)-337(\265)500(a)-331(partir)-337(de)-331(tribus)-337(donn)18(\266)475(ees.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 16.892 Td [(I)-30(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 20.25 Td [(1.)-289(T)90(race)-391(d'une)-385(tribu.)]TJ /F2 1 Tf -3.006 -1.6325 Td [(Soien)24(t)]TJ /F6 1 Tf 3.1024 0 Td [(T)]TJ /F2 1 Tf 1.2048 0 Td [(une)-415(tribu)-409(de)-403(parties)-415(de)]TJ /F8 1 Tf 10.723 0 Td [(X)]TJ /F2 1 Tf 1.3193 0 Td [(et)]TJ /F8 1 Tf 1.241 0 Td [(Y)]TJ /F2 1 Tf 1.2108 0 Td [(une)-409(partie)-415(de)]TJ /F8 1 Tf 6.3434 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(.)-668(Il)-409(est)-415(facile)-409(de)-409(mon)24(trer)-409(que)]TJ -26.048 -1.1988 Td [(l'ensem)30(ble)-343(des)]TJ /F8 1 Tf 6.5482 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.89759 0 Td [(Y)]TJ /F2 1 Tf 1.1386 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(A)]TJ /F2 1 Tf 1.0904 0 Td [(parcouran)24(t)]TJ /F6 1 Tf 5.0964 0 Td [(T)]TJ /F2 1 Tf 1.1386 0 Td [(est)-343(une)-343(tribu)-337(de)-337(parties)-343(de)]TJ /F8 1 Tf 11.94 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(.)-463(On)-337(l'app)-36(elle)-343(la)]TJ /F1 1 Tf -31.994 -1.1988 Td [(trace)]TJ /F2 1 Tf 2.8494 0 Td [(de)]TJ /F6 1 Tf 1.3373 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(Y)]TJ /F2 1 Tf 1.1386 0 Td [(ou)-331(la)]TJ /F1 1 Tf 2.5 0 Td [(tribu)-385(induite)]TJ /F2 1 Tf 6.7651 0 Td [(par)]TJ /F6 1 Tf 1.7831 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(Y)]TJ /F2 1 Tf 1.1386 0 Td [(et)-337(on)-331(la)-337(note)]TJ /F6 1 Tf 5.8916 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.931 22.995 Td [(Y)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.584 16.22 Td [(.)]TJ -30.253 -1.6325 Td [(Lorsque)]TJ /F8 1 Tf 3.8133 0 Td [(Y)]TJ /F2 1 Tf 1.1807 0 Td [(est)]TJ /F6 1 Tf 1.6024 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable,)-391(il)-373(est)-385(clair)-373(que)-379(la)-373(tribu)-379(induite)-373(est)-379(form)24(\266)469(ee)-373(par)-379(les)-379(parties)-373(de)]TJ /F8 1 Tf -7.3916 -1.1988 Td [(Y)]TJ /F2 1 Tf 1.1325 0 Td [(qui)-337(son)24(t)]TJ /F6 1 Tf 3.8434 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurables.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 8.6052 Td [(I)-30(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 10.316 Td [(2.)-289(Image)-385(directe)-385(et)-385(r)18(\266)548(ecipro)-36(que)-385(d'une)-391(tribu.)]TJ /F2 1 Tf -3.006 -1.6265 Td [(Soit)]TJ /F8 1 Tf 1.9578 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(:)]TJ /F8 1 Tf 0.55422 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(!)]TJ /F8 1 Tf 1.2771 0 Td [(Y)]TJ /F2 1 Tf 1.0422 0 Td [(une)-234(application.)-421(Si)]TJ /F6 1 Tf 8.3916 0 Td [(T)]TJ /F2 1 Tf 1.0361 0 Td [(est)-240(une)-240(tribu)-240(sur)]TJ /F8 1 Tf 7.247 0 Td [(X)]TJ /F2 1 Tf 1.1446 0 Td [(on)-240(v)48(\266)475(eri\257e)-240(ais)24(\266)469(emen)30(t)-240(que)-246(l'ensem)30(ble)]TJ -24.717 -1.1988 Td [(des)-337(parties)]TJ /F8 1 Tf 5.012 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(Y)]TJ /F2 1 Tf 1.1325 0 Td [(telles)-343(que)]TJ /F8 1 Tf 4.4277 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.078 11.306 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.994 7.4906 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-331(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 6.4217 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(est)-337(une)-337(tribu)-331(sur)]TJ /F8 1 Tf 7.6265 0 Td [(Y)]TJ /F2 1 Tf 0.80723 0 Td [(;)-331(on)-337(dit)-337(que)-337(c'est)]TJ -31.795 -1.2048 Td [(l')]TJ /F1 1 Tf 0.55422 0 Td [(image)-385(directe)]TJ /F2 1 Tf 7.1024 0 Td [(de)-337(la)-331(tribu)]TJ /F6 1 Tf 4.9518 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(par)]TJ /F8 1 Tf 1.7831 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)]TJ /F5 1 Tf -16.12 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(32)]TJ ET endstream endobj 131 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 130 0 R /ProcSet 2 0 R >> /Contents 129 0 R >> endobj 130 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R /F16 81 0 R /F17 97 0 R >> endobj 132 0 obj << /Length 15821 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F2 1 Tf 0 -2.4036 Td [(De)-265(m)24(^)475(eme,)-277(si)]TJ /F6 1 Tf 5.512 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.664 89.633 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.187 62.22 Td [(est)-265(une)-265(tribu)-259(sur)]TJ /F8 1 Tf 7.3373 0 Td [(Y)]TJ /F2 1 Tf 1.0663 0 Td [(on)-265(v)54(\266)469(eri\257e)-259(que)-265(l'ensem)24(ble)-259(des)]TJ /F8 1 Tf 12.38 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.034 89.633 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.645 62.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-265(p)-30(our)]TJ /F8 1 Tf 2.9398 0 Td [(A)]TJ /F2 1 Tf 1.012 0 Td [(parcouran)18(t)]TJ /F6 1 Tf -34.404 -1.1988 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7672 87.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.741 61.021 Td [(est)-343(une)-331(tribu)-337(sur)]TJ /F8 1 Tf 7.6325 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(;)-331(on)-337(dit)-337(que)-337(c'est)-343(l')]TJ /F1 1 Tf 8.259 0 Td [(image)-385(r)24(\266)548(ecipro)-36(que)]TJ /F2 1 Tf 8.9819 0 Td [(de)-337(la)-331(tribu)]TJ /F6 1 Tf 4.9518 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.767 87.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.886 61.021 Td [(par)]TJ /F8 1 Tf 1.7831 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 48.379 Td [(I)-30(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 57.997 Td [(3.)-289(In)30(tersection)-385(de)-391(tribus.)]TJ /F2 1 Tf -3.006 -1.6024 Td [(Soit)]TJ /F8 1 Tf 2.0361 0 Td [(X)]TJ /F2 1 Tf 1.2229 0 Td [(un)-319(ensem)24(ble.)-445(P)30(armi)-319(les)-319(ensem)24(bles)-319(de)-319(parties)-319(de)]TJ /F8 1 Tf 21.012 0 Td [(X)]TJ /F2 1 Tf 1.2229 0 Td [(les)-319(tribus)-319(son)24(t)-319(caract)24(\266)469(eris)24(\266)475(ees)-319(par)]TJ -25.494 -1.1988 Td [(des)-427(conditions)-421(de)-421(stabilit)24(\266)475(e)-433(sous)-421(certaines)-421(op)-6(\266)475(erations.)-716(Il)-421(est)-433(donc)-421(clair)-427(que)-427(l'in)24(tersection)]TJ 0 -1.1988 Td [(d'une)-337(famille)-337(de)-331(tribus)-337(sur)]TJ /F8 1 Tf 11.88 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(est)-343(encore)-337(une)-337(tribu)-331(sur)]TJ /F8 1 Tf 10.747 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -24.777 -1.6024 Td [(Ce)-337(r)30(\266)469(esultat)-337(a)-337(l'imp)-24(ortan)18(te)-331(application)-343(suiv)54(an)24(te.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 41.183 Td [(I)-30(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 49.37 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.4036 0.084339 Td [(\266)]TJ -0.0060244 -0.084339 Td [(eor)]TJ 1.8012 0.084339 Td [(\265)]TJ -0.0060244 -0.084339 Td [(eme.)]TJ /F12 1 Tf 2.7651 0 Td [(Etan)24(t)-361(donn)18(\266)475(es)-361(un)-367(ensem)24(ble)]TJ /F8 1 Tf 11.97 0 Td [(X)]TJ /F12 1 Tf 1.2651 0 Td [(et)-361(un)-367(ensem)24(ble)]TJ /F6 1 Tf 6.9518 0 Td [(S)]TJ /F12 1 Tf 1.0422 0 Td [(de)-355(parties)-367(de)]TJ /F8 1 Tf 6.0301 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)]TJ -38.886 -1.2048 Td [(l'in)24(tersection)-265(de)-265(toutes)-265(les)-271(tribus)-265(sur)]TJ /F8 1 Tf 15.873 0 Td [(X)]TJ /F12 1 Tf 1.1687 0 Td [(qui)-265(con)30(tiennen)24(t)]TJ /F6 1 Tf 6.9458 0 Td [(S)]TJ /F12 1 Tf 0.94578 0 Td [(est)-265(encore)-271(une)-265(tribu)-265(sur)]TJ /F8 1 Tf 10.392 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-421(C'est)]TJ -36.235 -1.1988 Td [(la)-331(plus)-337(p)-30(etite)-337(tribu)-337(qui)-331(con)24(tienne)]TJ /F6 1 Tf 14.765 0 Td [(S)]TJ /F12 1 Tf 0.68072 0 Td [(,)-331(on)-337(dit)-337(que)-337(c'est)-343(la)]TJ /F1 1 Tf 8.8133 0 Td [(tribu)-385(engendr)18(\266)548(ee)]TJ /F12 1 Tf 8.3133 0 Td [(par)]TJ /F6 1 Tf 1.7831 0 Td [(S)]TJ /F12 1 Tf 0.68072 0 Td [(.)]TJ /F1 1 Tf -35.036 -1.8976 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(En)-265(e\256et,)-277(cette)-265(in)24(tersection)]TJ /F6 1 Tf 11.59 0 Td [(T)]TJ /F2 1 Tf 1.0602 0 Td [(est)-271(une)-265(tribu)-265(et)-265(il)-265(est)-265(clair)-265(qu'elle)-265(p)-30(oss)24(\265)475(ede)]TJ -21.47 -1.2048 Td [(tous)-271(les)-240(\266)475(el)18(\266)475(emen)24(ts)-265(de)]TJ /F6 1 Tf 8.7831 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-421(P)24(ar)-265(ailleurs,)-283(si)]TJ /F6 1 Tf 7.1325 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.526 63.366 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.277 43.864 Td [(est)-271(une)-271(tribu)-265(qui)-271(con)30(tien)24(t)]TJ /F6 1 Tf 11.151 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(,)-283(c'est)-271(l'une)-271(des)-271(tribus)]TJ -29.777 -1.1988 Td [(don)24(t)-385(on)-391(prend)-385(l'in)24(tersection)-391(p)-30(our)-385(d)24(\266)475(e\257nir)]TJ /F6 1 Tf 18.253 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(,)-403(elle)-391(con)30(tien)24(t)-391(donc)]TJ /F6 1 Tf 8.8373 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)-602(On)-385(conclut)-391(que)]TJ /F6 1 Tf 8.0663 0 Td [(T)]TJ /F2 1 Tf 1.1807 0 Td [(est)]TJ -37.934 -1.1988 Td [(bien)-337(la)-331(plus)-337(p)-30(etite)-337(des)-337(tribus)-337(con)30(tenan)18(t)]TJ /F6 1 Tf 17.5 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)]TJ ET 512.64 494.38 m 519.84 494.38 l 519.84 502.81 l 512.64 502.81 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 39.563 Td [(Ainsi,)-409(une)-391(partie)-397(est)-397(mesurable)-391(p)-30(our)-391(la)-391(tribu)-397(engendr)24(\266)475(ee)-397(par)]TJ /F6 1 Tf 26.97 0 Td [(S)]TJ /F2 1 Tf 1.0723 0 Td [(si)-397(et)-391(seulemen)30(t)-397(si)-391(elle)-397(est)]TJ -28.042 -1.1988 Td [(mesurable)-331(p)-30(our)-337(toutes)-337(les)-337(tribus)-337(qui)-337(con)30(tiennen)24(t)]TJ /F6 1 Tf 21.542 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)]TJ -22.223 -1.6024 Td [(On)-409(retiendra)-403(que)-409(si)]TJ /F6 1 Tf 9.1084 0 Td [(T)]TJ /F2 1 Tf 1.2048 0 Td [(est)-409(la)-409(tribu)-403(engendr)24(\266)469(ee)-409(par)]TJ /F6 1 Tf 11.994 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(,)-421(non)-409(seulemen)24(t)]TJ /F6 1 Tf 7.4337 0 Td [(T)]TJ /F2 1 Tf 1.2108 0 Td [(est)-409(une)-403(tribu)-409(qui)]TJ -31.633 -1.1988 Td [(con)30(tien)24(t)]TJ /F6 1 Tf 3.8313 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(,)-337(mais)]TJ /F6 1 Tf 2.9518 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(est)-343(con)30(ten)24(ue)-337(dans)-331(toute)-337(tribu)-331(qui)-337(con)30(tien)18(t)]TJ /F6 1 Tf 18.717 0 Td [(S)]TJ /F2 1 Tf 0.68675 0 Td [(.)]TJ -28 -1.6024 Td [(En)-331(application)-337(de)-337(ce)-337(princip)-30(e)-343(prouv)30(ons)-337(par)-337(exemple)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 25.806 Td [(I)-30(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 30.936 Td [(5.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8554 0 Td [(Soien)24(t)]TJ /F8 1 Tf 2.994 0 Td [(X)]TJ /F12 1 Tf 1.2048 0 Td [(un)-301(ensem)24(ble,)]TJ /F8 1 Tf 5.9157 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.017 44.866 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 28.542 30.936 Td [(une)-301(partie)-301(de)]TJ /F8 1 Tf 6.006 0 Td [(X)]TJ /F12 1 Tf 1.2048 0 Td [(et)]TJ /F6 1 Tf 1.1325 0 Td [(S)]TJ /F12 1 Tf 0.98193 0 Td [(un)-295(ensem)18(ble)-295(de)]TJ -32.536 -1.1988 Td [(parties)-325(de)]TJ /F8 1 Tf 4.5964 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-439(Notons)]TJ /F6 1 Tf 4.1325 0 Td [(T)]TJ /F12 1 Tf 1.1205 0 Td [(la)-325(tribu)-319(sur)]TJ /F8 1 Tf 5.259 0 Td [(X)]TJ /F12 1 Tf 1.2289 0 Td [(engendr)24(\266)475(ee)-331(par)]TJ /F6 1 Tf 6.4277 0 Td [(S)]TJ /F12 1 Tf 1.006 0 Td [(et)]TJ /F6 1 Tf 1.1566 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.75 43.142 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.572 29.738 Td [(la)-319(trace)-325(sur)]TJ /F8 1 Tf 5.259 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.431 43.142 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 39.337 29.738 Td [(de)]TJ /F6 1 Tf 1.3253 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(.)-439(Alors)]TJ /F6 1 Tf -36.133 -1.1988 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7672 41.426 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 6.741 28.539 Td [(est)-343(aussi)-331(la)-337(tribu)-337(engendr)24(\266)475(ee)-337(par)-337(l'ensem)30(ble)]TJ /F6 1 Tf 18.904 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.672 41.426 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.94 28.539 Td [(des)]TJ /F8 1 Tf 1.7289 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.991 41.426 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.054 28.539 Td [(p)-24(our)]TJ /F8 1 Tf 2.3614 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(parcouran)24(t)]TJ /F6 1 Tf 5.0904 0 Td [(S)]TJ /F12 1 Tf 0.68073 0 Td [(.)]TJ /F1 1 Tf -35.94 -1.9036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Notons)]TJ /F6 1 Tf 3.3434 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.172 38.711 Td [(0)]TJ /F4 1 Tf -0.36207 -0.94828 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.825 26.635 Td [(la)-253(tribu)-253(engendr)24(\266)469(ee)-253(par)]TJ /F6 1 Tf 9.747 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.862 38.711 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.53 26.635 Td [(.)-421(P)30(ar)-253(d)24(\266)475(e\257nition)-259(la)-253(tribu)]TJ /F6 1 Tf 10.313 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.164 38.711 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.175 26.635 Td [(p)-30(oss)24(\265)475(ede)]TJ -35.843 -1.1988 Td [(tous)-337(les)-307(\266)475(el)18(\266)475(emen)24(ts)-331(de)]TJ /F6 1 Tf 9.0482 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.552 36.987 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.337 25.436 Td [(,)-337(par)-337(suite)]TJ /F6 1 Tf 4.7892 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.948 36.987 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.542 25.436 Td [(con)30(tien)18(t)]TJ /F6 1 Tf 3.8374 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.457 36.987 Td [(0)]TJ /F4 1 Tf -0.36207 -0.93965 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.458 25.436 Td [(.)]TJ -21.127 -1.6024 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-367(l'ensem)24(ble)]TJ /F6 1 Tf 12.578 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 26.414 33.892 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.259 23.834 Td [(des)-361(parties)]TJ /F8 1 Tf 5.0663 0 Td [(B)]TJ /F2 1 Tf 1.1687 0 Td [(de)]TJ /F8 1 Tf 1.3554 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(telles)-361(que)]TJ /F8 1 Tf 4.4699 0 Td [(B)]TJ /F6 1 Tf 1.0482 0 Td [(\\)]TJ /F8 1 Tf 0.90361 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.724 34.693 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.084 23.834 Td [(appartien)24(t)-355(\265)500(a)]TJ /F6 1 Tf 5.747 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61 34.693 Td [(0)]TJ /F4 1 Tf -0.36207 -0.93966 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 23.834 Td [(est)]TJ -37.934 -1.1988 Td [(une)-409(tribu)-403(sur)]TJ /F8 1 Tf 6.2892 0 Td [(X)]TJ /F2 1 Tf 1.3133 0 Td [(\(c'est)-409(l'image)-409(directe)-409(de)]TJ /F6 1 Tf 11.03 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.431 32.978 Td [(0)]TJ /F4 1 Tf -0.36207 -0.93966 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.452 22.635 Td [(par)-409(l'injection)-403(de)]TJ /F8 1 Tf 7.9759 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.129 32.978 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.018 22.635 Td [(dans)]TJ /F8 1 Tf 2.4157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\).)-668(Comme)]TJ /F6 1 Tf 5.0723 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 32.978 Td [(0)]TJ /F4 1 Tf -0.37069 -0.93966 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 21.43 Td [(con)30(tien)24(t)]TJ /F6 1 Tf 3.8313 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.086 31.262 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.127 21.43 Td [(,)]TJ /F6 1 Tf 0.61446 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.147 30.452 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.066 21.43 Td [(p)-30(oss)30(\265)469(ede)-337(tous)-337(les)-307(\266)469(el)24(\266)469(emen)24(ts)-331(de)]TJ /F6 1 Tf 12.705 0 Td [(S)]TJ /F2 1 Tf 1.012 0 Td [(donc)-337(con)24(tien)24(t)-337(la)-331(tribu)]TJ /F6 1 Tf 9.8434 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(qu'ils)-337(engendren)24(t.)]TJ -31.428 -1.1988 Td [(Ainsi)-331(p)-30(our)]TJ /F8 1 Tf 4.9458 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(dans)]TJ /F6 1 Tf 2.3313 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(on)-331(a)]TJ /F8 1 Tf 2.2108 0 Td [(B)]TJ /F6 1 Tf 1.0904 0 Td [(2)-277(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.147 28.737 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.446 20.232 Td [(et)-331(donc)]TJ /F8 1 Tf 3.5482 0 Td [(B)]TJ /F6 1 Tf 1.0241 0 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.353 29.547 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.361 20.232 Td [(2)-283(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.655 29.547 Td [(0)]TJ /F4 1 Tf -0.37069 -0.94828 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.386 20.232 Td [(,)-331(par)-331(suite)-331(la)-331(trace)]TJ /F6 1 Tf 8.3855 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.19 29.547 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.181 20.232 Td [(de)]TJ /F6 1 Tf 1.3253 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(sur)]TJ /F8 1 Tf 1.6687 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 29.547 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 19.033 Td [(est)-337(con)24(ten)24(ue)-331(dans)]TJ /F6 1 Tf 8.0663 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.319 27.823 Td [(0)]TJ /F4 1 Tf -0.36207 -0.93966 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.482 19.033 Td [(.)-445(Les)-331(deux)-337(inclusions)-331(obten)24(ues)-337(prouv)24(en)30(t)-337(l')24(\266)469(egalit)24(\266)475(e.)]TJ ET 512.64 226.26 m 519.84 226.26 l 519.84 234.68 l 512.64 234.68 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 12.163 Td [(I)-30(I)-35(I.3.)-241(La)-376(tribu)-376(des)-376(b)-35(or)20(\266)542(eliens.)]TJ 0 -2.5226 Td [(I)-30(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 11.557 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83132 0.084337 Td [(\266)]TJ -0.0060234 -0.084337 Td [(efinition.)]TJ /F12 1 Tf 5.259 0 Td [(Si)]TJ /F8 1 Tf 1.1205 0 Td [(X)]TJ /F12 1 Tf 1.1928 0 Td [(est)-295(un)-295(espace)-289(top)-30(ologique,)-301(la)]TJ /F1 1 Tf 12.693 0 Td [(tribu)-337(des)-337(b)-30(or)24(\266)548(eliens)]TJ /F12 1 Tf 9.6024 0 Td [(de)]TJ /F8 1 Tf 1.2892 0 Td [(X)]TJ /F12 1 Tf 1.1928 0 Td [(est)]TJ -37.934 -1.1988 Td [(la)-277(tribu)-277(engendr)24(\266)475(ee)-283(par)-277(l'ensem)30(ble)-277(des)-283(ouv)24(erts)-283(de)]TJ /F8 1 Tf 20.994 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-421(Les)-253(\266)475(el)18(\266)475(emen)24(ts)-277(de)-277(cette)-277(tribu)-277(s'app)-30(ellen)24(t)]TJ -21.904 -1.1988 Td [(les)]TJ /F1 1 Tf 1.4518 0 Td [(b)-30(or)24(\266)548(eliens)]TJ /F12 1 Tf 4.8373 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(ou)-331(les)]TJ /F1 1 Tf 2.8373 0 Td [(parties)-385(b)-36(or)24(\266)548(eliennes)]TJ /F12 1 Tf 9.8133 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)]TJ /F2 1 Tf -4.8434 -5.2711 Td [(33)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 134 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 133 0 R /ProcSet 2 0 R >> /Contents 132 0 R >> endobj 133 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 135 0 obj << /Length 15991 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I)-30(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 62.22 Td [(2.)-289(Remarques)-385(et)-391(exemples.)]TJ /F2 1 Tf -3.006 -1.5964 Td [(Si)]TJ /F8 1 Tf 1.1024 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.509 87.34 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8916 60.623 Td [(est)-277(une)-271(partie)-277(d'un)-277(espace)-271(top)-36(ologique)]TJ /F8 1 Tf 16.759 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(,)-289(la)-271(prop)-30(osition)]TJ /F1 1 Tf 6.8133 0 Td [(I)-30(I)-36(I.2.5.)]TJ /F2 1 Tf 3.753 0 Td [(mon)24(tre)-271(que)-277(la)-271(tribu)]TJ -30.789 -1.2048 Td [(des)-337(b)-30(or)30(\266)469(eliens)-337(de)]TJ /F8 1 Tf 7.2651 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.328 85.625 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.12 59.418 Td [(est)-337(la)-337(trace)-331(de)-337(la)-331(tribu)-337(des)-337(b)-30(or)30(\266)469(eliens)-337(de)]TJ /F8 1 Tf 17.392 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -27.09 -2.1988 Td [(P)30(ar)-397(d)30(\266)469(e\257nition)-397(la)-391(tribu)-397(des)-397(b)-24(or)24(\266)475(eliens)-397(p)-30(oss)24(\265)475(ede)-403(tous)-391(les)-397(ouv)24(erts)-403(de)]TJ /F8 1 Tf 28.651 0 Td [(X)]TJ /F2 1 Tf 0.90963 0 Td [(,)-409(mais)-391(la)-391(stabilit)24(\266)469(e)-397(par)]TJ -29.56 -1.1988 Td [(passage)-331(au)-337(compl)24(\266)469(emen)30(taire)-337(mon)24(tre)-337(qu'elle)-343(p)-24(oss)24(\265)475(ede)-343(aussi)-337(tous)-331(les)-337(ferm)24(\266)475(es.)]TJ 0 -1.6024 Td [(Ainsi,)-277(dans)]TJ /F3 1 Tf 5.0663 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-271(les)-259(in)24(terv)54(alles)-259(ouv)24(erts)-265(et)-259(les)-253(in)24(terv)54(alles)-259(ferm)24(\266)475(es)-259(son)24(t)-259(des)-259(b)-30(or)24(\266)475(eliens.)-427(La)-259(stabilit)24(\266)475(e)]TJ -5.7892 -1.1988 Td [(par)-295(in)24(tersection)-301(assure)-295(en)-295(fait)-301(que)-301(tous)-295(les)-301(in)30(terv)54(alles)-301(son)24(t)-301(des)-301(b)-24(or)24(\266)469(eliens)-295(de)]TJ /F3 1 Tf 32.446 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-433(La)-301(tribu)-301(des)]TJ -33.169 -1.1988 Td [(b)-30(or)30(\266)469(eliens)-337(con)30(tien)24(t)-337(donc)-337(la)-337(tribu)-331(engendr)24(\266)469(ee)-337(par)-337(les)-331(in)24(terv)54(alles)-337(de)]TJ /F3 1 Tf 28.012 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -28.735 -1.6024 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-439(soit)]TJ /F8 1 Tf 9.7831 0 Td [(U)]TJ /F2 1 Tf 1.2048 0 Td [(un)-415(ouv)24(ert)-415(de)]TJ /F3 1 Tf 6.1325 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-680(P)30(our)]TJ /F8 1 Tf 3.4699 0 Td [(a)]TJ /F2 1 Tf 0.93976 0 Td [(dans)]TJ /F8 1 Tf 2.4157 0 Td [(U)]TJ /F2 1 Tf 1.2048 0 Td [(il)-415(y)-415(a)-409(par)-415(d)24(\266)475(e\257nition)-415(un)]TJ /F8 1 Tf 10.729 0 Td [(")-409(>)]TJ /F2 1 Tf 2.0602 0 Td [(0)]TJ -38.663 -1.1988 Td [(tel)-391(que)-397(])]TJ /F8 1 Tf 3.6988 0 Td [(a)]TJ /F6 1 Tf 0.78916 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(")]TJ /F2 1 Tf 0.46988 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(a)]TJ /F2 1 Tf 0.78916 0 Td [(+)]TJ /F8 1 Tf 1.0422 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [([)]TJ /F6 1 Tf 0.66867 0 Td [(\275)]TJ /F8 1 Tf 1.1506 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(.)-614(On)-391(p)-30(eut)-391(alors)-397(trouv)24(er)-397(des)-391(rationnels)]TJ /F8 1 Tf 17.464 0 Td [(u)]TJ /F2 1 Tf 0.96386 0 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(v)]TJ /F2 1 Tf 0.90964 0 Td [(tels)-391(que)]TJ /F8 1 Tf 3.8133 0 Td [(a)]TJ /F6 1 Tf 0.78916 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(")-373(<)]TJ -37.548 -1.1988 Td [(u)-301(<)-301(a)-301(<)-301(v)-337(<)-301(a)]TJ /F2 1 Tf 6.512 0 Td [(+)]TJ /F8 1 Tf 1.012 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [(.)-487(Ainsi)]TJ /F8 1 Tf 3.3675 0 Td [(a)]TJ /F2 1 Tf 0.8735 0 Td [(appartien)24(t)-349(\265)499(a)-349(l'in)24(terv)54(alle)-349(ouv)24(ert)-349(])]TJ /F8 1 Tf 14.042 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(;)]TJ /F8 1 Tf 0.44579 0 Td [(v)]TJ /F2 1 Tf 0.51807 0 Td [([)-349(d'extr)24(\266)469(emit)24(\266)469(es)-343(rationnelles)]TJ -27.807 -1.2048 Td [(et)-403(con)24(ten)24(u)-403(dans)]TJ /F8 1 Tf 7.4398 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(.)-656(Il)-403(s'ensuit)-409(que)]TJ /F8 1 Tf 7.6024 0 Td [(U)]TJ /F2 1 Tf 1.1988 0 Td [(est)-409(la)-403(r)24(\266)475(eunion)-409(des)-409(in)24(terv)54(alles)-403(ouv)18(erts)-409(\265)500(a)-403(extr)30(\266)469(emit)24(\266)469(es)]TJ -17.036 -1.1988 Td [(rationnelles)-259(qu'il)-259(con)30(tien)24(t.)-427(En)-253(particulier)]TJ /F8 1 Tf 17.892 0 Td [(U)]TJ /F2 1 Tf 1.0482 0 Td [(est)-259(une)-259(r)24(\266)475(eunion)-259(d)24(\266)469(enom)24(brable)-259(d'in)24(terv)54(alles)-259(donc)]TJ -18.94 -1.1988 Td [(appartien)24(t)-343(\265)500(a)-349(la)-343(tribu)-349(engendr)24(\266)475(ee)-349(par)-349(les)-343(in)24(terv)54(alles.)-481(Cette)-349(tribu)-349(p)-30(oss)30(\266)469(edan)24(t)-349(tous)-343(les)-349(ouv)24(erts)]TJ 0 -1.2048 Td [(de)]TJ /F3 1 Tf 1.3313 0 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(con)24(tien)24(t)-337(la)-337(tribu)-331(qu'ils)-343(engendren)24(t,)-337(c'est-)0(\265)500(a-dire)-337(celles)-331(des)-337(b)-30(or)24(\266)475(eliens.)]TJ -2.3855 -1.5964 Td [(Nous)-331(v)18(enons)-337(donc)-337(de)-331(prouv)24(er)-337(:)]TJ /F15 1 Tf 0 -1.9036 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(La)-337(tribu)-337(des)-337(b)-30(or)30(\266)469(eliens)-337(de)]TJ /F3 1 Tf 11.229 0 Td [(R)]TJ /F12 1 Tf 1.0602 0 Td [(est)-337(engendr)24(\266)469(ee)-337(par)-331(les)-337(in)24(terv)54(alles)-337(de)]TJ /F3 1 Tf 15.53 0 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F2 1 Tf -35.41 -1.8976 Td [(De)-246(fa)0(\273)439(con)-246(analogue)-246(on)-240(mon)24(tre)-246(que)-246(la)-240(tribu)-246(des)-246(b)-30(or)30(\266)469(eliens)-246(de)]TJ /F3 1 Tf 24.904 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.302 54.711 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.747 37.816 Td [(est)-246(engendr)24(\266)475(ee)-246(par)-246(les)-240(pro)-30(duits)]TJ -26.416 -1.1988 Td [(d'in)24(terv)54(alles)-337(\(i.e.)-463(p)-30(our)]TJ /F8 1 Tf 10.06 0 Td [(n)]TJ /F2 1 Tf 0.88554 0 Td [(=)-283(2)-337(les)-343(rectangles)-337(\(resp.)-457(p)-30(our)]TJ /F8 1 Tf 13.313 0 Td [(n)]TJ /F2 1 Tf 0.88554 0 Td [(=)-283(3)-337(les)-343(parall)24(\266)475(el)18(\266)475(epip)-6(\265)475(edes\))-343(\265)500(a)-337(c)-6(^)500(ot)24(\266)469(es)]TJ -25.145 -1.1988 Td [(parall)24(\265)475(eles)-337(aux)-337(axes\).)]TJ 0 -2.1988 Td [(Concernan)24(t)-337(le)-337(cas)-337(de)]TJ ET 0.504 w 176.4 406.94 m 185.04 406.94 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 14.759 33.22 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(nous)-337(utiliserons)-337(l')30(\266)469(enonc)30(\266)469(e)-337(plus)-337(pr)24(\266)475(ecis)-337(suiv)54(an)24(t)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 25.188 Td [(I)-30(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 30.195 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.1988 0 Td [(La)-445(tribu)-439(des)-439(b)-30(or)24(\266)475(eliens)-445(de)]TJ ET 347.18 370.8 m 355.82 370.8 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 29.048 30.195 Td [(R)]TJ /F12 1 Tf 1.1627 0 Td [(est)-439(engendr)24(\266)469(ee)-439(par)-445(les)-439(in)24(terv)54(alles)]TJ /F2 1 Tf -24.88 -1.1988 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.60843 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(parcouran)24(t)]TJ /F3 1 Tf 5.0904 0 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F1 1 Tf -12.681 -1.9036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Les)-457(in)24(terv)54(alles)-457(])]TJ /F8 1 Tf 7.0542 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-463(son)24(t)-457(des)-463(ouv)24(erts)-463(de)]TJ ET 405.36 333.79 m 414 333.79 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 33.916 27.093 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-487(par)-457(suite)-463(la)-457(tribu)]TJ /F6 1 Tf 9.0602 0 Td [(T)]TJ /F2 1 Tf -38.367 -1.1988 Td [(qu'ils)-337(engendren)24(t)-337(est)-343(con)30(ten)24(ue)-337(dans)-337(la)-331(tribu)]TJ /F6 1 Tf 19.482 0 Td [(B)]TJ /F2 1 Tf 1.0241 0 Td [(engendr)24(\266)475(ee)-337(par)-337(tous)-337(les)-337(ouv)18(erts,)-331(c'est-)-6(\265)500(a-dire)]TJ -20.506 -1.1988 Td [(la)-337(tribu)-337(des)-343(b)-30(or)30(\266)469(eliens.)-463(P)30(our)-343(mon)24(trer)-337(l')24(\266)469(egalit)24(\266)475(e)-343(il)-343(su\261t)-343(donc)-337(de)-337(prouv)24(er)-343(que)]TJ /F6 1 Tf 32.633 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(con)24(tien)24(t)-343(une)]TJ -33.765 -1.2048 Td [(partie)-337(g)30(\266)469(en)24(\266)475(eratrice)-337(de)]TJ /F6 1 Tf 9.3434 0 Td [(B)]TJ /F2 1 Tf 0.68675 0 Td [(,)-337(par)-337(exemple)-337(que)]TJ /F6 1 Tf 8.1205 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(p)-30(oss)30(\265)469(ede)-337(tous)-337(les)-337(ouv)24(erts)-337(de)]TJ ET 439.06 290.73 m 447.7 290.73 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 36.735 23.491 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -32.127 -1.5964 Td [(T)78(out)-259(d'ab)-36(ord)]TJ /F6 1 Tf 5.9578 0 Td [(T)]TJ /F2 1 Tf 1.0602 0 Td [(p)-30(oss)30(\265)469(ede)-259(tous)-259(les)-259(in)24(terv)54(alles)-259(])]TJ /F8 1 Tf 11.976 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-253(p)-30(our)]TJ /F8 1 Tf 2.8193 0 Td [(a)]TJ /F2 1 Tf 0.78313 0 Td [(dans)]TJ /F3 1 Tf 2.2651 0 Td [(R)]TJ /F2 1 Tf 0.97591 0 Td [(mais)-259(aussi)-259(p)-30(our)]TJ /F8 1 Tf 6.9337 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)-277(+)]TJ /F6 1 Tf 1.8313 0 Td [(1)]TJ /F2 1 Tf -38.163 -1.2048 Td [(\(])-138(+)]TJ /F6 1 Tf 1.7169 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-277(=)]TJ /F6 1 Tf 1.6084 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(\))-295(et)-295(p)-24(our)]TJ /F8 1 Tf 4.1265 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(\2411)]TJ /F2 1 Tf 2.0663 0 Td [(car)-301(])]TJ /F6 1 Tf 2.0422 0 Td [(\241)-138(1)]TJ /F2 1 Tf 1.9157 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-289(est)-295(la)-289(r)24(\266)475(eunion)-295(de)-295(la)-289(famille)-289(d)24(\266)469(enom)24(brable)]TJ -21.283 -1.1988 Td [(des)-337(])]TJ /F6 1 Tf 2.2289 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2169 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(])-331(p)-30(our)]TJ /F8 1 Tf 2.9759 0 Td [(n)]TJ /F6 1 Tf 0.87349 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -11.572 -1.5964 Td [(P)30(ar)-427(ailleurs)-427(tout)-397(\266)469(el)24(\266)469(emen)24(t)]TJ /F8 1 Tf 11.566 0 Td [(a)]TJ /F2 1 Tf 0.95783 0 Td [(de)]TJ ET 230.4 223.77 m 239.04 223.77 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 19.277 17.894 Td [(R)]TJ /F2 1 Tf 1.1506 0 Td [(est)-433(la)-427(b)-30(orne)-421(sup)-6(\266)475(erieure)-433(des)-427(rationnels)-433(qui)-427(le)-427(minoren)18(t)]TJ -15.096 -1.2048 Td [(strictemen)24(t.)-608(Il)-391(s'ensuit)-385(que)-391(l'in)24(terv)54(alle)-391([)]TJ /F8 1 Tf 17.657 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2169 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(])-385(est)-391(l'in)24(tersection)-391(des)-391(])]TJ /F8 1 Tf 10.319 0 Td [(r)]TJ /F2 1 Tf 0.48193 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-385(p)-30(our)]TJ /F8 1 Tf 3.0843 0 Td [(r)]TJ /F2 1 Tf 0.86145 0 Td [(par-)]TJ -37.38 -1.1988 Td [(couran)24(t)-421(l'ensem)24(ble)-421(des)-421(rationnels)-427(strictemen)18(t)-421(inf)24(\266)475(erieurs)-427(\265)499(a)]TJ /F8 1 Tf 25.892 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-710(S'agissan)24(t)-421(de)-427(l'in)24(tersection)]TJ -26.422 -1.1988 Td [(d'une)-337(famille)-337(d)30(\266)469(enom)24(brable)-337(d')24(\266)475(el)24(\266)469(emen)24(ts)-331(de)]TJ /F6 1 Tf 18.157 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(on)-331(conclut)-343(que)-337([)]TJ /F8 1 Tf 7.0301 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-331(appartien)24(t)-337(\265)499(a)]TJ /F6 1 Tf 6.3072 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)]TJ -36.181 -1.6024 Td [(Ainsi)]TJ /F6 1 Tf 2.6084 0 Td [(T)]TJ /F2 1 Tf 1.1566 0 Td [(p)-30(oss)30(\265)469(ede)-361(tous)-355(les)-355(])]TJ /F8 1 Tf 7.6205 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-355(et)-361([)]TJ /F8 1 Tf 2.1024 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-355(p)-30(our)]TJ /F8 1 Tf 3.0181 0 Td [(a)]TJ /F6 1 Tf 0.84337 0 Td [(2)]TJ ET 348.55 161.56 m 357.19 161.56 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 29.163 12.689 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-506(P)24(ar)-355(passage)-355(au)-355(compl)24(\266)469(emen)24(taire)]TJ -24.554 -1.1988 Td [(on)-367(v)24(oit)-373(que)]TJ /F6 1 Tf 5.3614 0 Td [(T)]TJ /F2 1 Tf 1.1627 0 Td [(p)-30(oss)24(\265)475(ede)-379(aussi)-367(tous)-373(les)-373([)]TJ /F6 1 Tf 10.157 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(])-367(et)-373([)]TJ /F6 1 Tf 2.1265 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [([)-367(p)-30(our)]TJ /F8 1 Tf 3.0482 0 Td [(a)]TJ /F6 1 Tf 0.86747 0 Td [(2)]TJ ET 413.14 147.24 m 421.78 147.24 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 34.566 11.491 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-548(P)24(ar)-367(in)24(tersection)-373(on)]TJ -29.958 -1.2048 Td [(en)-331(d)24(\266)475(eduit)-337(que)]TJ /F6 1 Tf 6.3072 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(p)-30(oss)30(\265)469(ede)-337(tous)-337(les)-337(in)24(terv)60(alles)-337(et)-337(en)-337(particulier)-331(les)-337(singletons)]TJ /F6 1 Tf 25.428 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)]TJ -34.398 -1.5964 Td [(Reprenan)24(t)-451(l'argumen)24(t)-451(donn)18(\266)475(e)-457(pr)24(\266)475(ec)24(\266)469(edemmen)24(t)-457(p)-24(our)]TJ /F3 1 Tf 22.458 0 Td [(R)]TJ /F2 1 Tf 1.1747 0 Td [(on)-451(conclut)-457(que)]TJ /F6 1 Tf 7.1024 0 Td [(T)]TJ /F2 1 Tf 1.253 0 Td [(p)-30(oss)30(\265)469(ede)-451(tous)-457(les)]TJ -31.988 -1.1988 Td [(ouv)24(erts)-271(de)]TJ /F3 1 Tf 4.7108 0 Td [(R)]TJ /F2 1 Tf 0.98795 0 Td [(mais)-271(aussi)-271(tous)-271(les)-265(ouv)18(erts)-271(de)]TJ ET 285.7 99.428 m 294.34 99.428 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 23.904 7.4906 Td [(R)]TJ /F2 1 Tf 0.98795 0 Td [(car)-271(ceux-ci)-277(p)-30(euv)24(en)30(t)-271(s')24(\266)469(ecrire)-271(comme)-271(la)-265(r)24(\266)475(eunion)]TJ -19.56 -1.2048 Td [(d'un)-337(ouv)24(ert)-337(de)]TJ /F3 1 Tf 6.7289 0 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(et)-337(de)-331(z)24(\266)469(ero,)-331(un)-337(ou)-337(deux)-331(des)-337(singletons)]TJ /F6 1 Tf 16.5 0 Td [(f\2411g)]TJ /F2 1 Tf 3.1145 0 Td [(et)]TJ /F6 1 Tf 1.1627 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(+)]TJ /F6 1 Tf 0.78313 0 Td [(1g)]TJ /F2 1 Tf 1.5 0 Td [(.)]TJ ET 512.64 73.976 m 519.84 73.976 l 519.84 82.4 l 512.64 82.4 l f BT /F5 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 3.8882 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(34)]TJ ET endstream endobj 137 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 136 0 R /ProcSet 2 0 R >> /Contents 135 0 R >> endobj 136 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 138 0 obj << /Length 21819 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(I)-30(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 62.22 Td [(4.)-289(Le)-385(lemme)-385(des)-391(classes)-385(monotones.)]TJ /F2 1 Tf -3.006 -1.5843 Td [(Consid)30(\266)469(erons)-379(un)-373(ensem)24(ble)]TJ /F8 1 Tf 11.446 0 Td [(X)]TJ /F2 1 Tf 1.2831 0 Td [(et)]TJ /F6 1 Tf 1.2048 0 Td [(S)]TJ /F2 1 Tf 1.0542 0 Td [(un)-379(ensem)24(ble)-373(de)-373(parties)-379(de)]TJ /F8 1 Tf 11.861 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(.)-566(Dans)-379(une)-373(telle)-379(situation)]TJ -27.753 -1.1988 Td [(nous)-391(aurons)-391(souv)24(en)24(t)-391(\265)500(a)-397(prouv)30(er)-397(qu'une)-397(propri)24(\266)469(et)30(\266)469(e)-397(vraie)-397(p)-30(our)-391(les)-367(\266)469(el)24(\266)469(emen)24(ts)-391(de)]TJ /F6 1 Tf 33.687 0 Td [(S)]TJ /F2 1 Tf 1.0723 0 Td [(est)-397(encore)]TJ -34.759 -1.1988 Td [(vraie)-379(p)-24(our)-379(les)-349(\266)469(el)24(\266)469(emen)30(ts)-373(de)-379(la)-373(tribu)-379(engendr)30(\266)469(ee)-379(par)]TJ /F6 1 Tf 22.151 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(.)-572(Notan)24(t)]TJ /F6 1 Tf 4.2771 0 Td [(U)]TJ /F2 1 Tf 1.1024 0 Td [(l'ensem)30(ble)-379(des)-379(parties)-373(de)]TJ /F8 1 Tf -28.211 -1.2048 Td [(X)]TJ /F2 1 Tf 1.2169 0 Td [(a)30(y)24(an)24(t)-307(la)-313(propri)24(\266)469(et)30(\266)469(e)-313(consid)30(\266)469(er)24(\266)469(ee)-313(et)]TJ /F6 1 Tf 13.946 0 Td [(T)]TJ /F2 1 Tf 1.1084 0 Td [(la)-307(tribu)-313(engendr)24(\266)475(ee)-313(par)]TJ /F6 1 Tf 9.9699 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(,)-313(nous)-307(serons)-313(donc)-307(amen)24(\266)469(es)-307(\265)500(a)]TJ -26.922 -1.1988 Td [(prouv)30(er)]TJ /F6 1 Tf 3.6687 0 Td [(U)-379(\276)-283(T)]TJ /F2 1 Tf 3.1867 0 Td [(\265)500(a)-331(partir)-331(de)]TJ /F6 1 Tf 4.994 0 Td [(U)-385(\276)-277(S)]TJ /F2 1 Tf 2.741 0 Td [(.)-445(P)30(our)-331(cela)-337(la)-331(m)30(\266)469(etho)-30(de)-331(la)-331(plus)-331(simple)-331(consiste)-331(\265)500(a)-331(prouv)30(er)]TJ -14.59 -1.1988 Td [(que)]TJ /F6 1 Tf 1.8675 0 Td [(U)]TJ /F2 1 Tf 1.0723 0 Td [(est)-343(une)-349(tribu.)-469(En)-349(e\256et,)]TJ /F6 1 Tf 10.452 0 Td [(T)]TJ /F2 1 Tf 1.1145 0 Td [(\266)469(etan)24(t,)-349(par)-343(d)30(\266)469(e\257nition,)-349(la)-343(plus)-343(p)-30(etite)-349(tribu)-343(con)30(tenan)18(t)]TJ /F6 1 Tf 22.681 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(,)-349(si)]TJ /F6 1 Tf -37.867 -1.2048 Td [(U)]TJ /F2 1 Tf 1.0602 0 Td [(est)-337(une)-337(tribu)-331(elle)-337(est)-343(plus)-331(grande)-337(que)]TJ /F6 1 Tf 16.554 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(ce)-337(qui)-337(donne)-337(l'inclusion)-337(souhait)24(\266)469(ee.)]TJ -18.747 -1.5843 Td [(Dans)-391(de)-385(nom)24(breux)-385(cas)-385(l'ensem)24(ble)]TJ /F6 1 Tf 15.301 0 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(n'est)-385(malheureusemen)24(t)-391(pas)-391(une)-385(tribu,)-403(une)-385(m)24(\266)469(etho)-30(de)]TJ -16.416 -1.1988 Td [(e\261cace)-385(est)-391(alors)-385(d'utiliser)-385(le)]TJ /F1 1 Tf 12.886 0 Td [(lemme)-445(des)-445(classes)-439(monotones)]TJ /F2 1 Tf 15.289 0 Td [(don)24(t)-385(v)24(oici)-385(l'une)-385(des)-385(v)48(ari-)]TJ -28.175 -1.1988 Td [(an)30(tes.)]TJ /F15 1 Tf 0 -1.8675 Td [(Lemme.)]TJ /F12 1 Tf 4.0964 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.006 0 Td [(X)]TJ /F12 1 Tf 1.2169 0 Td [(un)-313(ensem)24(ble,)]TJ /F6 1 Tf 5.9398 0 Td [(S)]TJ /F12 1 Tf 0.98795 0 Td [(et)]TJ /F6 1 Tf 1.1446 0 Td [(U)]TJ /F12 1 Tf 1.0361 0 Td [(des)-313(ensem)24(bles)-313(de)-307(parties)-313(de)]TJ /F8 1 Tf 12.211 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-439(On)-307(supp)-30(ose)-313(que)]TJ /F6 1 Tf 7.9337 0 Td [(S)]TJ /F12 1 Tf -38.482 -1.2048 Td [(est)-343(stable)-337(par)-343(in)24(tersection)-337(\257nie)-343(et)-343(que)]TJ /F6 1 Tf 16.843 0 Td [(U)]TJ /F12 1 Tf 1.0602 0 Td [(con)24(tien)24(t)]TJ /F6 1 Tf 3.8434 0 Td [(S)]TJ /F12 1 Tf 0.68072 0 Td [(.)-457(Alors,)-343(p)-30(our)-337(que)]TJ /F6 1 Tf 7.9036 0 Td [(U)]TJ /F12 1 Tf 1.0663 0 Td [(con)30(tienne)-349(la)-337(tribu)]TJ -31.398 -1.1988 Td [(engendr)24(\266)475(ee)-337(par)]TJ /F6 1 Tf 6.4518 0 Td [(S)]TJ /F12 1 Tf 0.68072 0 Td [(,)-331(il)-337(su\261t)-337(que)-337(l'on)-337(ait)-337(:)]TJ /F1 1 Tf -6.3193 -1.5843 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(la)-337(partie)]TJ /F8 1 Tf 4.006 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 5.6988 0 Td [(U)]TJ /F12 1 Tf 1.0602 0 Td [(;)]TJ /F1 1 Tf -13.566 -1.5843 Td [(2.)]TJ /F12 1 Tf 1.5301 0 Td [(p)-30(our)-319(tout)]TJ /F8 1 Tf 4.5 0 Td [(A)]TJ /F12 1 Tf 1.0723 0 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(B)]TJ /F12 1 Tf 1.1265 0 Td [(dans)]TJ /F6 1 Tf 2.3253 0 Td [(U)]TJ /F12 1 Tf 0.72289 0 Td [(,)-325(si)]TJ /F8 1 Tf 1.5904 0 Td [(A)]TJ /F6 1 Tf 0.93976 0 Td [(\\)]TJ /F8 1 Tf 0.86145 0 Td [(B)]TJ /F12 1 Tf 1.1265 0 Td [(appartien)24(t)-319(\265)499(a)]TJ /F6 1 Tf 5.6687 0 Td [(U)]TJ /F12 1 Tf 0.72892 0 Td [(,)]TJ /F8 1 Tf 0.59639 0 Td [(A)]TJ /F6 1 Tf 0.93976 0 Td [(\\)]TJ /F8 1 Tf 0.86145 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.784 60.711 Td [(c)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 33.422 42.009 Td [(appartien)30(t)-295(\266)469(egalemen)24(t)-319(\265)500(a)]TJ /F6 1 Tf 10.349 0 Td [(U)]TJ /F12 1 Tf -38.44 -1.1988 Td [(\(o)-30(\265)530(u)]TJ /F8 1 Tf 1.7771 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.328 58.987 Td [(c)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6566 40.81 Td [(d)24(\266)469(esigne)-337(le)-337(compl)24(\266)475(emen)24(taire)-337(de)]TJ /F8 1 Tf 13.102 0 Td [(B)]TJ /F12 1 Tf 0.80723 0 Td [(\))-337(;)]TJ /F1 1 Tf -16.422 -1.5843 Td [(3.)]TJ /F12 1 Tf 1.5602 0 Td [(p)-30(our)-337(toute)-331(suite)-337(croissan)24(te)-331(d')24(\266)469(el)24(\266)469(emen)30(ts)]TJ /F8 1 Tf 16.898 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.276 55.918 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.229 39.226 Td [(de)]TJ /F6 1 Tf 1.3373 0 Td [(U)]TJ /F12 1 Tf 1.0542 0 Td [(la)-337(r)30(\266)469(eunion)-337(des)]TJ /F8 1 Tf 6.4578 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.276 55.918 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.705 39.226 Td [(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 5.7048 0 Td [(U)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F1 1 Tf -37.801 -1.8675 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Supp)-24(osons)-337(les)-337(propri)24(\266)469(et)24(\266)475(es)]TJ /F1 1 Tf 11.06 0 Td [(1.)]TJ /F2 1 Tf 0.89157 0 Td [(,)]TJ /F1 1 Tf 0.61446 0 Td [(2.)]TJ /F2 1 Tf 1.3373 0 Td [(et)]TJ /F1 1 Tf 1.1687 0 Td [(3.)]TJ /F2 1 Tf 1.3373 0 Td [(v)48(\266)475(eri\257)24(\266)469(ees.)]TJ -25.229 -1.5843 Td [(P)30(ar)-499(d)30(\266)469(e\257nition)-500(de)-493(la)-500(tribu)-493(engendr)24(\266)469(ee)-500(par)]TJ /F6 1 Tf 18.867 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(,)-536(celle-ci)-493(est)-499(con)24(ten)24(ue)-493(dans)-500(toute)-493(tribu)-500(qui)]TJ -19.548 -1.2048 Td [(con)30(tien)24(t)]TJ /F6 1 Tf 3.8253 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-439(P)24(our)-325(prouv)30(er)-331(ce)-331(lemme)-325(il)-325(su\261t)-325(donc)-331(de)-325(construire)-325(une)-325(tribu)-325(qui)-325(con)30(tien)18(t)]TJ /F6 1 Tf 32.819 0 Td [(S)]TJ /F2 1 Tf 1.006 0 Td [(et)]TJ -38.331 -1.1988 Td [(est)-337(con)24(ten)24(ue)-331(dans)]TJ /F6 1 Tf 8.0663 0 Td [(U)]TJ /F2 1 Tf 0.72892 0 Td [(.)]TJ -8.7952 -1.5843 Td [(In)30(tro)-36(duisons)-283(p)-30(our)-283(cela)-289(l'ensem)24(ble)]TJ /F6 1 Tf 14.741 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.698 46.073 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.313 31.786 Td [(obten)24(u)-283(en)-283(a)-54(joutan)18(t)-283(\265)500(a)]TJ /F6 1 Tf 9.3253 0 Td [(S)]TJ /F2 1 Tf 0.96386 0 Td [(l')30(\266)469(el)24(\266)469(emen)24(t)]TJ /F8 1 Tf 4.1988 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(\(s'il)-289(ne)-283(le)-283(p)-30(oss)24(\265)475(ede)]TJ -31.657 -1.1988 Td [(pas)-337(d)24(\266)475(ej)-6(\265)500(a\).)-445(Il)-337(est)-337(clair)-337(que)]TJ /F6 1 Tf 11.325 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.81 44.357 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.952 30.587 Td [(reste)-337(stable)-337(par)-331(in)24(tersection)-337(\257nie)-337(et)-337(est)-337(con)30(ten)24(u)-337(dans)]TJ /F6 1 Tf 23.41 0 Td [(U)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -36.753 -1.5843 Td [(Consid)30(\266)469(erons)-337(alors)-337(les)-337(ensem)24(bles)-337(suiv)48(an)30(ts)-337(:)]TJ /F6 1 Tf 3.9578 -1.2048 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.328 40.375 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.572 27.798 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(j)-277(8)]TJ /F8 1 Tf 1.1145 0 Td [(S)]TJ /F6 1 Tf 0.94578 0 Td [(2)-283(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.103 40.375 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.789 27.798 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(S)]TJ /F6 1 Tf 0.94578 0 Td [(2)-283(U)-102(g)]TJ /F2 1 Tf 3.506 0 Td [(et)]TJ /F6 1 Tf 2.1687 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.638 40.375 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.789 27.798 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(B)]TJ /F6 1 Tf 1.0904 0 Td [(j)-277(8)]TJ /F8 1 Tf 1.1084 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49 40.375 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.193 27.798 Td [(B)]TJ /F6 1 Tf 1.0301 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(A)]TJ /F6 1 Tf 1.0241 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.974 40.375 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 40.09 27.798 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)]TJ -35.259 -1.5843 Td [(Comme)]TJ /F8 1 Tf 3.6687 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(appartien)30(t)-343(\265)500(a)]TJ /F6 1 Tf 5.7048 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.802 38.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.91 26.214 Td [(,)-337(la)-337(d)24(\266)469(e\257nition)-337(de)]TJ /F6 1 Tf 7.5181 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.991 38.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.771 26.214 Td [(prouv)30(e)]TJ /F6 1 Tf 3.2831 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.621 38.107 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.343 26.214 Td [(\275)-283(U)]TJ /F2 1 Tf 2.1205 0 Td [(\(prendre)]TJ /F8 1 Tf 4.0663 0 Td [(S)]TJ /F2 1 Tf 0.95783 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(\))-343(et)]TJ /F8 1 Tf 1.8976 0 Td [(X)]TJ /F6 1 Tf 1.1928 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 38.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 25.015 Td [(\(puisque)]TJ /F6 1 Tf 4.0843 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.448 36.392 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.735 25.015 Td [(\275)-361(U)]TJ /F2 1 Tf 1.8614 0 Td [(\).)-596(Cette)-385(appartenance)-385(en)30(tra)108(^)385(\263ne)-379(alors)]TJ /F6 1 Tf 16.614 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.836 36.392 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.801 25.015 Td [(\275)-361(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.75 36.392 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.307 25.015 Td [(\275)-361(U)]TJ /F2 1 Tf 2.241 0 Td [(par)-385(d)24(\266)475(e\257nition)-385(de)]TJ /F6 1 Tf 7.7108 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.94 36.392 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 23.816 Td [(\(prendre)]TJ /F8 1 Tf 4.0602 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\).)]TJ -7.0542 -1.5843 Td [(Prouv)30(ons)-385(que)]TJ /F6 1 Tf 6.3614 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.767 32.4 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.307 22.232 Td [(con)30(tien)24(t)]TJ /F6 1 Tf 3.8795 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.569 32.4 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.524 22.232 Td [(\(et)-379(donc)]TJ /F6 1 Tf 4.0301 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(\).)-578(P)24(our)-379(cela)-379(prenons)]TJ /F8 1 Tf 9.5422 0 Td [(S)]TJ /F2 1 Tf 1.0482 0 Td [(dans)]TJ /F6 1 Tf 2.3795 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.793 32.4 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.548 22.232 Td [(et)]TJ /F8 1 Tf 1.2108 0 Td [(A)]TJ /F2 1 Tf 1.1265 0 Td [(dans)]TJ /F6 1 Tf 2.3795 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.526 32.4 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.271 22.232 Td [(,)-391(il)]TJ -37.94 -1.2048 Td [(s'agit)-295(de)-295(prouv)24(er)]TJ /F8 1 Tf 7.5602 0 Td [(S)]TJ /F6 1 Tf 0.81928 0 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.638 30.685 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.5 21.027 Td [(,)-301(c'est-)0(\265)500(a-dire)]TJ /F8 1 Tf 5.6627 0 Td [(S)]TJ /F6 1 Tf 0.81325 0 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(A)]TJ /F6 1 Tf 0.89759 0 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.871 30.685 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.723 21.027 Td [(2)-277(U)]TJ /F2 1 Tf 1.9639 0 Td [(p)-30(our)-295(tout)]TJ /F8 1 Tf 4.4518 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.819 30.685 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.386 21.027 Td [(dans)]TJ /F6 1 Tf 2.3012 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.905 30.685 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.645 21.027 Td [(.)-433(Cela)-301(r)30(\266)469(esulte)]TJ -33.313 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(S)]TJ /F6 1 Tf 0.8494 0 Td [(\\)]TJ /F8 1 Tf 0.8494 0 Td [(A)]TJ /F6 1 Tf 0.92771 0 Td [(\\)]TJ /F8 1 Tf 0.8494 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.44 28.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.349 19.828 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F6 1 Tf 0.93373 0 Td [(\\)]TJ /F8 1 Tf 0.84337 0 Td [(S)]TJ /F6 1 Tf 0.85542 0 Td [(\\)]TJ /F8 1 Tf 0.84337 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.69 28.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.145 19.828 Td [(,)]TJ /F8 1 Tf 0.59638 0 Td [(S)]TJ /F6 1 Tf 0.8494 0 Td [(\\)]TJ /F8 1 Tf 0.8494 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.776 28.969 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.669 19.828 Td [(2)-283(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.905 28.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.886 19.828 Td [(\(puisque)]TJ /F6 1 Tf 4.0181 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.474 28.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.175 19.828 Td [(est)-319(stable)-313(par)-313(in)24(tersection\))-313(et)]TJ /F8 1 Tf 13.06 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.871 28.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 19.828 Td [(.)]TJ -38.886 -1.5843 Td [(Il)-265(ne)-271(reste)-271(plus)-265(qu')0(\265)500(a)-271(prouv)30(er)-277(que)]TJ /F6 1 Tf 14.09 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.828 26.702 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.922 18.244 Td [(est)-277(une)-265(tribu,)-283(p)-30(our)-271(cela)-271(on)-271(v)54(a)-271(utiliser)-271(la)-265(caract)24(\266)469(erisation)]TJ -15.59 -1.1988 Td [(donn)24(\266)469(ee)-337(en)]TJ /F1 1 Tf 4.7229 0 Td [(I)-36(I)-30(I.1.2.)]TJ /F2 1 Tf 3.4819 0 Td [(.)]TJ -8.2048 -1.5843 Td [(La)-337(d)24(\266)475(e\257nition)-337(de)-337(l'ensem)30(ble)]TJ /F6 1 Tf 12.054 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.914 22.711 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.952 15.461 Td [(mon)24(tre)-337(imm)24(\266)469(ediatemen)24(t)-337(qu'il)-331(p)-30(oss)24(\265)475(ede)]TJ /F8 1 Tf 16.41 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -30.94 -1.5843 Td [(Soit)]TJ /F8 1 Tf 2.1205 0 Td [(B)]TJ /F2 1 Tf 1.2108 0 Td [(un)-379(\266)475(el)18(\266)475(emen)24(t)-403(de)]TJ /F6 1 Tf 6.6807 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.991 20.443 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.578 13.876 Td [(,)-421(prouv)30(ons)-409(que)-403(son)-403(compl)24(\266)469(emen)24(taire)]TJ /F8 1 Tf 16.102 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.922 20.443 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.295 13.876 Td [(appartien)24(t)-379(\266)475(egalemen)24(t)-403(\265)500(a)]TJ /F6 1 Tf -28.964 -1.2048 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 18.728 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5663 12.671 Td [(.)-439(P)30(ar)-319(d)24(\266)475(e\257nition,)-325(cela)-319(signi\257e)-319(que,)-325(p)-30(our)]TJ /F8 1 Tf 17.066 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.681 18.728 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.614 12.671 Td [(,)-319(on)-319(a)]TJ /F8 1 Tf 2.7831 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.233 18.728 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.801 12.671 Td [(\\)]TJ /F8 1 Tf 0.85542 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.164 18.728 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.639 12.671 Td [(,)-319(c'est-)0(\265)500(a-dire)-319(que,)-325(p)-30(our)]TJ /F8 1 Tf -29.307 -1.1988 Td [(S)]TJ /F6 1 Tf 1 0 Td [(2)-331(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.457 17.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2831 11.473 Td [(,)-373(on)-361(a)]TJ /F8 1 Tf 2.9337 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.207 17.012 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.675 11.473 Td [(\\)]TJ /F8 1 Tf 0.90964 0 Td [(A)]TJ /F6 1 Tf 0.99398 0 Td [(\\)]TJ /F8 1 Tf 0.90964 0 Td [(S)]TJ /F6 1 Tf 0.99398 0 Td [(2)-331(U)]TJ /F2 1 Tf 1.7229 0 Td [(.)-536(Remarquons)-361(alors)-367(que)-367(l'on)-367(a)]TJ /F8 1 Tf 13.777 0 Td [(A)]TJ /F6 1 Tf 0.98795 0 Td [(\\)]TJ /F8 1 Tf 0.90964 0 Td [(S)]TJ /F6 1 Tf 1 0 Td [(2)-331(U)]TJ /F2 1 Tf 2.0843 0 Td [(\(puisque)]TJ /F8 1 Tf 4.0663 0 Td [(A)]TJ /F6 1 Tf 1.0723 0 Td [(2)-337(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.716 17.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 11.473 Td [(\))]TJ -38.777 -1.1988 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(B)]TJ /F6 1 Tf 1.1506 0 Td [(2)-343(U)]TJ /F2 1 Tf 2.1024 0 Td [(\(puisque)]TJ /F6 1 Tf 4.0723 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.871 15.288 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.44 10.274 Td [(\275)-337(U)]TJ /F2 1 Tf 1.8434 0 Td [(\).)-560(Compte)-373(ten)24(u)-373(de)-367(la)-373(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 14.175 0 Td [(2.)]TJ /F2 1 Tf 1.4518 0 Td [(du)-373(lemme)-373(il)-373(su\261t)-373(donc)-373(de)]TJ -27.578 -1.1988 Td [(prouv)30(er)]TJ /F8 1 Tf 3.6747 0 Td [(B)]TJ /F6 1 Tf 1.0301 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(S)]TJ /F6 1 Tf 0.94578 0 Td [(2)-283(U)]TJ /F2 1 Tf 1.6687 0 Td [(,)-337(mais)-331(cela)-337(d)24(\266)469(ecoule)-337(de)]TJ /F8 1 Tf 9.8434 0 Td [(B)]TJ /F6 1 Tf 1.0904 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.078 13.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.518 9.075 Td [(.)]TJ -23.187 -1.5843 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.767 11.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.235 7.4906 Td [(son)24(t)-337(dans)]TJ /F6 1 Tf 4.488 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.103 11.306 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.958 7.4906 Td [(,)-337(p)-30(our)]TJ /F8 1 Tf 2.9759 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.966 11.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.253 7.4906 Td [(on)]TJ /F8 1 Tf 1.3916 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.991 11.306 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.958 7.4906 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.853 11.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.163 7.4906 Td [(et)-337(donc)]TJ /F8 1 Tf 3.5602 0 Td [(B)]TJ /F6 1 Tf 1.0301 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.733 11.306 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 35.958 7.4906 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.595 11.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.825 7.4906 Td [(.)-451(P)30(ar)-337(suite)]TJ /F8 1 Tf -34.494 -1.2048 Td [(B)]TJ /F6 1 Tf 1.0301 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.534 9.59 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6205 6.2858 Td [(2)-277(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.724 9.59 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.801 6.2858 Td [(.)]TJ 13.446 -2.3976 Td [(35)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 140 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 139 0 R /ProcSet 2 0 R >> /Contents 138 0 R >> endobj 139 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 141 0 obj << /Length 15371 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F2 1 Tf 0 -2.4036 Td [(En\257n,)-319(si)]TJ /F8 1 Tf 3.9217 0 Td [(B)]TJ /F2 1 Tf 1.1205 0 Td [(est)-313(la)-307(r)24(\266)475(eunion)-319(d'une)-313(suite)-307(croissan)24(te)-313(d')24(\266)475(el)18(\266)475(emen)24(ts)]TJ /F8 1 Tf 20.765 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.638 88.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.747 62.22 Td [(de)]TJ /F6 1 Tf 1.3133 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.776 89.633 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.295 62.22 Td [(,)-313(p)-30(our)]TJ /F8 1 Tf 2.9337 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.569 89.633 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.518 62.22 Td [(et)-313(tout)]TJ -36.187 -1.1988 Td [(en)30(tier)]TJ /F8 1 Tf 2.7711 0 Td [(n)]TJ /F2 1 Tf 0.89759 0 Td [(on)-301(a)]TJ /F8 1 Tf 2.1506 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.043 87.107 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.602 61.021 Td [(\\)]TJ /F8 1 Tf 0.81928 0 Td [(A)]TJ /F6 1 Tf 1.0241 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.069 87.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.699 61.021 Td [(et)-295(donc,)-307(p)-30(our)-301(tout)]TJ /F8 1 Tf 8.2289 0 Td [(S)]TJ /F2 1 Tf 0.96385 0 Td [(de)]TJ /F6 1 Tf 1.3012 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.888 87.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.151 61.021 Td [(,)]TJ /F8 1 Tf 0.58434 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.207 87.107 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.187 61.021 Td [(\\)]TJ /F8 1 Tf 0.81928 0 Td [(A)]TJ /F6 1 Tf 0.89759 0 Td [(\\)]TJ /F8 1 Tf 0.81928 0 Td [(S)]TJ /F6 1 Tf 0.94578 0 Td [(2)-283(U)]TJ /F2 1 Tf 1.6747 0 Td [(.)-427(Cep)-30(endan)24(t,)-307(la)-301(suite)]TJ -30.012 -1.1988 Td [(des)]TJ /F8 1 Tf 1.7711 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.25 85.392 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6627 59.822 Td [(\\)]TJ /F8 1 Tf 0.92169 0 Td [(A)]TJ /F6 1 Tf 1 0 Td [(\\)]TJ /F8 1 Tf 0.92169 0 Td [(S)]TJ /F2 1 Tf 1.0482 0 Td [(est)-385(une)-385(suite)-379(croissan)24(te)-379(d')24(\266)469(el)24(\266)469(emen)30(ts)-385(de)]TJ /F6 1 Tf 16.988 0 Td [(U)]TJ /F2 1 Tf 0.72289 0 Td [(,)-391(sa)-385(r)30(\266)469(eunion)]TJ /F8 1 Tf 5.6084 0 Td [(B)]TJ /F6 1 Tf 1.0602 0 Td [(\\)]TJ /F8 1 Tf 0.92169 0 Td [(A)]TJ /F6 1 Tf 1.006 0 Td [(\\)]TJ /F8 1 Tf 0.91566 0 Td [(S)]TJ /F2 1 Tf 1.0542 0 Td [(est)-385(donc)]TJ -35.5 -1.2048 Td [(encore)-343(dans)]TJ /F6 1 Tf 5.4639 0 Td [(U)]TJ /F2 1 Tf 0.72891 0 Td [(.)-463(Cela)-343(a)24(y)24(an)30(t)-343(lieu)-343(p)-30(our)-343(tout)]TJ /F8 1 Tf 12.199 0 Td [(S)]TJ /F2 1 Tf 1.012 0 Td [(dans)]TJ /F6 1 Tf 2.3434 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.724 84.478 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.38 58.617 Td [(en)24(tra)108(^)391(\263ne)]TJ /F8 1 Tf 3.8735 0 Td [(B)]TJ /F6 1 Tf 1.0361 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(A)]TJ /F6 1 Tf 1.0422 0 Td [(2)-289(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.81 84.478 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.181 58.617 Td [(.)-463(Cela)-349(a)30(y)24(an)24(t)-343(lieu)]TJ -31.849 -1.1988 Td [(p)-30(our)-331(tout)]TJ /F8 1 Tf 4.5301 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(dans)]TJ /F6 1 Tf 2.3434 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.052 82.762 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.627 57.418 Td [(en)30(tra)102(^)391(\263ne)]TJ /F8 1 Tf 3.8675 0 Td [(B)]TJ /F6 1 Tf 1.0843 0 Td [(2)-283(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.405 82.762 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.759 57.418 Td [(.)]TJ -16.428 -1.6506 Td [(Nous)-331(a)24(v)24(ons)-337(bien)-337(prouv)54(\266)469(e)-337(que)]TJ /F6 1 Tf 12.602 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.698 80.4 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.506 55.768 Td [(est)-337(une)-337(tribu,)-331(ce)-337(qui)-337(ac)24(h)24(\265)475(ev)24(e)-337(la)-337(preuv)30(e)-337(de)-337(ce)-337(lemme.)]TJ ET 512.64 665.31 m 519.84 665.31 l 519.84 673.74 l 512.64 673.74 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 53.467 Td [(Un)-415(exemple.)]TJ /F2 1 Tf 7.3675 0 Td [(Nous)-361(a)30(v)24(ons)-367(indiqu)24(\266)469(e)-361(en)-361(in)24(tro)-30(duction)-367(\(cf.)]TJ /F1 1 Tf 17.958 0 Td [(I.4.2.)]TJ /F2 1 Tf 2.5482 0 Td [(\))-367(que)-361(Borel)-367(s'in)30(t)24(\266)469(eressait)-361(\265)500(a)]TJ -27.873 -1.1988 Td [(l'ensem)30(ble)-337(des)-337(parties)-331("bien)-337(d)24(\266)469(e\257nies")-337(du)-337(segmen)24(t)-337([0;)-168(1].)]TJ 0 -1.6506 Td [(Il)-319(est)-319(alors)-325(facile)-325(de)-319(v)24(oir)-319(que)-319(le)-325(lemme)-319(pr)24(\266)469(ec)24(\266)475(eden)24(t)-319(s'applique)-325(en)-313(prenan)18(t)-319(p)-30(our)]TJ /F8 1 Tf 33.361 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(le)-319(segmen)18(t)]TJ -34.59 -1.2048 Td [([0;)-168(1],)-355(p)-30(our)]TJ /F6 1 Tf 5.0181 0 Td [(S)]TJ /F2 1 Tf 1.0301 0 Td [(l'ensem)24(ble)-355(des)-355(sous-segmen)24(ts)-349(de)-355([0;)-168(1])-349(et)-355(p)-30(our)]TJ /F6 1 Tf 20.319 0 Td [(U)]TJ /F2 1 Tf 1.0783 0 Td [(l'ensem)30(ble)-355(des)-355(parties)-355(bien)]TJ -27.446 -1.1988 Td [(d)30(\266)469(e\257nies)-337(au)-337(sens)-331(de)-337(Borel.)]TJ 0 -1.6506 Td [(En)-361(e\256et)-361(le)-367(p)-30(oin)24(t)]TJ /F1 1 Tf 7.5482 0 Td [(1.)]TJ /F2 1 Tf 1.4217 0 Td [(est)-343(\266)475(eviden)18(t,)-373(le)-367(p)-30(oin)24(t)]TJ /F1 1 Tf 9.1325 0 Td [(2.)]TJ /F2 1 Tf 1.4277 0 Td [(corresp)-30(ond)-367(\265)500(a)-361(la)-361(stabilit)24(\266)469(e)-367(par)-361(l'op)-6(\266)475(eration)-367(\(B\),)]TJ -19.53 -1.1988 Td [(quan)30(t)-295(au)-289(p)-30(oin)30(t)]TJ /F1 1 Tf 6.6988 0 Td [(3.)]TJ /F2 1 Tf 1.3193 0 Td [(il)-289(su\261t)-295(d'appliquer)-289(la)-283(stabilit)24(\266)469(e)-289(par)-289(l'op)-6(\266)469(eration)-289(\(A\))-295(en)-283(remarquan)24(t)-295(que)-289(la)]TJ -8.0181 -1.1988 Td [(r)30(\266)469(eunion)-271(croissan)24(te)-271(des)]TJ /F8 1 Tf 9.747 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.655 62.987 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.645 44.165 Td [(est)-277(aussi)-271(la)-271(r)30(\266)469(eunion)-271(disjoin)24(te)-271(des)-271(parties)]TJ /F8 1 Tf 17.319 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.69 62.987 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.536 44.165 Td [(d)24(\266)469(e\257nies)-271(par)]TJ /F8 1 Tf 5.2169 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 59.405 62.987 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.241 44.165 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 63.034 62.987 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 42.961 Td [(et,)-385(p)-30(our)]TJ /F8 1 Tf 3.9036 0 Td [(n)]TJ /F11 1 Tf 0.95181 0 Td [(>)]TJ /F2 1 Tf 1.1265 0 Td [(1,)]TJ /F8 1 Tf 1.1627 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.94 61.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.127 42.961 Td [(=)]TJ /F8 1 Tf 1.1265 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.905 61.262 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.801 42.961 Td [(\\)]TJ /F8 1 Tf 0.91566 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.422 62.073 Td [(c)]TJ 0 -0.94828 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.41 42.961 Td [(et)-379(que)-379(ces)]TJ /F8 1 Tf 4.7771 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.129 61.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.861 42.961 Td [(son)18(t)-373(bien)-379(d)24(\266)469(e\257nies)-379(\265)500(a)-373(cause)-379(de)-373(la)-379(stabilit)24(\266)475(e)]TJ -21.53 -1.1988 Td [(par)-331(l'op)-6(\266)469(eration)-337(\(B\).)]TJ 0 -1.6506 Td [(Il)-307(s'ensuit)-307(que)-307(l'ensem)24(ble)-307(des)-307(parties)-307(bien)-307(d)24(\266)469(e\257nies)-307(con)24(tien)24(t)-307(la)-307(tribu)-307(engendr)24(\266)469(ee)-307(par)-307(les)-307(sous-)]TJ 0 -1.1988 Td [(segmen)24(ts)-433(de)-433([0;)-168(1].)-753(En)-433(utilisan)24(t)-433(les)-409(\266)469(enonc)24(\266)475(es)]TJ /F1 1 Tf 19.849 0 Td [(I)-30(I)-36(I.2.5.)]TJ /F2 1 Tf 3.9157 0 Td [(et)]TJ /F1 1 Tf 1.2651 0 Td [(I)-36(I)-36(I.3.3.)]TJ /F2 1 Tf 3.9217 0 Td [(on)-433(en)-433(d)24(\266)469(eduit)-433(que)-439(cette)]TJ -28.952 -1.2048 Td [(tribu)-343(est)-349(la)-343(tribu)-343(des)-349(b)-30(or)30(\266)469(eliens)-349(de)-343([0;)-168(1].)-475(Ces)-343(b)-30(or)30(\266)469(eliens)-343(son)18(t)-343(donc)-349(des)-343(parties)-349(bien)-343(d)24(\266)475(e\257nies.)]TJ 0 -1.1988 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-289(les)-265(segmen)18(ts)-265(son)24(t)-271(des)-271(b)-30(or)30(\266)469(eliens)-271(et)-265(les)-271(op)-6(\266)475(erations)-271(\(A\))-271(et)-271(\(B\))-271(ne)-265(p)-30(ermetten)24(t)]TJ 0 -1.1988 Td [(pas)-367(de)-361(sortir)-367(de)-361(l'ensem)24(ble)-361(des)-367(b)-30(or)30(\266)469(eliens,)-373(on)-367(conclut)-367(que)-367(les)-367(parties)-367(bien)-361(d)24(\266)469(e\257nies)-367(de)-361([0;)-168(1])]TJ 0 -1.2048 Td [(son)24(t)-457(b)-30(or)24(\266)475(eliennes.)-819(P)30(ar)-457(suite)-457(les)-457(parties)-457(b)-30(or)30(\266)469(eliennes)-457(de)-457([0;)-168(1])-457(son)24(t)-457(exactemen)24(t)-457(les)-457(parties)]TJ 0 -1.1988 Td [(consid)30(\266)469(er)24(\266)469(ees)-331(par)-337(Borel)-337(ce)-337(qui)-331(justi\257e)-337(la)-337(terminologie)-331(in)24(tro)-30(duite.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 23.681 Td [(I)-30(I)-35(I.4.)-241(Espaces)-376(mesur)20(\266)537(es.)]TJ 0 -2.5879 Td 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1.012 0 Td [(alors)-337(:)]TJ /F5 1 Tf -12.271 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.271 0 Td [(36)]TJ ET endstream endobj 143 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 142 0 R /ProcSet 2 0 R >> /Contents 141 0 R >> endobj 142 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 144 0 obj << /Length 11809 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F12 1 Tf 0 -2.4036 Td [({)]TJ /F8 1 Tf 0.83133 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(B)]TJ /F12 1 Tf 1.1446 0 Td [(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 5.6988 0 Td [(S)]TJ /F12 1 Tf 1.012 0 Td [(;)]TJ -10.548 -1.5422 Td [({)]TJ /F8 1 Tf 0.90361 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.9914 87.418 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6566 60.677 Td [(\\)]TJ /F8 1 Tf 0.93976 0 Td [(B)]TJ /F12 1 Tf 1.2108 0 Td [(est)-409(la)-403(r)30(\266)469(eunion)-409(d'un)-403(nom)18(bre)-403(\257ni)-403(d')24(\266)475(el)18(\266)475(emen)24(ts)-403(de)]TJ /F6 1 Tf 20.693 0 Td [(S)]TJ /F12 1 Tf 1.0843 0 Td [(deux)-403(\265)500(a)-403(deux)-403(disjoin)30(ts)-403(\(o)-30(\265)524(u)]TJ /F8 1 Tf 11.753 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.095 87.418 Td [(c)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 59.479 Td [(d)30(\266)469(esigne)-337(le)-337(compl)24(\266)469(emen)24(taire)-337(de)]TJ /F8 1 Tf 13.096 0 Td [(A)]TJ /F12 1 Tf 0.75301 0 Td [(\).)]TJ /F1 1 Tf -13.849 -2.0301 Td [(Exemples.)]TJ 0.81325 -1.5422 Td [(1.)]TJ /F2 1 Tf 1.5602 0 Td [(Les)-331(tribus)-343(son)24(t)-337(des)-337(semi-anneaux.)]TJ /F1 1 Tf -1.5602 -1.5482 Td [(2.)]TJ /F2 1 Tf 1.4217 0 Td [(L'ensem)24(ble)-265(des)-271(in)30(terv)54(alles)-271(de)]TJ /F3 1 Tf 12.681 0 Td [(R)]TJ /F2 1 Tf 0.98795 0 Td [(est)-265(un)-271(semi-anneau.)-421(De)-265(m)24(^)469(eme,)-277(dans)-265(l'espace)-271(euclidien)]TJ /F3 1 Tf -15.904 -1.1988 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 76.668 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7952 53.159 Td [(\(resp.)]TJ /F3 1 Tf 2.8855 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.888 76.668 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.849 53.159 Td [(\))-301(l'ensem)30(ble)-295(des)-295(rectangles)-295(\(resp.)-433(parall)24(\266)469(el)24(\266)469(epip)0(\265)469(edes\))-301(\265)500(a)-289(c)-6(^)500(ot)24(\266)469(es)-295(parall)24(\265)475(eles)-295(aux)-301(axes)]TJ -5.5181 -1.1988 Td [(est)-337(un)-337(semi-anneau.)]TJ 0 -1.1446 Td [(Sur)-337(ces)-343(semi-anneaux)-337(on)-343(p)-30(eut)-343(ais)24(\266)475(emen)24(t)-343(d)30(\266)469(e\257nir)-343(la)-337(longueur,)-337(\(resp.)-463(l'aire,)-343(le)-343(v)24(olume\).)-457(Cela)]TJ 0 -1.1988 Td 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[(Si)-271(l'on)-277(observ)24(e)-271(le)-277(cas)-246(\266)469(etudi)30(\266)469(e)-277(par)-271(Borel)-277(\(cf.)]TJ /F1 1 Tf 18.187 0 Td [(I.4.2.)]TJ /F2 1 Tf 2.5482 0 Td [(\))-277(et,)-283(plus)-277(g)30(\266)469(en)30(\266)469(eralemen)30(t,)-289(le)-277(cas)-277(des)-271(aires)-277(par)]TJ -20.735 -1.2048 Td [(exemple,)-397(on)-385(constate)-391(qu')0(\265)499(a)-385(partir)-391(de)-385(parties)-385(don)24(t)-391(on)-385(conna)108(^)385(\263t)-391(la)-385(mesure)-391(\(les)-385(segmen)24(ts,)-403(les)]TJ 0 -1.1988 Td [(rectangles\))-301(on)-307(essaie)-307(de)-301(reconstruire)-301(les)-307(parties)-301(don)24(t)-307(on)-301(v)24(eut)-283(\266)475(ev)54(aluer)-307(la)-301(mesure)-307(en)-301(utilisan)24(t)]TJ 0 -1.1988 Td [(les)-337(op)0(\266)475(erations)-337(\(A\))-337(et)-337(\(B\))-337(consid)24(\266)475(er)18(\266)475(ees)-337(par)-331(Borel.)]TJ 0 -1.5422 Td [(P)30(our)-457(une)-451(pr)24(\266)469(e-mesure)-451(d)24(\266)475(e\257nie)-457(sur)-457(ensem)24(ble)]TJ /F6 1 Tf 19.271 0 Td [(S)]TJ /F2 1 Tf 1.1325 0 Td [(de)-451(parties,)-487(l'id)24(\266)469(ee)-457(est)-457(donc)-457(de)-451(pro)-30(c)24(\266)469(eder)-457(de)]TJ -20.404 -1.2048 Td [(m)24(^)475(eme.)-427(Cep)-30(endan)24(t)-283(on)-277(observ)24(e)-283(que,)-295(si)]TJ /F6 1 Tf 16.157 0 Td [(S)]TJ /F2 1 Tf 0.95783 0 Td [(est)-283(une)-283(tribu,)-289(ce)-283(pro)-30(c)24(\266)469(ed)30(\266)469(e)-283(ne)-277(p)-30(ermet)-283(pas)-283(d'atteindre)]TJ -17.114 -1.1988 Td [(de)-433(nouv)24(elles)-439(parties.)-753(Plus)-439(g)24(\266)475(en)24(\266)475(eralemen)24(t)-439(si)]TJ /F6 1 Tf 19.452 0 Td [(S)]TJ /F2 1 Tf 1.1145 0 Td [(est)-445(con)30(ten)24(u)-439(dans)-433(une)-439(tribu)]TJ /F6 1 Tf 12.536 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(,)-463(le)-445(pro)-24(c)18(\266)475(ed)24(\266)475(e)]TJ -33.898 -1.1988 Td [(ne)-469(p)-30(ermettra)-475(pas)-481(d'atteindre)-481(des)-475(parties)-475(qui)-475(n'appartiennen)30(t)-475(pas)-481(\265)499(a)]TJ /F6 1 Tf 30.928 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(,)-512(tout)-475(au)-475(plus)-475(il)]TJ -31.723 -1.2048 Td [(p)-30(ermettra)-397(d'atteindre)-397(les)-373(\266)475(el)18(\266)475(emen)24(ts)-397(de)-391(la)-397(tribu)-397(engendr)24(\266)469(ee)-397(par)]TJ /F6 1 Tf 27.319 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(.)-632(Les)-397(tribus)-397(apparaissen)18(t)]TJ -28 -1.1988 Td [(ainsi)-325(comme)-325(des)-331(domaines)-325(de)-325(d)24(\266)469(e\257nitions)-325("maximaux")-325(p)-30(our)-331(une)-325(pr)24(\266)475(e-mesure.)-445(Cela)-331(conduit)]TJ 0 -1.1988 Td [(\265)499(a)-331(la)-337(d)30(\266)469(e\257nition)-337(suiv)54(an)24(te)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 22.676 Td [(I)-30(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 27.183 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83132 0.084337 Td [(\266)]TJ -0.0060234 -0.084337 Td [(efinition.)]TJ /F12 1 Tf 5.2651 0 Td [(On)-325(app)-30(elle)]TJ /F1 1 Tf 5.0602 0 Td [(mesure)]TJ /F12 1 Tf 3.8976 0 Td [(une)-325(pr)30(\266)469(e-mesure)-319(qui)-319(est)-325(d)24(\266)475(e\257nie)-325(sur)-319(une)-325(tribu.)]TJ -19.807 -1.5422 Td [(On)-373(app)-30(elle)]TJ /F1 1 Tf 5.1627 0 Td [(espace)-427(mesur)18(\266)548(e)]TJ /F12 1 Tf 7.5964 0 Td [(un)-379(triplet)]TJ /F2 1 Tf 4.5843 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))]TJ /F12 1 Tf 0.76506 0 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.4277 0 Td [(X)]TJ /F12 1 Tf 1.2771 0 Td [(est)-379(un)-373(ensem)18(ble,)]TJ /F6 1 Tf 7.6687 0 Td [(T)]TJ /F12 1 Tf 1.1747 0 Td [(est)-379(une)-373(tribu)]TJ -33.464 -1.2048 Td [(de)-367(parties)-367(de)]TJ /F8 1 Tf 6.0542 0 Td [(X)]TJ /F12 1 Tf 1.2711 0 Td [(et)]TJ /F8 1 Tf 1.1988 0 Td [(m)]TJ /F12 1 Tf 1.247 0 Td [(est)-373(une)-367(mesure)-373(d)30(\266)469(e\257nie)-367(sur)]TJ /F6 1 Tf 11.861 0 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(.)-548(On)-367(dit)-373(alors)-367(que)-373(la)-367(mesure)-373(est)]TJ /F1 1 Tf 14.608 0 Td [(\257nie)]TJ /F12 1 Tf -37.036 -1.1988 Td [(lorsque)]TJ /F8 1 Tf 3.4458 0 Td [(X)]TJ /F12 1 Tf 1.2651 0 Td [(est)-361(de)-355(mesure)-361(\257nie.)-512(On)-355(dit)-361(que)-361(la)-355(mesure)-361(est)]TJ /F8 1 Tf 20.271 0 Td [(\276)]TJ /F1 1 Tf 0.60843 0 Td [(-\257nie)]TJ /F12 1 Tf 2.8614 0 Td [(lorsque)]TJ /F8 1 Tf 3.4518 0 Td [(X)]TJ /F12 1 Tf 1.259 0 Td [(est)-361(la)-355(r)24(\266)475(eunion)]TJ -33.163 -1.1988 Td [(d'une)-337(famille)-337(d)30(\266)469(enom)24(brable)-337(de)-331(parties)-337(mesurables)-337(de)-337(mesure)-337(\257nie.)]TJ /F1 1 Tf 0 -2.0301 Td [(Exemples.)]TJ 0.81325 -1.5422 Td [(1.)]TJ /F2 1 Tf 1.4639 0 Td [(La)-289(pr)24(\266)469(e-mesure)]TJ /F8 1 Tf 6.488 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.81 26.211 Td [(Y)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.518 18.467 Td [(d)24(\266)469(e\257nie)-283(dans)-289(les)-289(exemples)-289(pr)30(\266)469(ec)24(\266)469(eden)30(ts)-289(est)-289(une)-289(mesure.)-433(Il)-289(s'agit)-283(d'une)]TJ -10.187 -1.2048 Td [(mesure)-337(\257nie)-337(\(resp.)]TJ /F8 1 Tf 8.4639 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nie\))-337(si)-337(et)-331(seulemen)24(t)-337(si)]TJ /F8 1 Tf 10.717 0 Td [(Y)]TJ /F2 1 Tf 1.1325 0 Td [(est)-343(\257ni)-337(\(resp.)-445(d)24(\266)475(enom)24(brable\).)]TJ -20.922 -1.5422 Td [(En)-331(particulier)]TJ 0 -1.5422 Td [({)-602(si)]TJ /F8 1 Tf 2.0723 0 Td [(Y)]TJ /F2 1 Tf 1.0843 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-307(la)-301(mesure)-307(obten)24(ue)-301(est)-307(l'application)]TJ /F8 1 Tf 15.97 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(7!)]TJ /F2 1 Tf 1.2771 0 Td [(#)]TJ /F8 1 Tf 0.83132 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)-307(c'est)-307(la)]TJ /F1 1 Tf 3.9157 0 Td [(mesure)-349(de)-349(d)24(\266)548(ecompte)]TJ /F2 1 Tf -28.898 -1.1988 Td [(ou)]TJ /F1 1 Tf 1.3855 0 Td [(mesure)-385(de)-391(comptage)]TJ /F2 1 Tf 10.645 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(;)]TJ -14.946 -1.5482 Td [({)-692(si)]TJ /F8 1 Tf 2.2169 0 Td [(Y)]TJ /F2 1 Tf 1.1506 0 Td [(est)-349(un)-349(singleton)]TJ /F6 1 Tf 7.2771 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(,)-349(la)-349(mesure)-355(est)-349(aussi)-355(not)24(\266)475(ee)]TJ /F8 1 Tf 11.898 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.716 16.142 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.331 11.43 Td [(.)-487(C'est)-355(la)]TJ /F1 1 Tf 4.4699 0 Td [(mesure)-397(de)-403(Dirac)]TJ /F2 1 Tf 8.6386 0 Td [(ou)]TJ /F1 1 Tf -38.108 -1.1988 Td [(masse)-385(de)-385(Dirac)]TJ /F2 1 Tf 7.9639 0 Td [(au)-337(p)-30(oin)24(t)]TJ /F8 1 Tf 4.006 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(.)-445(On)-337(a)-331(alors)]TJ /F8 1 Tf 5.6265 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.19 14.426 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.38 10.232 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-283(=)-277(1)-331(si)]TJ /F8 1 Tf 3.5602 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(et)-337(0)-331(sinon.)]TJ /F1 1 Tf -25.771 -1.5422 Td [(2.)]TJ /F2 1 Tf 1.5361 0 Td [(La)-331(mesure)-325(de)-319(Borel)-331(et)-319(la)-325(mesure)-325(de)-325(Leb)-30(esgue)-325(\(cf.)]TJ /F1 1 Tf 21.946 0 Td [(I.4.2.)]TJ /F2 1 Tf 2.8675 0 Td [(et)]TJ /F1 1 Tf 1.1566 0 Td [(I.4.3.)]TJ /F2 1 Tf 2.5422 0 Td [(\))-325(son)18(t)-325(des)-325(mesures.)]TJ -30.861 -1.1988 Td [(Elles)-325(son)24(t)-331(\257nies)-331(si)-325(on)-325(les)-325(consid)24(\265)475(ere)-331(sur)-325(un)-331(segmen)24(t)-325(et)]TJ /F8 1 Tf 23.361 0 Td [(\276)]TJ /F2 1 Tf 0.60241 0 Td [(-\257nies)-331(si)-325(on)-325(les)-325(consid)24(\265)475(ere)-331(sur)]TJ /F3 1 Tf 12.711 0 Td [(R)]TJ /F2 1 Tf 1.0422 0 Td [(\(ou)]TJ /F3 1 Tf -37.717 -1.2048 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 9.59 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5964 6.2858 Td [(\).)]TJ 17.651 -2.3976 Td [(37)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 146 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 145 0 R /ProcSet 2 0 R >> /Contents 144 0 R >> endobj 145 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F12 35 0 R /F15 68 0 R >> endobj 147 0 obj << /Length 17807 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F2 1 Tf 0 -2.4036 Td [(Le)-331(fait)-331(que)-337(les)-331(tribus)-331(soien)24(t)-337(stables)-331(par)-331(passage)-331(au)-331(compl)24(\266)469(emen)30(taire)-337(p)-24(ermet)-337(de)-325(donner)-337(une)]TJ 0 -1.1988 Td [(autre)-337(caract)24(\266)475(erisation)-337(des)-337(mesures)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 48.324 Td [(I)-30(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 57.93 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.1506 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.1205 0 Td [(X)]TJ /F12 1 Tf 1.3253 0 Td [(un)-427(ensem)24(ble)-427(et)]TJ /F6 1 Tf 7.1386 0 Td [(T)]TJ /F12 1 Tf 1.2229 0 Td [(une)-427(tribu)-421(de)-427(parties)-421(de)]TJ /F8 1 Tf 10.795 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-716(Une)]TJ -36.422 -1.1988 Td [(application)]TJ /F8 1 Tf 5.1687 0 Td [(m)]TJ /F12 1 Tf 1.2108 0 Td [(de)]TJ /F6 1 Tf 1.3313 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(dans)]TJ /F2 1 Tf 2.3434 0 Td [([0)]TJ /F8 1 Tf 0.77711 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.61446 0 Td [(est)-337(une)-337(mesure)-337(si)-337(et)-337(seulemen)30(t)-337(si)-337(:)]TJ /F1 1 Tf -13.988 -1.6446 Td [(1.)]TJ /F8 1 Tf 1.5602 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(\))-283(=)-277(0)]TJ /F12 1 Tf 2.5602 0 Td [(;)]TJ /F1 1 Tf -5.8855 -1.6446 Td [(2.)]TJ /F12 1 Tf 1.5602 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(B)]TJ /F12 1 Tf 1.1446 0 Td [(disjoin)30(ts)-337(appartenan)24(t)-337(\265)500(a)]TJ /F6 1 Tf 10.434 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(on)-337(a)]TJ /F8 1 Tf 2.5602 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [([)]TJ /F8 1 Tf 0.88554 0 Td [(B)]TJ /F2 1 Tf 0.81325 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ /F1 1 Tf -33.536 -1.8434 Td [(3.)]TJ /F12 1 Tf 1.5602 0 Td [(p)-30(our)-337(toute)-331(suite)-337(croissan)24(te)-331(d')24(\266)469(el)24(\266)469(emen)30(ts)]TJ /F8 1 Tf 16.898 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.276 73.625 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.229 51.599 Td [(de)]TJ /F6 1 Tf 1.3373 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(on)-331(a)]TJ /F8 1 Tf 2.5542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F9 1 Tf 0.39157 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.44 73.625 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.096 51.599 Td [(\))-277(=)-283(sup)]TJ /F8 1 Tf 3.3916 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.966 73.625 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.054 51.599 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)]TJ /F1 1 Tf -36.108 -2.1084 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Si)]TJ /F8 1 Tf 1.1024 0 Td [(m)]TJ /F2 1 Tf 1.1506 0 Td [(a)-271(les)-271(propri)24(\266)469(et)24(\266)469(es)-271(ci-dessus,)-283(mon)24(trons)-277(que)-271(c'est)-277(une)-277(mesure,)-289(c'est-)0(\265)499(a-)]TJ -11.072 -1.1988 Td [(dire)-295(qu'elle)-307(est)-301(d)24(\266)475(enom)24(brablemen)24(t)-301(additiv)30(e.)-439(Prenons)-301(donc)-301(une)-301(famille)-295(d)24(\266)469(enom)24(brable)-295(\()]TJ /F8 1 Tf 36.41 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.81 68.892 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.825 48.292 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.836 68.892 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 46.888 Td [(d')24(\266)475(el)24(\266)469(emen)24(ts)-331(deux)-337(\265)499(a)-331(deux)-331(disjoin)24(ts)-331(de)]TJ /F6 1 Tf 15.886 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(.)-445(Il)-337(s'agit)-337(de)-331(prouv)24(er)]TJ /F8 1 Tf 9.3795 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F9 1 Tf 0.38554 0.95181 Td [([)]TJ /F8 1 Tf 1.2831 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.638 66.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.018 46.888 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.776 66.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.705 46.888 Td [(\).)]TJ -35.373 -2.1446 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(,)]TJ /F9 1 Tf 0.61446 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.44 63.814 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.102 44.744 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(,)]TJ /F9 1 Tf 0.61446 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.612 63.814 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.229 44.744 Td [(\))-283(=)-277(0)-337(et)-337(la)-331(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 9.1265 0 Td [(1.)]TJ /F2 1 Tf 1.3374 0 Td [(assure)-331(l')24(\266)469(egalit)24(\266)475(e.)]TJ -23.361 -1.7831 Td [(Si)]TJ /F8 1 Tf 1.1566 0 Td [(I)]TJ /F2 1 Tf 0.84337 0 Td [(est)-331(\257ni)-331(non)-325(vide,)-325(l')24(\266)469(egalit)24(\266)475(e)-331(se)-325(d)24(\266)469(eduit)-325(ais)24(\266)475(emen)24(t)-331(par)-325(r)30(\266)469(ecurrence)-331(sur)-325(le)-331(cardinal)-331(de)]TJ /F8 1 Tf 34.217 0 Td [(I)]TJ /F2 1 Tf 0.84337 0 Td [(de)-325(la)]TJ -37.06 -1.2048 Td [(propri)24(\266)469(et)30(\266)469(e)]TJ /F1 1 Tf 4.2831 0 Td [(2.)]TJ /F2 1 Tf -4.2831 -1.6446 Td [(Si)]TJ /F8 1 Tf 1.1506 0 Td [(I)]TJ /F2 1 Tf 0.83735 0 Td [(est)-325(in\257ni)-319(on)-319(p)-30(eut,)-325(par)-319(c)24(hangemen)18(t)-319(d'indices,)-325(supp)-30(oser)]TJ /F8 1 Tf 23.813 0 Td [(I)]TJ /F2 1 Tf 0.8012 0 Td [(=)]TJ /F3 1 Tf 1.0542 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)-439(Si)-319(l'on)-325(p)-30(ose)-319(alors)]TJ /F8 1 Tf 8.4217 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.379 57.193 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.717 40.111 Td [(=)]TJ /F9 1 Tf -38.175 -0.39157 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.6293 53.762 Td [(p)]TJ /F17 1 Tf 0.58621 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0301 38.768 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.129 55.262 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.241 38.768 Td [(,)-313(on)-307(obtien)30(t)-307(une)-307(suite)-307(croissan)24(te)-307(d')24(\266)475(el)18(\266)475(emen)24(ts)-307(de)]TJ /F6 1 Tf 20.337 0 Td [(T)]TJ /F2 1 Tf 1.1084 0 Td [(p)-24(our)-307(laquelle)]TJ /F8 1 Tf 5.9458 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.888 55.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.205 38.768 Td [(\))-283(=)]TJ /F9 1 Tf 1.7651 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.138 53.762 Td [(p)]TJ /F17 1 Tf 0.58621 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.627 38.768 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.457 55.262 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 38.768 Td [(\))]TJ -38.777 -2.2831 Td [(\(d'apr)30(\265)469(es)-439(le)-439(cas)-439(pr)24(\266)469(ec)24(\266)475(eden)24(t\).)-759(La)-439(r)30(\266)469(eunion)-439(des)]TJ /F8 1 Tf 19.62 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.793 51.995 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.693 36.485 Td [(est)-439(aussi)-439(celle)-439(des)]TJ /F8 1 Tf 8.3795 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.267 51.995 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.723 36.485 Td [(et)-439(la)-439(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 6.8795 0 Td [(3.)]TJ /F2 1 Tf -38.271 -1.4398 Td [(assure)]TJ /F8 1 Tf 3.012 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F9 1 Tf 0.39157 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.655 49.935 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.102 35.045 Td [(\))-283(=)-277(sup)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.509 49.012 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 15.542 35.997 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.181 48.435 Td [(p)]TJ /F17 1 Tf 0.58621 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.199 35.045 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.5 49.935 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.681 35.045 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.819 49.935 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.494 35.045 Td [(\).)]TJ -20.163 -3.0482 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-379(si)]TJ /F8 1 Tf 8.7892 0 Td [(m)]TJ /F2 1 Tf 1.241 0 Td [(est)-373(une)-361(mesure)-373(elle)-367(a)-337(\266)469(evidemmen)30(t)-373(la)-361(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 20.723 0 Td [(1.)]TJ /F2 1 Tf 1.4277 0 Td [(\(additivit)24(\266)469(e)-367(p)-30(our)]TJ -32.181 -1.2048 Td [(une)-234(famille)-240(vide\))-234(et)-240(la)-234(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 13.645 0 Td [(2.)]TJ /F2 1 Tf 1.3072 0 Td [(\(additivit)24(\266)475(e)-240(p)-30(our)-240(une)-234(famille)-234(\265)499(a)-240(deux)-234(indices\).)-415(Prouv)30(ons)-240(la)]TJ -14.952 -1.1988 Td [(propri)24(\266)469(et)30(\266)469(e)]TJ /F1 1 Tf 4.2711 0 Td [(3.)]TJ /F2 1 Tf 0.89759 0 Td [(.)-439(Consid)24(\266)475(erons)-325(donc)-325(une)-325(suite)-325(croissan)24(te)-319(d')24(\266)469(el)24(\266)469(emen)30(ts)]TJ /F8 1 Tf 22.476 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.267 42.133 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.59 29.593 Td [(de)]TJ /F6 1 Tf 1.3253 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)-439(P)30(osons)]TJ /F8 1 Tf 4.0361 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 59.405 42.133 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.241 29.593 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 63.034 42.133 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 28.394 Td [(et,)-427(p)-30(our)]TJ /F8 1 Tf 3.9699 0 Td [(n)]TJ /F11 1 Tf 1 0 Td [(>)]TJ /F2 1 Tf 1.1807 0 Td [(1,)-427(notons)]TJ /F8 1 Tf 4.506 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.966 40.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.699 28.394 Td [(le)-409(compl)24(\266)469(emen)30(taire)-409(de)]TJ /F8 1 Tf 9.8072 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.44 40.418 Td [(n)]TJ /F10 1 Tf 0.70689 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.229 28.394 Td [(dans)]TJ /F8 1 Tf 2.4157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.793 40.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.94 28.394 Td [(;)-445(on)-403(obtien)24(t)-409(une)-409(famille)]TJ -28.608 -1.1988 Td [(d)30(\266)469(enom)24(brable)-367(de)-367(parties)-367(deux)-367(\265)500(a)-367(deux)-367(disjoin)30(tes)-373(qui)-367(son)24(t)-367(des)-343(\266)475(el)18(\266)475(emen)24(ts)-367(de)]TJ /F6 1 Tf 32 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(.)-548(En)-367(outre)]TJ /F8 1 Tf 5.0723 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 38.702 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 25.846 Td [(est)-307(la)-307(r)30(\266)469(eunion)-307(des)]TJ /F8 1 Tf 7.8976 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.017 36.771 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.759 25.846 Td [(p)-30(our)]TJ /F8 1 Tf 2.3313 0 Td [(p)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(.)-433(On)-307(a)-307(donc)-307(par)-307(additivit)24(\266)469(e,)]TJ /F8 1 Tf 12.054 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.086 36.771 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.145 25.846 Td [(\))-283(=)]TJ /F9 1 Tf 1.7711 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.328 35.271 Td [(p)]TJ /F17 1 Tf 0.59483 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.572 25.846 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.664 36.771 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.06 25.846 Td [(\).)-439(Comme)]TJ -34.729 -2.5843 Td [(la)-367(r)30(\266)469(eunion)-367(des)]TJ /F8 1 Tf 6.5542 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.078 33.073 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.542 23.262 Td [(est)-373(aussi)-367(celle)-373(des)]TJ /F8 1 Tf 8.1024 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.06 33.073 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.235 23.262 Td [(on)-367(a)-361(aussi)]TJ /F8 1 Tf 4.7771 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F9 1 Tf 0.39157 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.802 33.073 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.849 23.262 Td [(\))-337(=)]TJ /F9 1 Tf 1.8313 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.405 33.073 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.783 23.262 Td [(\))-367(et)-373(l')30(\266)469(egalit)24(\266)469(e)-367(de)]TJ /F1 1 Tf -32.452 -1.3434 Td [(3.)]TJ /F2 1 Tf 1.3373 0 Td [(en)-337(r)30(\266)469(esulte.)]TJ ET 512.64 260.74 m 519.84 260.74 l 519.84 269.17 l 512.64 269.17 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 19.31 Td [(Remarque.)]TJ /F2 1 Tf 6.4398 0 Td [(La)-385(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 5.8373 0 Td [(3.)]TJ /F2 1 Tf 1.4819 0 Td [(dans)-385(la)-379(prop)-30(osition)-385(pr)24(\266)475(ec)24(\266)469(eden)24(te)-379(s'app)-30(elle)-385(la)]TJ /F1 1 Tf 19.175 0 Td [(propri)18(\266)548(et)24(\266)548(e)-445(de)]TJ -32.934 -1.2048 Td [(Borel)]TJ /F2 1 Tf 2.7108 0 Td [(.)]TJ -2.7108 -2.2831 Td [(Comme)-403(nous)-403(l'a)30(v)24(ons)-409(dit,)-415(dans)-403(la)-403(pratique,)-421(il)-403(est)-409(fr)24(\266)469(equen)24(t)-403(que)-403(l'on)-409(ne)-397(disp)-30(ose)-403(que)-403(d'une)]TJ 0 -1.1988 Td [(pr)30(\266)469(e-mesure)-409(d)24(\266)475(e\257nie)-409(sur)-415(un)-409(ensem)18(ble)-409(de)-409(parties)]TJ /F6 1 Tf 20.819 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-668(Le)-415(probl)24(\265)475(eme)-409(est)-415(alors)-415(de)-403(la)-409(prolonger)]TJ -21.5 -1.2048 Td [(en)-433(une)-433(mesure)-439(d)24(\266)469(e\257nie)-433(sur)-439(la)-433(tribu)-439(engendr)24(\266)475(ee)-439(par)]TJ /F6 1 Tf 22.446 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-746(Celui-ci)-439(se)-433(scinde)-439(en)-433(fait)-439(en)-433(deux)]TJ -23.127 -1.1988 Td [(sous-probl)24(\265)475(emes)-337(:)-445(l'existence)-337(d'un)-337(tel)-337(prolongemen)24(t)-337(et)-331(l'unicit)24(\266)469(e)-337(de)-337(ce)-337(prolongemen)30(t.)]TJ 0 -1.6446 Td [(Le)-337(probl)24(\265)475(eme)-337(de)-331(l'unicit)24(\266)469(e)-337(est)-337(essen)18(tiellemen)30(t)-337(r)30(\266)469(esolu)-337(par)-337(l')30(\266)469(enonc)24(\266)475(e)-337(suiv)48(an)24(t.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 6.2485 Td [(I)-30(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 7.4906 Td [(5.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8614 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.006 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F12 1 Tf 0.69879 0 Td [(un)-319(espace)-313(mesurable,)]TJ /F6 1 Tf 9.5181 0 Td [(S)]TJ /F12 1 Tf 0.99398 0 Td [(une)-313(partie)-319(g)30(\266)469(en)30(\266)469(eratrice)-313(de)]TJ -28.325 -1.2048 Td [(la)-331(tribu)]TJ /F6 1 Tf 3.6145 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(et)-337(deux)-331(mesures)]TJ /F8 1 Tf 7.3795 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.241 9.59 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 18.614 6.2858 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.776 9.59 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.952 6.2858 Td [(d)30(\266)469(e\257nies)-337(sur)]TJ /F6 1 Tf 5.2349 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(et)-337(qui)-337(co)108(\304)391(\263nciden)18(t)-337(sur)]TJ /F6 1 Tf 9.2952 0 Td [(S)]TJ /F12 1 Tf 0.68073 0 Td [(.)-445(Alors,)-331(si)]TJ /F5 1 Tf -31.964 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(38)]TJ ET endstream endobj 149 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 148 0 R /ProcSet 2 0 R >> /Contents 147 0 R >> endobj 148 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F17 97 0 R >> endobj 150 0 obj << /Length 28118 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ 0.81325 -2.4036 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(l'ensem)30(ble)]TJ /F6 1 Tf 4.8133 0 Td [(S)]TJ /F12 1 Tf 1.012 0 Td [(est)-343(stable)-331(par)-337(in)24(tersection)-337(\257nie,)]TJ /F1 1 Tf -7.3855 -1.5843 Td [(2.)]TJ /F8 1 Tf 1.6747 0 Td [(X)]TJ /F12 1 Tf 1.2952 0 Td [(est)-397(r)24(\266)475(eunion)-397(d'une)-391(suite)-391(d')24(\266)469(el)24(\266)469(emen)30(ts)]TJ /F8 1 Tf 15.506 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.31 86.556 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.307 60.635 Td [(de)]TJ /F6 1 Tf 1.3916 0 Td [(S)]TJ /F12 1 Tf 1.0723 0 Td [(sur)-397(lesquels)]TJ /F8 1 Tf 5.4398 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.216 87.357 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 35.759 60.635 Td [(\(et)-397(donc)]TJ /F8 1 Tf 4.0602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.241 87.357 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 41.211 60.635 Td [(\))-397(prend)]TJ -35.88 -1.2048 Td [(une)-331(v)48(aleur)-337(\257nie,)]TJ 0 -1.5843 Td [(les)-337(mesures)]TJ /F8 1 Tf 5.241 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.388 83.366 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 12.066 57.846 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.198 83.366 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.958 57.846 Td [(son)24(t)-307(\266)469(egales.)]TJ /F1 1 Tf -9.6265 -1.8735 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Notons)]TJ /F6 1 Tf 3.4398 0 Td [(U)]TJ /F2 1 Tf 1.0783 0 Td [(l'ensem)30(ble)-355(des)-325(\266)469(el)24(\266)469(emen)24(ts)-349(de)]TJ /F6 1 Tf 12.036 0 Td [(T)]TJ /F2 1 Tf 1.1566 0 Td [(sur)-355(lesquels)]TJ /F8 1 Tf 5.3614 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.526 80.685 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.735 55.973 Td [(et)]TJ /F8 1 Tf 1.1867 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.379 80.685 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.663 55.973 Td [(co)108(\304)391(\263nci-)]TJ -36.331 -1.1988 Td [(den)30(t.)-451(Cet)-337(ensem)24(ble)-331(con)24(tien)24(t)]TJ /F6 1 Tf 12.62 0 Td [(S)]TJ /F2 1 Tf 1.0181 0 Td [(par)-331(h)24(yp)-30(oth)18(\265)475(ese)-337(et)-337(il)-331(s'agit)-337(de)-331(mon)18(trer)-331(qu'il)-337(est)-313(\266)475(egal)-337(\265)500(a)]TJ /F6 1 Tf 23.06 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)]TJ -37.5 -1.5904 Td [(Supp)-24(osons)-337(tout)-337(d'ab)-30(ord)]TJ /F8 1 Tf 10.795 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(2)-283(U)]TJ /F2 1 Tf 2 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.569 76.702 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.645 53.183 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-337(\()-337(=)]TJ /F8 1 Tf 2.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.664 76.702 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.831 53.183 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-337(\))-337(\257ni.)]TJ -22.795 -1.5843 Td [(Nous)-283(dev)24(ons)-283(donc)-283(mon)24(trer)-283(que)]TJ /F6 1 Tf 13.62 0 Td [(U)]TJ /F2 1 Tf 1.006 0 Td [(con)30(tien)24(t)-283(la)-283(tribu)-283(engendr)24(\266)469(ee)-283(par)]TJ /F6 1 Tf 13.639 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-427(Comme)]TJ /F6 1 Tf 4.3193 0 Td [(S)]TJ /F2 1 Tf 0.95783 0 Td [(est)-289(supp)-30(os)24(\266)475(e)]TJ -34.223 -1.2048 Td [(stable)-246(par)-246(in)24(tersection)-253(\257nie,)-265(nous)-246(p)-30(ouv)24(ons)-253(utiliser)-246(p)-30(our)-253(cela)-246(le)-253(lemme)-253(des)-246(classes)-246(monotones)]TJ 0 -1.1988 Td [(\(cf.)]TJ /F1 1 Tf 1.8614 0 Td [(I)-36(I)-30(I.3.4.)]TJ /F2 1 Tf 3.4819 0 Td [(\).)]TJ -5.3434 -1.5843 Td [(Nous)-331(v)18(enons)-337(de)-331(supp)-30(oser)]TJ /F8 1 Tf 11.307 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(2)-283(U)]TJ /F2 1 Tf 1.6687 0 Td [(.)]TJ -14.163 -1.5904 Td [(Supp)-24(osons)-331(que)]TJ /F6 1 Tf 6.7771 0 Td [(U)]TJ /F2 1 Tf 1.0482 0 Td [(p)-30(oss)24(\265)475(ede)]TJ /F8 1 Tf 3.6446 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)]TJ /F8 1 Tf 0.60241 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(A)]TJ /F6 1 Tf 0.95783 0 Td [(\\)]TJ /F8 1 Tf 0.87349 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(.)-445(En)-325(notan)24(t)]TJ /F8 1 Tf 5.4699 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.034 66.452 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.102 46.021 Td [(le)-331(compl)24(\266)469(emen)24(taire)-325(de)]TJ /F8 1 Tf 9.5723 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(,)]TJ /F8 1 Tf 0.60241 0 Td [(A)]TJ /F2 1 Tf 1.0783 0 Td [(est)-331(la)]TJ -36.831 -1.1988 Td [(r)30(\266)469(eunion)-379(disjoin)24(te)-373(de)]TJ /F8 1 Tf 9.0843 0 Td [(A)]TJ /F6 1 Tf 1 0 Td [(\\)]TJ /F8 1 Tf 0.92169 0 Td [(B)]TJ /F2 1 Tf 1.1807 0 Td [(et)-379(de)]TJ /F8 1 Tf 2.5904 0 Td [(A)]TJ /F6 1 Tf 1 0 Td [(\\)]TJ /F8 1 Tf 0.91566 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.681 64.728 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.62 44.822 Td [(et)-379(toutes)-379(ces)-379(parties)-379(son)24(t)-379(dans)]TJ /F6 1 Tf 13.819 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(,)-385(on)-379(en)-379(d)24(\266)475(eduit)]TJ /F8 1 Tf -32.91 -1.2048 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 63.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.488 43.617 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-506(=)]TJ /F8 1 Tf 2.1687 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.276 63.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.952 43.617 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 1.0602 0 Td [(\\)]TJ /F8 1 Tf 0.98193 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-313(+)]TJ /F8 1 Tf 1.7892 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.129 63.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.139 43.617 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 1.0602 0 Td [(\\)]TJ /F8 1 Tf 0.98193 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.164 63.012 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.789 43.617 Td [(\).)-843(Cep)-30(endan)24(t)]TJ /F8 1 Tf 6.6988 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.595 63.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.651 43.617 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 1.0602 0 Td [(\\)]TJ /F8 1 Tf 0.98193 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-469(inf)24(\266)475(erieur)-469(\265)499(a)]TJ /F8 1 Tf 5.9337 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.379 63.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.982 43.617 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-469(est)-469(\257ni.)]TJ -34.946 -1.1988 Td [(On)-403(p)-30(eut)-409(donc)-379(\266)469(ecrire)]TJ /F8 1 Tf 9.3795 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.302 61.297 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.867 42.418 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 1.0181 0 Td [(\\)]TJ /F8 1 Tf 0.93373 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.224 61.297 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.428 42.418 Td [(\))-403(=)]TJ /F8 1 Tf 1.9639 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.871 61.297 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.548 42.418 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F6 1 Tf 0.65663 0 Td [(\241)]TJ /F8 1 Tf 1.0482 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.603 61.297 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.554 42.418 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 1.0181 0 Td [(\\)]TJ /F8 1 Tf 0.93976 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\).)-662(De)-409(m)24(^)475(eme)-409(on)-403(a)]TJ /F8 1 Tf 8.2651 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.603 61.297 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.367 42.418 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 1.0181 0 Td [(\\)]TJ /F8 1 Tf 0.93976 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.853 61.297 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.934 42.418 Td [(\))-397(=)]TJ /F8 1 Tf -37.602 -1.1988 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 59.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7169 41.22 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F6 1 Tf 0.62651 0 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.871 59.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.904 41.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.99398 0 Td [(\\)]TJ /F8 1 Tf 0.90964 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\).)-542(P)30(ar)-367(h)30(yp)-36(oth)24(\265)469(ese)-361(les)-367(deuxi)24(\265)475(emes)-367(mem)24(bres)-367(de)-367(ces)-361(relations)-367(son)24(t)-343(\266)475(egaux,)]TJ -8.6687 -1.1988 Td [(les)-337(premiers)-331(le)-337(son)24(t)-337(donc)-313(\266)475(egalemen)24(t,)-337(on)-337(en)-331(d)24(\266)475(eduit)]TJ /F8 1 Tf 21.922 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.819 57.857 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.608 40.021 Td [(2)-283(U)]TJ /F2 1 Tf 1.6687 0 Td [(.)]TJ -26.946 -1.5904 Td [(Si)]TJ /F8 1 Tf 1.247 0 Td [(A)]TJ /F2 1 Tf 1.1627 0 Td [(est)-421(la)-415(r)24(\266)475(eunion)-421(d'une)-421(suite)-415(croissan)24(te)-415(d')24(\266)475(el)18(\266)475(emen)24(ts)-415(de)]TJ /F6 1 Tf 22.922 0 Td [(U)]TJ /F2 1 Tf 0.72289 0 Td [(,)-439(la)-415(propri)24(\266)475(et)24(\266)469(e)-415(de)-415(Borel)-421(assure)]TJ /F8 1 Tf -26.054 -1.1988 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 53.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.488 37.232 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-499(=)-493(lim)]TJ /F8 1 Tf 3.7108 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.483 53.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.5 37.232 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.517 53.064 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.181 37.232 Td [(\))-469(et)]TJ /F8 1 Tf 2.1506 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.629 53.866 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.723 37.232 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-493(=)-493(lim)]TJ /F8 1 Tf 3.7108 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.552 53.866 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.958 37.232 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.914 53.064 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.645 37.232 Td [(\).)-837(Les)]TJ /F8 1 Tf 3.4277 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.672 53.064 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.801 37.232 Td [(\266)475(etan)18(t)-463(dans)]TJ /F6 1 Tf 5.2108 0 Td [(U)]TJ /F2 1 Tf 0.72289 0 Td [(,)-499(on)-463(a)]TJ /F8 1 Tf 3.259 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.483 53.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.151 37.232 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.517 53.064 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.837 37.232 Td [(\))-493(=)]TJ /F8 1 Tf -37.506 -1.2048 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 52.15 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7169 36.027 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.25 51.349 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4036 36.027 Td [(\))-337(et)-337(donc)]TJ /F8 1 Tf 4.2771 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.405 52.15 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.843 36.027 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.164 52.15 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.096 36.027 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)]TJ -13.904 -1.5843 Td [(Le)-337(lemme)-337(des)-337(classes)-331(monotones)-343(assure)-331(bien)-337(alors)-337(que)]TJ /F6 1 Tf 23.934 0 Td [(U)]TJ /F2 1 Tf 1.0602 0 Td [(con)24(tien)24(t)]TJ /F6 1 Tf 3.8374 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(donc)-331(est)-313(\266)469(egal)-331(\265)500(a)]TJ /F6 1 Tf 6.8434 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(.)]TJ -37.602 -1.8735 Td [(Lorsqu'on)-427(ne)-427(supp)-30(ose)-427(plus)-427(que)]TJ /F8 1 Tf 14.199 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.207 47.202 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.114 32.569 Td [(et)]TJ /F8 1 Tf 1.259 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.276 47.202 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.187 32.569 Td [(prennen)18(t)-427(une)-427(m)24(^)475(eme)-427(v)54(aleur)-427(\257nie)-427(sur)]TJ /F8 1 Tf 16.355 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-451(on)-427(se)]TJ -36.114 -1.1988 Td [(ram)24(\265)475(ene)-337(\265)500(a)-337(ce)-337(cas)-337(en)-331(utilisan)24(t)-331(l')24(\266)469(enonc)30(\266)469(e)]TJ /F1 1 Tf 16.325 0 Td [(2.)]TJ /F2 1 Tf 1.3374 0 Td [(de)-337(la)-331(prop)-36(osition.)]TJ -17.663 -1.5904 Td [(Plus)-265(pr)24(\266)469(ecis)30(\266)469(emen)24(t,)-277(p)-30(our)-265(toute)-265(mesure)]TJ /F8 1 Tf 16.042 0 Td [(m)]TJ /F2 1 Tf 1.1386 0 Td [(d)24(\266)469(e\257nie)-265(sur)-265(\()]TJ /F8 1 Tf 5.0964 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(\))-265(et)]TJ /F8 1 Tf 1.747 0 Td [(A)]TJ /F2 1 Tf 1.012 0 Td [(appartenan)24(t)-271(\265)500(a)]TJ /F6 1 Tf 6.3373 0 Td [(T)]TJ /F2 1 Tf 1.0663 0 Td [(d)30(\266)469(e\257nissons)]TJ /F8 1 Tf -34.536 -1.1988 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 40.685 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0964 28.581 Td [(sur)]TJ /F6 1 Tf 1.5843 0 Td [(T)]TJ /F2 1 Tf 1.0361 0 Td [(par)]TJ /F8 1 Tf 1.6867 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.578 40.685 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.934 28.581 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.78313 0 Td [(\\)]TJ /F8 1 Tf 0.70482 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\).)-415(Il)-240(est)-246(clair)-240(que)]TJ /F8 1 Tf 7.3253 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.483 40.685 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.482 28.581 Td [(est)-240(encore)-246(d)24(\266)475(enom)24(brablemen)24(t)-240(additiv)30(e)]TJ -23.151 -1.1988 Td [(donc)-367(est)-373(une)-373(mesure)-373(sur)]TJ /F6 1 Tf 11.09 0 Td [(T)]TJ /F2 1 Tf 1.1627 0 Td [(p)-30(our)-373(laquelle)]TJ /F8 1 Tf 6.0783 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.121 38.969 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.193 27.382 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-337(=)]TJ /F8 1 Tf 1.8373 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\).)-554(Lorsque)]TJ /F8 1 Tf 5.0241 0 Td [(A)]TJ /F6 1 Tf 1.0843 0 Td [(2)-343(S)]TJ /F2 1 Tf 1.6867 0 Td [(,)-379(comme)]TJ /F6 1 Tf 4.0783 0 Td [(S)]TJ /F2 1 Tf 1.0482 0 Td [(est)]TJ -37.934 -1.2048 Td [(stable)-259(par)-253(in)24(tersection,)-277(on)-259(v)24(oit)-259(que)]TJ /F8 1 Tf 15.127 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.534 38.056 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.241 26.177 Td [(co)102(\304)391(\263ncide)-259(a)30(v)24(ec)]TJ /F8 1 Tf 5.8795 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.491 38.056 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.904 26.177 Td [(sur)]TJ /F6 1 Tf 1.5964 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-421(Si)-259(alors)]TJ /F8 1 Tf 4.1084 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.19 38.056 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.446 26.177 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-259(\()-259(=)]TJ /F8 1 Tf 2.3494 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.836 38.056 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.319 26.177 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-259(\))]TJ -38.133 -1.1988 Td [(est)-337(\257ni,)-337(le)-337(r)24(\266)469(esultat)-337(pr)24(\266)475(ec)24(\266)469(eden)30(t)-337(prouv)24(e)]TJ /F8 1 Tf 16.108 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.94 36.34 Td [(0)]TJ /F7 1 Tf 0 -0.94827 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.247 24.979 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.034 36.34 Td [(00)]TJ /F7 1 Tf 0 -0.94827 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.837 24.979 Td [(.)-445(En)-331(particulier)-337(on)-331(a)-337(donc)]TJ /F8 1 Tf 11.747 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.034 36.34 Td [(0)]TJ /F7 1 Tf 0 -0.94827 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 78.12 49.258 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.88 24.979 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.836 36.34 Td [(00)]TJ /F7 1 Tf 0 -0.94827 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 84.831 49.258 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.958 24.979 Td [(.)]TJ -37.627 -1.5843 Td [(F)84(aisons)-337(alors)-337(deux)-337(remarques)-337(imp)-30(ortan)24(tes.)]TJ 0 -1.5904 Td [(Soien)24(t)]TJ /F8 1 Tf 3.1024 0 Td [(A)]TJ /F2 1 Tf 1.1566 0 Td [(et)]TJ /F8 1 Tf 1.241 0 Td [(B)]TJ /F2 1 Tf 1.2169 0 Td [(dans)]TJ /F6 1 Tf 2.4157 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)-662(P)24(our)]TJ /F8 1 Tf 3.4518 0 Td [(U)]TJ /F6 1 Tf 1.1928 0 Td [(2)-403(T)]TJ /F2 1 Tf 1.8675 0 Td [(,)-427(\()]TJ /F8 1 Tf 1.0964 0 Td [(A)]TJ /F6 1 Tf 1.0181 0 Td [([)]TJ /F8 1 Tf 0.93976 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))]TJ /F6 1 Tf 0.66265 0 Td [(\\)]TJ /F8 1 Tf 0.93976 0 Td [(U)]TJ /F2 1 Tf 1.1988 0 Td [(est)-415(la)-409(r)30(\266)469(eunion)-409(disjoin)24(te)-409(de)]TJ /F8 1 Tf 12 0 Td [(B)]TJ /F6 1 Tf 1.0843 0 Td [(\\)]TJ /F8 1 Tf 0.93976 0 Td [(U)]TJ /F2 1 Tf 1.1988 0 Td [(et)]TJ -38.331 -1.1988 Td [(de)]TJ /F8 1 Tf 1.4398 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.845 30.073 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 20.605 Td [(\\)]TJ /F2 1 Tf 0.96386 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 1.0482 0 Td [(\\)]TJ /F8 1 Tf 0.95783 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-445(\(o)-30(\265)524(u)]TJ /F8 1 Tf 2.7169 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.819 30.073 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.795 20.605 Td [(est)-445(le)-445(compl)24(\266)475(emen)24(taire)-439(de)]TJ /F8 1 Tf 11.572 0 Td [(B)]TJ /F2 1 Tf 0.81325 0 Td [(\).)-765(L'additivit)24(\266)469(e)-445(des)-439(mesures)-445(donne)]TJ -23.849 -1.2048 Td [(alors)]TJ /F8 1 Tf 2.4578 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.405 27.556 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [([)]TJ /F7 1 Tf 0.76724 0 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.88 19.4 Td [(=)]TJ /F8 1 Tf 1.1566 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.483 27.556 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.867 19.4 Td [(+)]TJ /F9 1 Tf 1.0422 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.241 27.556 Td [(A)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 16.898 20.214 Td [(\242)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.836 27.193 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 35.988 38.403 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.416 19.4 Td [(.)-632(En)-397(particulier,)-409(cette)-397(form)24(ule)-397(prouv)30(e)-403(que)-397(si)]TJ /F8 1 Tf 20.253 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.595 28.357 Td [(0)]TJ /F7 1 Tf 0 -0.94827 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.578 19.4 Td [(=)]TJ /F8 1 Tf 1.1566 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.983 28.357 Td [(00)]TJ /F7 1 Tf 0 -0.94827 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 19.4 Td [(et)]TJ /F8 1 Tf -38.331 -1.3012 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 26.495 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1928 18.099 Td [(=)]TJ /F8 1 Tf 1.0783 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.086 26.495 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.181 18.099 Td [(alors)]TJ /F8 1 Tf 2.4096 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.276 26.495 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [([)]TJ /F7 1 Tf 0.76724 0 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.596 18.099 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.112 26.495 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [([)]TJ /F7 1 Tf 0.76724 0 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.38 18.099 Td [(,)-349(l'ensem)30(ble)-349(des)]TJ /F8 1 Tf 7.1867 0 Td [(A)]TJ /F2 1 Tf 1.0964 0 Td [(tels)-343(que)]TJ /F8 1 Tf 3.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.172 26.495 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.217 18.099 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.328 26.495 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.163 18.099 Td [(est)-355(donc)-343(stable)-349(par)]TJ -30.831 -1.1988 Td [(r)30(\266)469(eunion)-337(\257nie.)]TJ 0 -1.5904 Td [(Si)]TJ /F8 1 Tf 1.241 0 Td [(B)]TJ /F2 1 Tf 1.2108 0 Td [(est)-415(la)-409(r)30(\266)469(eunion)-409(d'une)-409(suite)-409(croissan)24(te)-409(d')30(\266)469(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 21.446 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.914 21.702 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.94 15.31 Td [(de)]TJ /F6 1 Tf 1.4096 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(,)-427(p)-30(our)]TJ /F8 1 Tf 3.1446 0 Td [(U)]TJ /F6 1 Tf 1.1928 0 Td [(2)-403(T)]TJ /F2 1 Tf 1.8675 0 Td [(,)]TJ /F8 1 Tf 0.6988 0 Td [(B)]TJ /F6 1 Tf 1.0843 0 Td [(\\)]TJ /F8 1 Tf 0.93373 0 Td [(U)]TJ /F2 1 Tf 1.1988 0 Td [(est)]TJ -37.934 -1.1988 Td [(la)-439(r)30(\266)469(eunion)-445(de)-439(la)-439(suite)-439(croissan)24(te)-445(des)]TJ /F8 1 Tf 16.639 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.526 19.978 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.566 14.111 Td [(\\)]TJ /F8 1 Tf 0.96385 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(.)-765(La)-445(propri)24(\266)469(et)24(\266)475(e)-445(de)-439(Bepp)-36(o)-439(Levi)-439(assure)-439(alors)]TJ /F8 1 Tf -19.988 -1.1988 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 18.262 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8976 12.912 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-283(=)-277(lim)]TJ /F8 1 Tf 3.2771 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.509 18.262 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 25.687 25.319 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.38 12.912 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\).)-451(Autremen)24(t)-331(dit)]TJ /F8 1 Tf 7.7169 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.138 18.262 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.12 12.912 Td [(=)-283(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.017 18.193 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.114 12.912 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.052 18.262 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 57.193 25.319 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.139 12.912 Td [(.)-439(En)-331(particulier,)-331(si)-331(l'on)-331(a)-325(p)-30(our)-331(tout)]TJ -23.807 -1.1988 Td [(en)30(tier)]TJ /F8 1 Tf 2.8072 0 Td [(n)-337(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.241 17.349 Td [(0)]TJ /F7 1 Tf 0 -0.93966 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 21.108 22.728 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.373 11.714 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.043 17.349 Td [(00)]TJ /F7 1 Tf 0 -0.93966 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 27.831 22.728 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.452 11.714 Td [(,)-337(alors)]TJ /F8 1 Tf 3.012 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.25 17.349 Td [(0)]TJ /F7 1 Tf 0 -0.93966 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.307 11.714 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.405 17.349 Td [(00)]TJ /F7 1 Tf 0 -0.93966 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.934 11.714 Td [(.)]TJ -16.602 -1.8855 Td [(P)30(osons)-253(alors)]TJ /F8 1 Tf 5.5663 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.681 13.849 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.482 9.828 Td [(=)]TJ /F9 1 Tf 1.2771 0.95181 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.371 12.34 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.265 9.828 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.914 13.849 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.506 9.828 Td [(,)]TJ /F8 1 Tf 0.54819 0 Td [(X)]TJ /F2 1 Tf 1.1566 0 Td [(est)-259(donc)-259(la)-253(r)30(\266)469(eunion)-253(de)-253(la)-253(suite)-259(croissan)30(te)-259(des)]TJ /F8 1 Tf 19.12 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.509 13.849 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.639 9.828 Td [(.)-415(La)-259(premi)24(\265)469(ere)]TJ -33.307 -2.3373 Td [(remarque)-331(assure,)-331(p)-30(our)-331(tout)-331(en)30(tier)]TJ /F8 1 Tf 15.042 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.138 11.306 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 46.133 14.282 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.886 7.4906 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.948 11.306 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 52.843 14.282 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.289 7.4906 Td [(et)-337(la)-331(deuxi)30(\265)469(eme)-325(assure)]TJ /F8 1 Tf 9.6988 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.19 11.306 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(X)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.91 7.4906 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.448 11.306 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(X)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.602 7.4906 Td [(,)-331(c'est-)]TJ -36.271 -1.2048 Td [(\265)499(a-dire)]TJ /F8 1 Tf 2.8373 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.948 9.59 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6024 6.2858 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.509 9.59 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.048 6.2858 Td [(.)]TJ ET 512.64 73.976 m 519.84 73.976 l 519.84 82.4 l 512.64 82.4 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.247 3.8882 Td [(39)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 152 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 151 0 R /ProcSet 2 0 R >> /Contents 150 0 R >> endobj 151 0 obj << /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F13 39 0 R /F17 97 0 R >> endobj 153 0 obj << /Length 12222 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F2 1 Tf 0 -2.4036 Td [(Le)-415(probl)24(\265)469(eme)-415(de)-415(l'existence)-415(du)-421(prolongemen)30(t)-421(d'une)-415(pr)24(\266)469(e-mesure)-415(en)-415(une)-415(mesure)-421(est)-415(r)24(\266)475(esolu)]TJ 0 -1.1988 Td [(par)-331(l')24(\266)469(enonc)30(\266)469(e)-337(suiv)54(an)24(t)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 48.429 Td [(I)-30(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 58.057 Td [(6.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.4036 0.084339 Td [(\266)]TJ -0.0060244 -0.084339 Td [(eor)]TJ 1.8012 0.084339 Td [(\265)]TJ -0.0060244 -0.084339 Td [(eme.)]TJ /F12 1 Tf 2.6807 0 Td [(\()]TJ /F1 1 Tf 0.38554 0 Td [(de)-391(Carath)24(\266)548(eo)-36(dory)]TJ /F12 1 Tf 8.494 0 Td [(\))]TJ -19.512 -1.5663 Td [(Soit)]TJ /F8 1 Tf 2.1024 0 Td [(m)]TJ /F12 1 Tf 1.253 0 Td [(une)-385(pr)30(\266)469(e-mesure)-379(d)24(\266)475(e\257nie)-385(sur)-379(un)-385(semi-anneau)]TJ /F6 1 Tf 19.313 0 Td [(S)]TJ /F12 1 Tf 0.68072 0 Td [(.)-578(Alors)]TJ /F8 1 Tf 3.5542 0 Td [(m)]TJ /F12 1 Tf 1.253 0 Td [(p)-30(eut)-385(se)-379(prolonger)-385(en)-379(une)]TJ -28.157 -1.1988 Td [(mesure)-337(d)24(\266)475(e\257nie)-337(sur)-337(la)-337(tribu)-331(engendr)24(\266)469(ee)-337(par)]TJ /F6 1 Tf 18.307 0 Td [(S)]TJ /F12 1 Tf 0.68674 0 Td [(.)]TJ -18.994 -1.5723 Td [(En)-234(outre)-240(il)-234(n'y)-240(a)-234(qu'un)-234(tel)-240(prolongemen)24(t)-234(lorsque)]TJ /F8 1 Tf 20.657 0 Td [(X)]TJ /F12 1 Tf 1.1446 0 Td [(est)-240(la)-234(r)30(\266)469(eunion)-240(d'une)-240(famille)-234(d)24(\266)469(enom)24(brable)]TJ -21.801 -1.1988 Td [(d'el)30(\266)469(emen)24(ts)-331(de)]TJ /F6 1 Tf 6.259 0 Td [(S)]TJ /F12 1 Tf 1.012 0 Td [(de)-337(pr)30(\266)469(e-mesure)-331(\257nie.)]TJ /F2 1 Tf -7.2711 -2.1024 Td [(La)-337(partie)-337("unicit)24(\266)475(e")-343(de)-331(ce)-337(th)24(\266)469(eor)24(\265)469(eme)-331(d)24(\266)469(ecoule)-337(du)-337(r)30(\266)469(esultat)-337(pr)24(\266)475(ec)24(\266)469(eden)30(t.)]TJ 0 -1.5723 Td [(Nous)-331(prouv)24(erons)-337(la)-337(partie)-337("existence")-337(dans)-331(le)-343(c)24(hapitre)]TJ /F1 1 Tf 24.181 0 Td [(VI)-36(I)]TJ /F2 1 Tf 1.7771 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 38.279 Td [(I)-30(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 45.888 Td [(7.)-289(Application)-379(\265)566(a)-391(la)-385(mesure)-385(de)-391(Borel.)]TJ /F2 1 Tf -3.006 -1.5723 Td [(Les)-487(in)24(terv)54(alles)-493(de)]TJ /F3 1 Tf 8.3313 0 Td [(R)]TJ /F2 1 Tf 1.2108 0 Td [(formen)24(t)-493(clairemen)24(t)-487(un)-493(semi-anneau)-493(don)24(t)-493(nous)-493(a)30(v)24(ons)-493(prouv)54(\266)469(e)-493(qu'il)]TJ -9.5422 -1.1988 Td [(engendre)-487(la)-487(tribu)-493(des)-487(b)-30(or)24(\266)475(eliens.)-915(Sur)-493(celui-ci)-487(on)-487(sait)-493(d)30(\266)469(e\257nir)-493(la)-487(longueur)-493(don)24(t)-487(on)-493(p)-30(eut)]TJ 0 -1.1988 Td [(mon)24(trer)-319(qu'elle)-325(est)-319(additiv)24(emen)30(t)-325(d)30(\266)469(enom)24(brable,)-325(il)-319(s'agit)-319(donc)-319(d'une)-325(pr)30(\266)469(e-mesure.)-439(Comme)]TJ /F3 1 Tf 38.44 0 Td [(R)]TJ /F2 1 Tf -38.44 -1.1988 Td [(est)-295(la)-295(r)24(\266)475(eunion)-301(de)-295(la)-289(famille)-301(d)30(\266)469(enom)24(brable)-295(des)-295([)]TJ /F6 1 Tf 19.59 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(])-295(\()]TJ /F8 1 Tf 0.95783 0 Td [(n)]TJ /F2 1 Tf 0.89759 0 Td [(parcouran)24(t)]TJ /F3 1 Tf 5.0482 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(\))-295(qui)-295(son)24(t)-301(de)-289(longueur)]TJ -29.645 -1.2048 Td [(\257nie,)-337(le)-337(th)24(\266)469(eor)24(\265)469(eme)-331(de)-337(Carath)24(\266)469(eo)-30(dory)-331(s'applique,)-337(on)-337(en)-331(d)24(\266)475(eduit)-337(:)]TJ /F12 1 Tf 0 -1.5663 Td [(Il)-289(y)-295(a)-289(sur)-295(l'ensem)24(ble)-289(des)-295(b)-30(or)30(\266)469(eliens)-295(de)]TJ /F3 1 Tf 16.078 0 Td [(R)]TJ /F12 1 Tf 1.012 0 Td [(une)-295(mesure)-295(et)-289(une)-295(seule)-289(telle)-295(que)-295(la)-289(mesure)-295(de)-295(tout)]TJ -17.09 -1.1988 Td [(in)24(terv)60(alle)-337(soit)-337(sa)-331(longueur.)]TJ /F2 1 Tf 0 -2.1084 Td [(Cette)-337(mesure)-337(est)-337(la)-337(mesure)-337(de)-337(Borel.)]TJ 0 -2.1024 Td [(Les)-325(parado)24(xes)-331(pr)24(\266)469(esen)30(t)24(\266)469(es)-331(dans)-325(la)-325(section)]TJ /F1 1 Tf 17.548 0 Td [(I.4.4.)]TJ /F2 1 Tf 2.8675 0 Td [(pro)24(viennen)24(t)-331(du)-325(fait)-331(que)-331(les)-331(parties)-325(utilis)24(\266)469(ees)]TJ -20.416 -1.2048 Td [(ne)-331(son)24(t)-337(pas)-343(b)-24(or)24(\266)475(eliennes.)-451(Ils)-337(prouv)24(en)30(t)-337(en)24(tre)-337(autre)-337(l'existence)-337(de)-331(parties)-337(non)-337(b)-30(or)30(\266)469(eliennes.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 22.485 Td [(I)-30(I)-35(I.5.)-241(Applications)-376(et)-376(fonctions)-376(mesurables.)]TJ 0 -2.4724 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 23.991 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83132 0.084337 Td [(\266)]TJ -0.0060234 -0.084337 Td [(efinition.)]TJ /F12 1 Tf 5.5663 0 Td [(Etan)24(t)-427(donn)24(\266)469(es)-427(deux)-427(espaces)-427(mesurables)]TJ /F2 1 Tf 17.681 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79519 0 Td [(\))]TJ /F12 1 Tf 0.81927 0 Td [(et)]TJ /F2 1 Tf 1.259 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.267 34.926 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.295 23.991 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.448 34.926 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.825 23.991 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(,)-451(une)]TJ -36.88 -1.1988 Td [(application)]TJ /F8 1 Tf 5.3012 0 Td [(f)]TJ /F12 1 Tf 1.0602 0 Td [(de)]TJ /F8 1 Tf 1.4699 0 Td [(X)]TJ /F12 1 Tf 1.3735 0 Td [(dans)]TJ /F8 1 Tf 2.4699 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.638 33.211 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 18.663 22.792 Td [(est)-469(dite)]TJ /F1 1 Tf 3.8253 0 Td [(mesurable)]TJ /F12 1 Tf 5.5663 0 Td [(lorsque,)-500(p)-30(our)-469(tout)]TJ /F8 1 Tf 8.6687 0 Td [(A)]TJ /F6 1 Tf 1.6807 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.103 33.211 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 39.482 22.792 Td [(-mesurable,)]TJ -34.151 -1.1988 Td [(l'image)-337(r)30(\266)469(ecipro)-30(que)]TJ /F8 1 Tf 8.3434 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.422 31.487 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.343 21.593 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)]TJ /F6 1 Tf 1.5663 0 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(-mesurable.)]TJ -14.235 -1.5723 Td [(Lorsque)]TJ /F8 1 Tf 3.8795 0 Td [(X)]TJ /F12 1 Tf 1.3434 0 Td [(et)]TJ /F8 1 Tf 1.2711 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.224 29.245 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 13.452 20.021 Td [(son)24(t)-439(des)-439(espaces)-445(top)-30(ologiques)-445(une)-439(application)-439(de)]TJ /F8 1 Tf 22.295 0 Td [(X)]TJ /F12 1 Tf 1.3434 0 Td [(dans)]TJ /F8 1 Tf 2.4458 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.879 29.245 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 41.163 20.021 Td [(est)-439(dite)]TJ /F1 1 Tf -35.831 -1.1988 Td [(b)-30(or)24(\266)548(elienne)]TJ /F12 1 Tf 5.5542 0 Td [(lorsqu'elle)-337(est)-337(mesurable)-337(si)-337(on)-331(m)24(unit)]TJ /F8 1 Tf 16.313 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.664 27.53 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 31.127 18.822 Td [(de)-331(leur)-337(tribu)-337(b)-30(or)30(\266)469(elienne.)]TJ -25.795 -1.5663 Td [(On)-337(app)-30(elle)]TJ /F1 1 Tf 5.0843 0 Td [(fonction)-391(mesurable)]TJ /F12 1 Tf 9.8735 0 Td [(sur)-343(un)-331(espace)-337(mesurable)]TJ /F2 1 Tf 10.976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(toute)-337(application)-337(de)]TJ /F8 1 Tf 9.1205 0 Td [(X)]TJ /F12 1 Tf -38.259 -1.2048 Td [(dans)]TJ ET 0.504 w 91.656 201.81 m 100.3 201.81 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6687 16.051 Td [(R)]TJ /F12 1 Tf 1.0542 0 Td [(qui)-337(est)-337(mesurable)-337(lorsque)]TJ ET 240.77 201.81 m 249.41 201.81 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 20.145 16.051 Td [(R)]TJ /F12 1 Tf 1.0542 0 Td [(est)-337(m)24(uni)-337(de)-337(sa)-331(tribu)-337(des)-337(b)-30(or)30(\266)469(eliens.)]TJ /F1 1 Tf -15.867 -1.8373 Td [(Remarque.)]TJ /F2 1 Tf 6.3675 0 Td [(Il)-361(est)-367(clair)-367(que,)-373(p)-30(our)]TJ /F8 1 Tf 9.4157 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(mesurable)-361(de)-361(\()]TJ /F8 1 Tf 6.512 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))-367(dans)-361(\()]TJ /F8 1 Tf 3.506 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.233 20.935 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.386 14.214 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.414 20.935 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.91 14.214 Td [(\))-361(et)]TJ /F8 1 Tf 1.9458 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.457 20.935 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.096 14.214 Td [(mesurable)]TJ -34.765 -1.1988 Td [(de)-331(\()]TJ /F8 1 Tf 1.7229 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.388 19.219 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.241 13.015 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.569 19.219 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7651 13.015 Td [(\))-337(dans)-331(\()]TJ /F8 1 Tf 3.4518 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.216 19.219 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.639 13.015 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.724 19.219 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.392 13.015 Td [(\),)-337(la)-337(comp)-30(os)24(\266)475(ee)]TJ /F8 1 Tf 6.5904 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.741 19.219 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.861 13.015 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(mesurable)-337(de)-331(\()]TJ /F8 1 Tf 8.0181 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))-337(dans)-337(\()]TJ /F8 1 Tf 3.4518 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.905 19.219 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.277 13.015 Td [(;)]TJ /F6 1 Tf 0.44579 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.414 19.219 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.03 13.015 Td [(\).)]TJ -36.699 -2.1024 Td [(Le)-337(lien)-331(en)24(tre)-337(partie)-337(mesurable)-337(et)-331(fonction)-343(mesurable)-331(est)-343(donn)24(\266)475(e)-343(par)-331(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 6.6304 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 7.9485 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.012 0 Td [(P)24(our)-379(qu'une)-379(partie)]TJ /F8 1 Tf 8.7169 0 Td [(A)]TJ /F12 1 Tf 1.1325 0 Td [(d'un)-385(espace)-379(mesurable)-379(soit)-385(mesurable)-379(il)]TJ -21.62 -1.1988 Td [(faut)-337(et)-337(il)-331(su\261t)-337(que)-337(sa)-337(fonction)-343(indicatrice)]TJ /F2 1 Tf 18.494 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.897 9.4435 Td [(A)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.373 6.7497 Td [(soit)-331(mesurable.)]TJ /F5 1 Tf -20.042 -2.8614 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(40)]TJ ET endstream endobj 155 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 154 0 R /ProcSet 2 0 R >> /Contents 153 0 R >> endobj 154 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 156 0 obj << /Length 18348 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ 0 -2.4036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.5843 Td [(Si)-397(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.19 86.564 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1627 60.635 Td [(est)-403(mesurable,)]TJ /F8 1 Tf 6.7108 0 Td [(A)]TJ /F2 1 Tf 1.1446 0 Td [(est)-397(l'image)-397(r)24(\266)475(ecipro)-30(que)-397(du)-397(b)-30(or)30(\266)469(elien)]TJ /F6 1 Tf 15.458 0 Td [(f)]TJ /F2 1 Tf 0.50602 0 Td [(1)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.89157 0 Td [(\(c'est)-397(un)-397(ferm)30(\266)469(e\))-397(donc)-391(est)]TJ -28.042 -1.1988 Td [(mesurable.)]TJ 0 -1.5843 Td [(Si)]TJ /F8 1 Tf 1.1024 0 Td [(A)]TJ /F2 1 Tf 1.0181 0 Td [(est)-277(mesurable,)-283(p)-30(our)-271(toute)-271(partie)]TJ /F8 1 Tf 14.139 0 Td [(U)]TJ /F2 1 Tf 1.0602 0 Td [(de)]TJ ET 0.504 w 285.91 701.42 m 294.55 701.42 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 23.922 57.852 Td [(R)]TJ /F2 1 Tf 0.98795 0 Td [(\(et)-277(en)-271(particulier)-271(si)]TJ /F8 1 Tf 8.4819 0 Td [(U)]TJ /F2 1 Tf 1.0602 0 Td [(est)-277(b)-30(or)30(\266)469(elienne\),)-289(suiv)54(an)24(t)]TJ -29.12 -1.1988 Td [(que)]TJ /F8 1 Tf 1.8916 0 Td [(U)]TJ /F2 1 Tf 1.1566 0 Td [(p)-30(oss)24(\265)475(ede)-373(ou)-367(non)-367(0)-367(et)-367(1,)-373(l'ensem)30(ble)]TJ /F6 1 Tf 15.145 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.90362 0 Td [(j)]TJ /F2 1 Tf 0.60843 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.345 80.857 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.753 56.653 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.72289 0 Td [(2)]TJ /F8 1 Tf 0.99398 0 Td [(U)]TJ /F6 1 Tf 0.79518 0 Td [(g)]TJ /F2 1 Tf 0.86145 0 Td [(est)-373(vide,)]TJ /F8 1 Tf 4.0542 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)-373(le)-367(compl)24(\266)469(emen)24(taire)]TJ -30.56 -1.1988 Td [(de)]TJ /F8 1 Tf 1.4639 0 Td [(A)]TJ /F2 1 Tf 1.2108 0 Td [(ou)-463(tout)-469(l'espace.)-831(Il)-463(s'agit)-469(dans)-463(tous)-463(les)-463(cas)-463(d'une)-469(partie)-469(mesurable)-463(donc)-463(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.319 79.142 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 55.454 Td [(est)]TJ -37.934 -1.1988 Td [(mesurable.)]TJ ET 512.64 647.24 m 519.84 647.24 l 519.84 655.66 l 512.64 655.66 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 52.105 Td [(La)-373(mesurabilit)30(\266)469(e)-373(d'une)-373(application)-367(est)-373(souv)24(en)30(t)-373(prouv)54(\266)469(ee)-367(en)-367(utilisan)24(t)-373(une)-367(partie)-373(g)30(\266)469(en)30(\266)469(eratrice)]TJ 0 -1.1988 Td [(de)-331(la)-337(tribu)]TJ /F6 1 Tf 4.9458 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.853 73.443 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.361 50.906 Td [(.)-445(Plus)-337(pr)24(\266)475(ecis)24(\266)475(emen)24(t)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 39.972 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 47.918 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8373 0 Td [(Soien)24(t)]TJ /F2 1 Tf 2.9337 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.63253 0 Td [(et)]TJ /F2 1 Tf 1.0723 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.267 69.168 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.627 47.918 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.448 69.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.151 47.918 Td [(\))]TJ /F12 1 Tf 0.62651 0 Td [(des)-240(espaces)-246(mesurables)-246(et)]TJ /F6 1 Tf 11.163 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.698 69.168 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 40.139 47.918 Td [(une)-240(partie)]TJ -34.807 -1.1988 Td [(g)30(\266)469(en)30(\266)469(eratrice)-343(de)-349(la)-343(tribu)]TJ /F6 1 Tf 10.114 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.25 67.443 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 16.524 46.72 Td [(.)-481(Alors)-343(p)-30(our)-349(qu'une)-349(application)]TJ /F8 1 Tf 14.241 0 Td [(f)]TJ /F12 1 Tf 0.93976 0 Td [(de)]TJ /F8 1 Tf 1.3494 0 Td [(X)]TJ /F12 1 Tf 1.247 0 Td [(dans)]TJ /F8 1 Tf 2.3554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.75 67.443 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.187 46.72 Td [(soit)-349(mesurable)]TJ -32.855 -1.1988 Td [(il)-331(faut)-337(et)-337(il)-337(su\261t)-337(que)]TJ /F8 1 Tf 9.3976 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.931 65.728 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.398 45.521 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(soit)]TJ /F6 1 Tf 1.8916 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurable)-337(p)-30(our)-331(tous)-337(les)]TJ /F8 1 Tf 11.054 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(de)]TJ /F6 1 Tf 1.3313 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.241 65.728 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 35.392 45.521 Td [(.)]TJ /F1 1 Tf -30.06 -1.8675 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Notons)]TJ /F6 1 Tf 3.3916 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.241 63.064 Td [(0)]TJ /F4 1 Tf -0.36207 -0.94827 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.922 43.653 Td [(l'ensem)30(ble)-307(des)-301(parties)]TJ /F8 1 Tf 9.7289 0 Td [(A)]TJ /F2 1 Tf 1.0542 0 Td [(de)]TJ /F8 1 Tf 1.3012 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.672 63.064 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.494 43.653 Td [(don)24(t)-307(l'image)-301(r)24(\266)475(ecipro)-30(que)-307(par)]TJ /F8 1 Tf -27.163 -1.1988 Td [(f)]TJ /F2 1 Tf 0.96385 0 Td [(est)]TJ /F6 1 Tf 1.5964 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable.)-554(La)-373(prop)-30(osition)-343(\266)469(enonce)-367(alors)-373(que)]TJ /F6 1 Tf 19.994 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.19 61.34 Td [(0)]TJ /F4 1 Tf -0.36207 -0.93966 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.133 42.454 Td [(con)30(tien)18(t)]TJ /F6 1 Tf 3.8675 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.802 61.34 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.452 42.454 Td [(si)-373(et)-367(seulemen)30(t)-373(si)]TJ /F6 1 Tf 7.9639 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 61.34 Td [(0)]TJ /F4 1 Tf -0.37069 -0.93966 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 41.25 Td [(con)30(tien)24(t)]TJ /F6 1 Tf 3.8193 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.069 59.625 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.114 41.25 Td [(.)-439(Or)-319(nous)-319(a)24(v)24(ons)-325(remarqu)30(\266)469(e)-325(que)]TJ /F6 1 Tf 13.536 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.983 59.625 Td [(0)]TJ /F4 1 Tf -0.36207 -0.94828 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.048 41.25 Td [(est)-325(une)-325(tribu)-319(sur)]TJ /F8 1 Tf 7.5783 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.991 59.625 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.133 41.25 Td [(\(c'est)-325(l'image)-325(directe)-319(de)]TJ /F6 1 Tf -28.801 -1.1988 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(par)]TJ /F8 1 Tf 1.7771 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\),)-337(l')30(\266)469(equiv)54(alence)-343(d)30(\266)469(ecoule)-337(donc)-337(du)-337(fait)-337(que)]TJ /F6 1 Tf 17.898 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.405 57.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.151 40.051 Td [(est)-343(engendr)30(\266)469(ee)-337(par)]TJ /F6 1 Tf 8.0181 0 Td [(S)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.733 57.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.127 40.051 Td [(.)]TJ ET 512.64 477.54 m 519.84 477.54 l 519.84 485.96 l 512.64 485.96 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 37.906 Td [(On)-337(en)-331(d)24(\266)475(eduit)-337(imm)24(\266)469(ediatemen)30(t)-337(:)]TJ /F15 1 Tf 0 -1.8675 Td [(Cor)24(ollaire.)]TJ /F12 1 Tf 6.8494 0 Td [(Si)]TJ /F8 1 Tf 1.259 0 Td [(X)]TJ /F12 1 Tf 1.3373 0 Td [(et)]TJ /F8 1 Tf 1.259 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.25 52.168 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.657 36.039 Td [(son)24(t)-427(des)-433(espaces)-427(top)-36(ologiques)-433(toute)-427(application)-427(con)24(tin)24(ue)-427(est)]TJ -12.325 -1.1988 Td [(b)-30(or)30(\266)469(elienne.)-626(Mieux,)-415(si)]TJ /F8 1 Tf 9.8614 0 Td [(X)]TJ /F12 1 Tf 1.3012 0 Td [(est)-397(s)24(\266)469(epar)30(\266)469(e,)-415(toute)-397(application)-397(don)24(t)-397(l'ensem)30(ble)-397(des)-397(p)-30(oin)24(ts)-391(de)-397(dis-)]TJ -11.163 -1.1988 Td [(con)30(tin)18(uit)24(\266)469(e)-337(est)-337(d)24(\266)475(enom)24(brable)-337(est)-337(b)-30(or)24(\266)475(elienne.)]TJ /F1 1 Tf 0 -1.8675 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-301(tribu)-301(b)-30(or)30(\266)469(elienne)-301(de)]TJ /F8 1 Tf 9.9639 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.81 46.064 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.572 31.774 Td [(\266)469(etan)24(t)-301(engendr)24(\266)469(ee)-301(par)-301(ses)-301(ouv)18(erts,)-301(p)-30(our)-301(prou-)]TJ -20.241 -1.1988 Td [(v)24(er)-367(la)-367(mesurabilit)30(\266)469(e)-367(d'une)-367(application)]TJ /F8 1 Tf 16.5 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(de)]TJ /F8 1 Tf 1.3614 0 Td [(X)]TJ /F2 1 Tf 1.2711 0 Td [(dans)]TJ /F8 1 Tf 2.3735 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.078 44.349 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.349 30.575 Td [(il)-367(su\261t)-367(de)-367(prouv)30(er,)-379(p)-30(our)]TJ /F8 1 Tf 11.211 0 Td [(U)]TJ /F2 1 Tf 1.1506 0 Td [(ouv)18(ert)]TJ -36.38 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.836 42.625 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8494 29.376 Td [(,)-337(que)]TJ /F8 1 Tf 2.4759 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.629 42.625 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.994 29.376 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-337(est)-337(b)-30(or)30(\266)469(elienne.)]TJ -7.8494 -1.5843 Td [(Lorsque)]TJ /F8 1 Tf 3.7831 0 Td [(f)]TJ /F2 1 Tf 0.94578 0 Td [(est)-355(con)30(tin)18(ue,)-349(on)-349(sait)-349(que)-349(cette)-349(image)-349(r)30(\266)469(ecipro)-30(que)-349(est)-355(un)-349(ouv)24(ert)-349(et)-349(le)-355(r)30(\266)469(esultat)-349(en)]TJ -4.7289 -1.1988 Td [(d)30(\266)469(ecoule.)]TJ 0 -1.5843 Td [(Lorsque)]TJ /F8 1 Tf 3.9157 0 Td [(f)]TJ /F2 1 Tf 1.0723 0 Td [(est)-481(con)30(tin)18(ue)-475(en)-475(dehors)-481(d'une)-475(partie)-481(d)24(\266)469(enom)24(brable)]TJ /F8 1 Tf 22.608 0 Td [(D)]TJ /F2 1 Tf 0.85542 0 Td [(,)-512(remarquons)-475(que,)-518(p)-30(our)]TJ /F8 1 Tf -28.452 -1.1988 Td [(a)]TJ /F6 1 Tf 0.81928 0 Td [(2)]TJ /F8 1 Tf 0.96386 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.034 34.659 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.7831 23.81 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\),)-355(si)]TJ /F8 1 Tf 2.0301 0 Td [(a)]TJ /F2 1 Tf 0.87349 0 Td [(est)-349(un)-343(p)-30(oin)24(t)-349(de)-343(con)30(tin)18(uit)24(\266)469(e,)]TJ /F8 1 Tf 11.952 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.371 34.659 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.488 23.81 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-343(est)-349(un)-349(v)24(oisinage)-349(de)]TJ /F8 1 Tf 9.3976 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(.)-475(Il)-343(s'ensuit)-343(que)]TJ /F8 1 Tf -32.265 -1.2048 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4828 32.943 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7 22.605 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-409(est)-403(la)-403(r)30(\266)469(eunion)-403(de)-403(son)-403(in)24(t)24(\266)469(erieur)-397(et)-403(de)-403(l'ensem)24(ble)-397(des)-409(p)-24(oin)24(ts)-403(de)-403(discon)30(tin)24(uit)18(\266)475(e)-409(qu'il)]TJ -2.8494 -1.1988 Td [(p)-30(oss)30(\265)469(ede.)-421(Il)-259(s'agit)-265(donc)-259(de)-265(la)-259(r)24(\266)475(eunion)-265(d'un)-271(ouv)24(ert)-265(et)-259(d'une)-265(partie)-271(de)]TJ /F8 1 Tf 28.753 0 Td [(D)]TJ /F2 1 Tf 0.85542 0 Td [(.)-421(Le)-265(premier)-265(ensem)24(ble)]TJ -29.608 -1.1988 Td [(est)-373(b)-30(or)30(\266)469(elien)-373(parce)-373(qu'ouv)24(ert,)-379(le)-373(deuxi)24(\265)469(eme)-367(est)-373(b)-30(or)24(\266)475(elien)-373(parce)-373(que)-379(r)30(\266)469(eunion)-373(d)24(\266)475(enom)24(brable)-373(de)]TJ 0 -1.1988 Td [(singletons)-409(qui)-409(son)24(t)-415(des)-415(ferm)30(\266)469(es)-409(\(puisque)]TJ /F8 1 Tf 17.837 0 Td [(X)]TJ /F2 1 Tf 1.3193 0 Td [(est)-415(s)24(\266)475(epar)24(\266)475(e\).)-680(Comme)-415(r)30(\266)469(eunion)-415(de)-409(deux)-409(parties)]TJ -19.157 -1.2048 Td [(b)-30(or)30(\266)469(eliennes,)]TJ /F8 1 Tf 5.4819 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.328 26.073 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.482 17.804 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-331(est)-343(bien)-337(b)-24(or)24(\266)469(elienne.)]TJ ET 512.64 211.64 m 519.84 211.64 l 519.84 220.06 l 512.64 220.06 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 15.659 Td [(Cette)-337(prop)-30(osition)-337(p)-30(ermet)-337(aussi)-337(d')24(\266)475(enoncer)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 10.57 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 12.671 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.012 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.0783 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(\))]TJ /F12 1 Tf 0.77108 0 Td [(un)-379(espace)-385(mesurable)-379(et)]TJ /F8 1 Tf 10.645 0 Td [(f)]TJ /F12 1 Tf 0.9759 0 Td [(une)-385(application)-379(de)]TJ /F8 1 Tf 8.5361 0 Td [(X)]TJ /F12 1 Tf -38.259 -1.1988 Td [(dans)]TJ ET 91.656 147.02 m 100.3 147.02 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6687 11.473 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)-445(Alors,)]TJ -3.0602 -1.5843 Td [({)-668(si)]TJ /F8 1 Tf 2.1747 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-337(mesurable,)-337(p)-30(our)-337(tout)]TJ /F8 1 Tf 11.108 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(dans)]TJ ET 271.8 128.08 m 280.44 128.08 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 22.741 9.8882 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(,)-337(les)-331(ensem)18(bles)-337(suiv)54(an)24(ts)-331(son)24(t)-337(mesurables)-343(:)]TJ /F8 1 Tf -12.699 -1.1988 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.534 13.021 Td [(\241)]TJ /F7 1 Tf -0.15517 -0.93965 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.506 8.6894 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(<)-277(a)]TJ /F6 1 Tf 1.5843 0 Td [(g)]TJ /F12 1 Tf 0.5 0 Td [(,)]TJ /F8 1 Tf 5.7108 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.681 13.021 Td [(\241)]TJ /F7 1 Tf -0.31897 -0.93965 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.38 8.6894 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(g)]TJ /F12 1 Tf 0.5 0 Td [(,)]TJ /F8 1 Tf -28.012 -1.1988 Td [(U)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.543 11.306 Td [(+)]TJ /F7 1 Tf -0.15517 -0.94828 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.506 7.4906 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.66265 0 Td [(>)-283(a)]TJ /F6 1 Tf 1.5904 0 Td [(g)]TJ /F12 1 Tf 0.5 0 Td [(,)]TJ /F8 1 Tf 5.7169 0 Td [(V)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.69 11.306 Td [(+)]TJ /F7 1 Tf -0.31897 -0.94828 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.373 7.4906 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(g)]TJ /F12 1 Tf 0.5 0 Td [(,)]TJ /F8 1 Tf -19.56 -1.2048 Td [(W)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.853 8.7797 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.928 6.2858 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(a)]TJ /F6 1 Tf 0.5241 0 Td [(g)]TJ /F12 1 Tf 0.5 0 Td [(.)]TJ /F2 1 Tf -6.0783 -2.3976 Td [(41)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 158 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 157 0 R /ProcSet 2 0 R >> /Contents 156 0 R >> endobj 157 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 159 0 obj << /Length 19776 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F12 1 Tf 0 -2.4036 Td [({)-385(r)24(\266)475(ecipro)-30(quemen)24(t,)-403(si)-391(p)-30(our)-391(tout)]TJ /F8 1 Tf 14.03 0 Td [(a)]TJ /F12 1 Tf 0.91566 0 Td [(dans)]TJ /F3 1 Tf 2.3976 0 Td [(R)]TJ /F12 1 Tf 1.1084 0 Td [(l'ensem)24(ble)]TJ /F8 1 Tf 4.8735 0 Td [(U)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 42.138 89.633 Td [(+)]TJ /F7 1 Tf -0.15517 -0.94828 Td [(a)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 30.5 62.22 Td [(est)-391(mesurable,)-403(la)-391(fonction)]TJ /F8 1 Tf 11.783 0 Td [(f)]TJ /F12 1 Tf 0.98193 0 Td [(est)]TJ -37.934 -1.1988 Td [(mesurable.)]TJ /F1 1 Tf 0 -1.9699 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-445(premi)24(\265)469(ere)-445(partie)-451(de)-439(cet)-421(\266)475(enonc)24(\266)475(e)-451(d)30(\266)469(ecoule)-451(du)-445(fait)-445(que)-445(les)-445(ensem)24(bles)]TJ -8.8193 -1.1988 Td [(consid)30(\266)469(er)24(\266)469(es)-343(son)24(t)-343(les)-343(images)-349(r)30(\266)469(ecipro)-30(ques)-343(par)]TJ /F8 1 Tf 18.741 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(de)-337([)]TJ /F6 1 Tf 1.6205 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [([,)-349([)]TJ /F6 1 Tf 1.1807 0 Td [(\2411)]TJ /F2 1 Tf 1.7771 0 Td [(;)]TJ /F8 1 Tf 0.44579 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(],)-343(])]TJ /F8 1 Tf 1.1747 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(],)-343([)]TJ /F8 1 Tf 1.1747 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-343(et)]TJ /F6 1 Tf 1.7952 0 Td [(f)]TJ /F8 1 Tf 0.49999 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(g)]TJ /F2 1 Tf -38.663 -1.1988 Td [(qui)-331(son)24(t)-337(ouv)18(erts)-337(ou)-337(ferm)30(\266)469(es)-331(donc)-337(b)-30(or)30(\266)469(eliens.)]TJ 0 -1.6506 Td [(La)-349(deuxi)30(\265)469(eme)-349(partie)-349(d)30(\266)469(ecoule)-349(du)-349(fait)-349(que)-349(les)]TJ /F8 1 Tf 18.988 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.94 79.306 Td [(\241)]TJ /F7 1 Tf -0.15517 -0.94827 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.133 55.003 Td [(son)24(t)-349(les)-349(images)-349(r)24(\266)469(ecipro)-24(ques)-349(des)-349(in)24(terv)54(alles)]TJ -20.801 -1.1988 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-331(don)24(t)-337(on)-337(a)-331(vu)-337(qu'ils)-343(engendraien)24(t)-337(la)-331(tribu)-337(des)-337(b)-30(or)30(\266)469(eliens)-337(de)]TJ ET 0.504 w 410.76 652.96 m 419.4 652.96 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 34.367 53.804 Td [(R)]TJ /F2 1 Tf 1.0602 0 Td [(\(cf.)]TJ /F1 1 Tf 1.8614 0 Td [(I)-36(I)-30(I.3.3.)]TJ /F2 1 Tf 3.4819 0 Td [(\).)]TJ ET 512.64 641.84 m 519.84 641.84 l 519.84 650.26 l 512.64 650.26 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 51.515 Td [(On)-337(en)-331(d)24(\266)475(eduit)-337(un)-337(premier)-307(\266)475(enonc)24(\266)469(e)-337(de)-331(stabilit)24(\266)469(e)-337(de)-331(l'ensem)24(ble)-331(des)-337(fonctions)-337(mesurables.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 40.394 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 48.424 Td [(5.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Soit)]TJ /F2 1 Tf 2.0542 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(un)-337(espace)-331(mesurable.)-451(Alors)-331(:)]TJ /F1 1 Tf -16.398 -1.6506 Td [(\(i\))]TJ /F12 1 Tf 1.8795 0 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-343(mesurable)-331(la)-337(fonction)-337(opp)-30(os)24(\266)469(ee)]TJ /F6 1 Tf 15.03 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-343(mesurable.)]TJ /F1 1 Tf -21.03 -1.6446 Td [(\(ii\))]TJ /F12 1 Tf 2.2892 0 Td [(Si)]TJ /F2 1 Tf 1.2108 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.138 64.366 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.217 45.129 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.172 64.366 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 12.265 45.129 Td [(est)-379(une)-379(famille)-379(d)24(\266)475(enom)24(brable)]TJ ET 228.24 537.76 m 295.06 537.76 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.066 45.129 Td [(de)-373(fonctions)-379(mesurables,)-397(les)-379(fonctions)]TJ /F2 1 Tf 16.934 0 Td [(sup)]TJ /F8 1 Tf 1.6747 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 64.366 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 43.93 Td [(et)]TJ /F2 1 Tf 1.1687 0 Td [(inf)]TJ /F8 1 Tf 1.3795 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.983 62.65 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0361 43.93 Td [(son)24(t)-337(mesurables.)]TJ /F1 1 Tf -3.8494 -1.6446 Td [(\(iii\))]TJ /F12 1 Tf 2.6506 0 Td [(Si)]TJ /F8 1 Tf 1.2289 0 Td [(f)]TJ /F12 1 Tf 1 0 Td [(est)-403(une)-403(fonction)-403(mesurable,)-415(sa)-403(partie)-403(p)-30(ositiv)24(e)]TJ /F8 1 Tf 20.627 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.621 60.297 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 31.843 42.286 Td [(,)-415(sa)-403(partie)-403(n)24(\266)469(egativ)30(e)]TJ /F8 1 Tf 8.9578 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.095 60.297 Td [(\241)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 42.367 42.286 Td [(et)-403(sa)]TJ -37.036 -1.2048 Td [(v)54(aleur)-337(absolue)]TJ /F6 1 Tf 6.5361 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F12 1 Tf 0.60843 0 Td [(son)24(t)-337(mesurables.)]TJ /F1 1 Tf -8.1386 -1.6446 Td [(\(iv\))]TJ /F12 1 Tf 2.6747 0 Td [(P)30(our)-433(toute)-427(suite)-427(de)-427(fonctions)-427(mesurables)]TJ /F8 1 Tf 18.723 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.784 56.219 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 27.645 39.436 Td [(,)-451(les)-427(fonctions)]TJ /F2 1 Tf 6.6265 0 Td [(lim)-168(sup)]TJ /F8 1 Tf 3.2289 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.362 56.219 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.53 39.436 Td [(,)]TJ /F2 1 Tf 0.72892 0 Td [(lim)-174(inf)]TJ /F8 1 Tf 2.9398 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.095 56.219 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 39.436 Td [(et)]TJ /F2 1 Tf -38.331 -1.1988 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.6121 54.435 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.2651 38.238 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.095 54.504 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6325 38.238 Td [(si)-337(elle)-337(existe)-337(son)24(t)-337(mesurables.)]TJ /F1 1 Tf -3.3012 -1.9699 Td [(D)24(\266)548(emonstration.)]TJ 0.49398 -1.6446 Td [(\(i\))]TJ /F2 1 Tf 2.0783 0 Td [(P)30(our)]TJ /F8 1 Tf 2.5301 0 Td [(a)]TJ /F2 1 Tf 0.95783 0 Td [(dans)]TJ ET 165.31 423.72 m 173.95 423.72 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 13.831 34.623 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-457(la)-433(condition)]TJ /F6 1 Tf 6.4337 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.8253 0 Td [(>)-445(a)]TJ /F2 1 Tf 2.1566 0 Td [(\266)475(equiv)48(aut)-433(\265)500(a)]TJ /F8 1 Tf 5.1145 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.83133 0 Td [(<)]TJ /F6 1 Tf 1.2229 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-734(elle)-439(d)30(\266)469(e\257nit)-433(donc)]TJ -31.006 -1.1988 Td [(une)-379(partie)-385(mesurable)-385(de)]TJ /F8 1 Tf 11.036 0 Td [(X)]TJ /F2 1 Tf 1.2892 0 Td [(d'apr)24(\265)475(es)-385(la)-379(prop)-30(osition)-385(pr)24(\266)475(ec)24(\266)469(eden)24(te.)-584(La)-385(mesurabilit)24(\266)469(e)-379(de)]TJ /F6 1 Tf 24.084 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F2 1 Tf 0.97591 0 Td [(en)]TJ -38.163 -1.1988 Td [(d)30(\266)469(ecoule.)]TJ /F1 1 Tf 0.1747 -1.6506 Td [(\(ii\))]TJ /F2 1 Tf 2.253 0 Td [(Si)-367(\()]TJ /F8 1 Tf 1.5843 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.078 43.538 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.169 30.575 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.112 43.538 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.199 30.575 Td [(est)-367(une)-367(famille)-361(de)-361(fonctions)-367(mesurables,)-373(p)-30(our)-361(tout)]TJ /F8 1 Tf 22.47 0 Td [(a)]TJ /F2 1 Tf 0.89157 0 Td [(dans)]TJ ET 453.31 375.4 m 461.95 375.4 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 37.928 30.575 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-367(la)-367(condition)]TJ -33.319 -1.1988 Td [(sup)]TJ /F8 1 Tf 1.6687 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.724 41.823 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8253 29.376 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.93976 0 Td [(>)-548(a)]TJ /F2 1 Tf 2.3253 0 Td [(\266)475(equiv)48(aut)-500(\265)500(a)-500(l'existence)-500(d'un)-499(indice)]TJ /F8 1 Tf 15.717 0 Td [(i)]TJ /F2 1 Tf 0.84337 0 Td [(tel)-500(que)]TJ /F8 1 Tf 3.6325 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.853 41.823 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.072 29.376 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.93976 0 Td [(>)-554(a)]TJ /F2 1 Tf 1.8614 0 Td [(.)-933(Il)-499(s'ensuit)-493(que)]TJ -31.5 -1.1988 Td [(l'ensem)30(ble)]TJ /F6 1 Tf 4.9036 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 1.006 0 Td [(2)]TJ /F8 1 Tf 1.1024 0 Td [(X)]TJ /F6 1 Tf 1.3434 0 Td [(j)]TJ /F2 1 Tf 0.71084 0 Td [(sup)]TJ /F8 1 Tf 1.6747 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.414 40.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.392 28.177 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.8253 0 Td [(>)-433(a)]TJ /F6 1 Tf 1.741 0 Td [(g)]TJ /F2 1 Tf 0.92771 0 Td [(est)-427(la)-427(r)24(\266)469(eunion)-427(des)-433(ensem)24(bles)]TJ /F6 1 Tf 13.139 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 1.006 0 Td [(2)]TJ /F8 1 Tf 1.0964 0 Td [(X)]TJ /F6 1 Tf 1.3434 0 Td [(j)]TJ /F8 1 Tf 0.71084 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.44 40.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.47 28.177 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F8 1 Tf 0.8253 0 Td [(>)-433(a)]TJ /F6 1 Tf 1.741 0 Td [(g)]TJ /F2 1 Tf -38.663 -1.2048 Td [(qui)-433(son)24(t)-439(mesurables)-439(puisque)-439(les)]TJ /F8 1 Tf 14.572 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.181 38.392 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.157 26.973 Td [(le)-445(son)24(t.)-753(Ainsi)-439(lorsque)-433(l'ensem)24(ble)]TJ /F8 1 Tf 15.133 0 Td [(I)]TJ /F2 1 Tf 0.95181 0 Td [(est)-439(d)24(\266)469(enom)24(brable)]TJ -31.91 -1.1988 Td [(on)-385(conclut)-385(que)]TJ /F6 1 Tf 6.8976 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.93373 0 Td [(2)]TJ /F8 1 Tf 1.0301 0 Td [(X)]TJ /F6 1 Tf 1.2651 0 Td [(j)]TJ /F2 1 Tf 0.63855 0 Td [(sup)]TJ /F8 1 Tf 1.6747 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.845 36.668 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.09 25.774 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.74699 0 Td [(>)-361(a)]TJ /F6 1 Tf 1.6687 0 Td [(g)]TJ /F2 1 Tf 0.88554 0 Td [(est)-385(mesurable)-385(et)-385(donc)-385(que)-385(la)-385(fonction)-385(sup)]TJ /F8 1 Tf 18.705 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.888 36.668 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 25.774 Td [(est)]TJ -37.934 -1.1988 Td [(mesurable.)]TJ 0 -1.6446 Td [(De)-421(m)24(^)475(eme,)-439(sous)-421(ces)-415(h)24(yp)-36(oth)24(\265)469(eses,)-439(la)-421(fonction)-421(inf)]TJ /F8 1 Tf 20.873 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.198 32.599 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.446 22.93 Td [(est)-421(mesurable.)-698(En)-421(e\256et,)-439(on)-421(a)-415(inf)]TJ /F8 1 Tf 15.03 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.483 32.599 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.717 22.93 Td [(=)]TJ /F6 1 Tf -38.386 -1.2048 Td [(\241)]TJ /F9 1 Tf 0.77711 0.81325 Td [(\241)]TJ /F2 1 Tf 0.62651 -0.81325 Td [(sup)0(\()]TJ /F6 1 Tf 1.8916 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.164 30.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.229 21.726 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F2 1 Tf 0.8735 -0.81325 Td [(et)-421(les)-421(r)24(\266)475(esultats)-421(pr)24(\266)469(ec)24(\266)475(eden)24(ts)-415(mon)18(tren)24(t)-421(successiv)30(emen)24(t)-421(que)-421(les)-421(fonctions)]TJ /F6 1 Tf 31.12 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.802 30.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 21.726 Td [(,)]TJ -38.886 -1.3012 Td [(sup)0(\()]TJ /F6 1 Tf 1.8916 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.155 29.021 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8253 20.424 Td [(\))-337(et)]TJ /F6 1 Tf 1.8916 0 Td [(\241)]TJ /F9 1 Tf 0.77711 0.81325 Td [(\241)]TJ /F2 1 Tf 0.6265 -0.81325 Td [(sup)0(\()]TJ /F6 1 Tf 1.8916 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.871 29.021 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.614 20.424 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F2 1 Tf 0.78916 -0.81325 Td [(son)18(t)-331(mesurables.)]TJ /F1 1 Tf -11.608 -1.6446 Td [(\(iii\))]TJ /F2 1 Tf 2.7952 0 Td [(En)-475(particulier,)-506(p)-30(our)]TJ /F8 1 Tf 9.506 0 Td [(f)]TJ /F2 1 Tf 1.0663 0 Td [(mesurable,)-512(le)-475(r)24(\266)475(esultat)-481(pr)30(\266)469(ec)24(\266)475(eden)24(t)-475(mon)24(tre)-475(que)-475(les)-475(fonctions)]TJ /F8 1 Tf -13.223 -1.1988 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3276 24.943 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.759 17.581 Td [(=)-283(sup)0(\()]TJ /F8 1 Tf 2.9518 0 Td [(f)-54(;)]TJ /F2 1 Tf 0.98795 0 Td [(0\),)]TJ /F8 1 Tf 1.5 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.164 24.943 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.645 17.581 Td [(=)-277(sup)0(\()]TJ /F6 1 Tf 2.9518 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)-54(;)]TJ /F2 1 Tf 0.98795 0 Td [(0\))-337(et)]TJ /F6 1 Tf 2.3855 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.55422 0 Td [(=)-277(sup)0(\()]TJ /F8 1 Tf 2.9518 0 Td [(f)-54(;)]TJ /F6 1 Tf 0.98795 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-337(son)24(t)-337(mesurables.)]TJ /F1 1 Tf -22.277 -1.6446 Td [(\(iv\))]TJ /F2 1 Tf 2.512 0 Td [(De)-343(m)24(^)469(eme,)-349(p)-24(our)-349(une)-343(suite)-349(de)-343(fonctions)-349(mesurables)]TJ /F8 1 Tf 22.169 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.483 22.59 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.928 15.936 Td [(,)-349(les)-349(fonctions)]TJ /F8 1 Tf 6.3554 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.207 23.392 Td [(0)]TJ /F7 1 Tf -0.15517 -0.93965 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.53 15.936 Td [(=)-313(sup)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.681 21.599 Td [(n)]TJ /F17 1 Tf 0.7069 0 Td [(>)]TJ /F7 1 Tf 0.88793 0 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.307 15.936 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.81 22.59 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.681 15.936 Td [(son)18(t)]TJ -37.349 -1.9819 Td [(mesurables)-343(ainsi)-337(donc)-343(que)-343(lim)-168(sup)]TJ /F8 1 Tf 14.97 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.75 19.754 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.62 13.954 Td [(=)-289(inf)]TJ /F8 1 Tf 2.4518 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.302 20.556 Td [(0)]TJ /F7 1 Tf -0.15517 -0.93966 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.024 13.954 Td [(.)-463(De)-343(la)-337(m)24(^)469(eme)-337(fa)-6(\273)439(con)-337(on)-343(prouv)30(e)-343(que)-343(lim)-174(inf)]TJ /F8 1 Tf 18.434 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 19.754 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 12.75 Td [(est)-301(mesurable,)-301(d'ailleurs)-301(on)-295(a)-295(lim)-168(inf)]TJ /F8 1 Tf 15.837 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.991 18.038 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.482 12.75 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(\241)]TJ /F2 1 Tf 0.94578 0 Td [(lim)-168(sup)0(\()]TJ /F6 1 Tf 3.4518 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.784 18.038 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.747 12.75 Td [(\).)-433(Si)-301(en)-295(outre)-295(la)-295(suite)-301(de)-295(fonctions)]TJ -24.416 -1.1988 Td [(admet)-331(une)-337(limite)-331(simple,)-337(celle-ci)-337(est)-307(\266)469(egale)-337(\265)499(a)-331(lim)-174(sup)]TJ /F8 1 Tf 22.584 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.647 16.314 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.283 11.551 Td [(donc)-331(est)-343(mesurable.)]TJ ET 512.64 136.9 m 519.84 136.9 l 519.84 145.33 l 512.64 145.33 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 9.581 Td [(Remarque.)]TJ /F2 1 Tf 6.4096 0 Td [(Les)-343(\266)469(enonc)24(\266)475(es)-373(\(i\))-379(et)-373(\(iii\))-373(de)-373(la)-373(prop)-36(osition)-373(pr)24(\266)469(ec)24(\266)475(eden)24(te)-373(p)-30(euv)24(en)30(t)-379(se)-373(d)24(\266)475(eduire)-379(du)]TJ -6.4096 -1.1988 Td [(fait)-271(que)-271(la)-271(mesurabilit)30(\266)469(e)-271(se)-271(conserv)24(e)-271(par)-271(comp)-30(osition)-271(des)-271(applications)-271(et)-271(que)-271(les)-271(applications)]TJ /F8 1 Tf 0 -1.1988 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)-277(\241)]TJ /F8 1 Tf 2.0542 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.345 10.064 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.783 7.1834 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.034 10.064 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.102 7.1834 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)-277(j)]TJ /F8 1 Tf 1.5542 0 Td [(x)]TJ /F6 1 Tf 0.57229 0 Td [(j)]TJ /F2 1 Tf 0.61446 0 Td [(son)24(t)-337(mesurables)-337(de)]TJ ET 364.18 95.756 m 372.82 95.756 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 30.47 7.1834 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(dans)]TJ ET 404.71 95.756 m 413.35 95.756 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 33.861 7.1834 Td [(R)]TJ /F2 1 Tf 1.0602 0 Td [(parce)-337(que)-337(con)24(tin)24(ues.)]TJ /F5 1 Tf -29.59 -3.2952 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(42)]TJ ET endstream endobj 161 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 160 0 R /ProcSet 2 0 R >> /Contents 159 0 R >> endobj 160 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R /F17 97 0 R >> endobj 162 0 obj << /Length 17734 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F2 1 Tf 0 -2.4036 Td [(V)84(oici)-337(un)-337(exemple)-337(imp)-36(ortan)24(t)-337(de)-331(fonctions)-337(mesurables.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 49.454 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 59.286 Td [(6.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83132 0.084335 Td [(\266)]TJ -0.0060234 -0.084335 Td [(efinition.)]TJ /F12 1 Tf 6.247 0 Td [(Soit)]TJ /F2 1 Tf 2.3675 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 1.0301 0 Td [(un)-650(espace)-644(mesurable.)-1373(On)-644(app)-30(elle)]TJ /F1 1 Tf 16.94 0 Td [(fonctions)]TJ /F6 1 Tf -34.651 -1.1988 Td [(T)]TJ /F1 1 Tf 0.79518 0 Td [(-)18(\266)548(etag)24(\266)548(ees)]TJ /F12 1 Tf 4.4639 0 Td [(\(ou)-469(simplemen)24(t)]TJ /F1 1 Tf 7.3313 0 Td [(\266)548(etag)24(\266)548(ees)]TJ /F12 1 Tf 3.6386 0 Td [(\))-469(sur)]TJ /F8 1 Tf 2.6566 0 Td [(X)]TJ /F12 1 Tf 1.3735 0 Td [(les)-463(com)30(binaisons)-463(lin)24(\266)469(eaires)-463(\(\257nies\))-463(de)-463(fonc-)]TJ -20.259 -2.1687 Td [(tions)-319(indicatrices)-319(de)-319(parties)]TJ /F6 1 Tf 12.295 0 Td [(T)]TJ /F12 1 Tf 0.80121 0 Td [(-mesurables)-313(de)]TJ /F8 1 Tf 6.759 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-439(Autremen)24(t)-319(dit,)-325(ce)-319(son)24(t)-319(les)]TJ /F8 1 Tf 12.349 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(=)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.828 81.814 Td [(i)]TJ /F4 1 Tf 0.39655 0 Td [(=)]TJ /F7 1 Tf 0.87931 0 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 40.38 56.87 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.888 78.331 Td [(i)]TJ /F4 1 Tf 0.40517 0 Td [(=1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.994 55.918 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.009 79.814 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.964 55.918 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.284 79.814 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 88.253 111.34 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 53.846 Td [(o)-24(\265)524(u)-331(les)]TJ /F8 1 Tf 2.8373 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.603 76.84 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 9.4759 53.846 Td [(son)24(t)-337(des)-337(r)24(\266)475(eels)-337(et)-337(les)]TJ /F8 1 Tf 8.7771 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.198 76.84 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 19.675 53.846 Td [(son)24(t)-337(des)-337(parties)-337(mesurables.)]TJ /F2 1 Tf -14.343 -2.0663 Td [(Remarquons)-331(que)-337(les)-337(fonctions)-307(\266)469(etag)24(\266)469(ees)-331(son)24(t)-337(des)-337(fonctions)-337(\265)499(a)-331(v)48(aleurs)-337(\257nies.)]TJ 0 -2.0663 Td [(Un)-337(outil)-331(cl)18(\266)475(e)-343(p)-24(our)-337(l')24(\266)475(etude)-343(de)-331(ces)-331(fonctions)-337(est)-343(l')30(\266)469(enonc)30(\266)469(e)-337(suiv)48(an)30(t)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 39.022 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 46.78 Td [(7.)]TJ /F15 1 Tf 1.753 0 Td [(Lemme.)]TJ /F12 1 Tf 4.5723 0 Td [(\()]TJ /F1 1 Tf 0.39157 0 Td [(de)-554(d)24(\266)548(ecomp)-36(osition)]TJ /F12 1 Tf 8.9337 0 Td [(\))-1000(P)24(our)-481(toute)-481(famille)-487(\257nie)]TJ /F2 1 Tf 12.446 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.767 66.728 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.91 46.78 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.802 66.728 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 40.06 46.78 Td [(de)-481(parties)]TJ /F6 1 Tf -34.729 -1.2048 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(-mesurables)-439(on)-445(p)-30(eut)-439(trouv)24(er)-445(une)-439(famille)-439(\257nie)-445(de)-439(parties)]TJ /F8 1 Tf 25.548 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.414 65.004 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.295 45.575 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurables,)-463(non)-445(vides,)]TJ -28.765 -1.1988 Td [(deux)-295(\265)499(a)-301(deux)-301(disjoin)30(tes)-307(qui)-301(recouvren)24(t)]TJ /F8 1 Tf 16.512 0 Td [(X)]TJ /F12 1 Tf 1.2108 0 Td [(et)-301(telle)-301(que)-301(c)18(haque)]TJ /F8 1 Tf 8.3916 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.078 63.288 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.831 44.376 Td [(soit)-301(la)-301(r)30(\266)469(eunion)-301(de)-301(certaines)]TJ -27.5 -1.1988 Td [(des)]TJ /F8 1 Tf 1.7289 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.19 61.573 Td [(j)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.241 43.177 Td [(.)]TJ /F1 1 Tf -2.9096 -1.8133 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(In)30(tro)-36(duisons)-313(l'application)]TJ /F8 1 Tf 11.446 0 Td [(\301)]TJ /F2 1 Tf 0.90361 0 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(X)]TJ /F2 1 Tf 1.2169 0 Td [(dans)-313(l'ensem)30(ble)]TJ /F6 1 Tf 7.1084 0 Td [(P)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-313(des)-313(parties)-313(de)]TJ /F8 1 Tf -32.494 -1.1988 Td [(I)]TJ /F2 1 Tf 0.96988 0 Td [(d)30(\266)469(e\257nie)-451(par)]TJ /F8 1 Tf 5.6325 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-475(=)]TJ /F6 1 Tf 2.1145 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(i)]TJ /F6 1 Tf 0.81928 0 Td [(2)]TJ /F8 1 Tf 1.1386 0 Td [(I)]TJ /F6 1 Tf 0.99398 0 Td [(j)]TJ /F8 1 Tf 0.75301 0 Td [(x)]TJ /F6 1 Tf 1.0422 0 Td [(2)]TJ /F8 1 Tf 1.1446 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.552 57.262 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.084 40.165 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)-795(Les)-451(images)-457(r)24(\266)475(ecipro)-30(ques)]TJ /F8 1 Tf 11.801 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.724 57.262 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.392 40.165 Td [(=)]TJ /F8 1 Tf 1.253 0 Td [(\301)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.155 58.064 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.313 40.165 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(K)]TJ /F2 1 Tf 0.91566 0 Td [(\))-457(p)-30(our)]TJ /F8 1 Tf -36.289 -1.2048 Td [(K)]TJ /F2 1 Tf 1.3012 0 Td [(parcouran)24(t)]TJ /F6 1 Tf 5.1386 0 Td [(P)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.5241 0 Td [(\))-379(son)24(t)-355(\266)469(evidemmen)24(t)-385(deux)-379(\265)499(a)-379(deux)-379(disjoin)24(tes)-385(et)-379(recouvren)24(t)]TJ /F8 1 Tf 25.09 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)-584(Mon)24(trons)]TJ -34.127 -1.1988 Td [(qu'elles)-337(son)24(t)-337(mesurables.)]TJ 0 -1.5542 Td [(P)30(ar)-301(d)30(\266)469(e\257nition)]TJ /F8 1 Tf 6.253 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.233 51.599 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.205 36.207 Td [(signi\257e)-301(que)]TJ /F8 1 Tf 5.1325 0 Td [(x)]TJ /F2 1 Tf 0.87349 0 Td [(est)-301(dans)]TJ /F8 1 Tf 3.8253 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.905 51.599 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.422 36.207 Td [(si)]TJ /F8 1 Tf 0.96988 0 Td [(i)]TJ /F6 1 Tf 0.62048 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(K)]TJ /F2 1 Tf 1.2169 0 Td [(et)-301(est)-307(en)-295(dehors)-307(de)]TJ /F8 1 Tf 8.3976 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.276 51.599 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.952 36.207 Td [(sinon.)-439(P)30(ar)]TJ -34.62 -1.1988 Td [(suite)]TJ /F8 1 Tf 2.3916 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.121 49.883 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.5723 35.009 Td [(est)-331(l'in)24(tersection)-337(de)-325(tous)-331(les)]TJ /F8 1 Tf 12.422 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.552 50.685 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.41 35.009 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.3855 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.552 50.685 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.157 35.009 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.017 49.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.627 35.009 Td [(si)]TJ /F8 1 Tf 1 0 Td [(i)]TJ /F6 1 Tf 0.62048 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(K)]TJ /F2 1 Tf 1.253 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.164 50.685 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.018 35.009 Td [(est)-337(le)-331(compl)24(\266)469(emen)24(taire)]TJ -29.687 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.586 48.159 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.0422 33.804 Td [(sinon.)-445(De)-319(toutes)-319(fa)0(\273)439(cons)-319(ces)]TJ /F8 1 Tf 12.133 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.948 48.969 Td [(0)]TJ /F7 1 Tf 0 -0.94828 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.572 33.804 Td [(son)24(t)-319(mesurables)-319(ainsi)-319(donc)-319(que)-319(leur)-313(in)24(tersection)]TJ /F8 1 Tf 21.114 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.172 48.159 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 33.804 Td [(.)]TJ -38.886 -1.5542 Td [(P)30(ar)-307(ailleurs,)-307(p)-30(our)]TJ /F8 1 Tf 7.8916 0 Td [(i)]TJ /F6 1 Tf 0.62048 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)]TJ /F8 1 Tf 0.59036 0 Td [(x)]TJ /F2 1 Tf 0.87349 0 Td [(est)-307(dans)]TJ /F8 1 Tf 3.8434 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.569 45.935 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22 32.25 Td [(si)-307(et)-307(seulemen)30(t)-307(si)]TJ /F8 1 Tf 7.7048 0 Td [(i)]TJ /F6 1 Tf 0.6265 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(\301)]TJ /F2 1 Tf 0.59036 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)-439(Cette)-307(derni)24(\265)469(ere)-307(condition)]TJ -27.5 -1.5 Td [(s')24(\266)475(ecrit)-343(aussi)]TJ /F6 1 Tf 5.4277 0 Td [(9)]TJ /F8 1 Tf 0.55422 0 Td [(K)]TJ /F6 1 Tf 1.2108 0 Td [(2)-295(P)]TJ /F2 1 Tf 1.7349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.72891 1.1084 Td [(\263)]TJ /F8 1 Tf 0.59639 -1.1084 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(=)]TJ /F8 1 Tf 1.747 0 Td [(K)]TJ /F2 1 Tf 1.2651 0 Td [(et)]TJ /F8 1 Tf 1.1747 0 Td [(i)]TJ /F6 1 Tf 0.63855 0 Td [(2)]TJ /F8 1 Tf 0.95783 0 Td [(K)]TJ /F9 1 Tf 0.92169 1.1084 Td [(\264)]TJ /F2 1 Tf 0.59639 -1.1084 Td [(,)-343(ou)-343(encore)]TJ /F6 1 Tf 5.1386 0 Td [(9)]TJ /F8 1 Tf 0.56024 0 Td [(K)]TJ /F6 1 Tf 1.2108 0 Td [(2)-295(P)]TJ /F2 1 Tf 1.7349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.72892 1.1084 Td [(\263)]TJ /F8 1 Tf 0.59639 -1.1084 Td [(x)]TJ /F6 1 Tf 0.86145 0 Td [(2)]TJ /F8 1 Tf 0.96386 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.552 43.788 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.928 30.75 Td [(et)]TJ /F8 1 Tf 1.1747 0 Td [(i)]TJ /F6 1 Tf 0.63855 0 Td [(2)]TJ /F8 1 Tf 0.95783 0 Td [(K)]TJ /F9 1 Tf 0.92168 1.1084 Td [(\264)]TJ /F2 1 Tf 0.59639 -1.1084 Td [(.)]TJ -38.886 -1.4458 Td [(On)-355(en)-355(d)24(\266)469(eduit)-355(que)]TJ /F8 1 Tf 8.0602 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.241 41.719 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.831 29.304 Td [(est)-361(la)-355(r)24(\266)475(eunion)-361(des)]TJ /F8 1 Tf 8.1084 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.914 41.719 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.825 29.304 Td [(p)-24(our)]TJ /F8 1 Tf 2.3855 0 Td [(K)]TJ /F2 1 Tf 1.2771 0 Td [(parcouran)24(t)-355(l'ensem)24(ble)-355(des)-361(parties)-355(de)]TJ /F8 1 Tf -23.157 -1.1988 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(qui)-337(p)-24(oss)24(\265)475(eden)24(t)]TJ /F8 1 Tf 6.2651 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(.)]TJ -7.4578 -1.5602 Td [(Il)-403(est)-409(alors)-409(clair)-403(que,)-427(si)-409(l'on)-403(enl)24(\265)475(ev)24(e)-409(de)-403(la)-403(famille)-409(des)-409(parties)]TJ /F8 1 Tf 26.898 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.207 37.771 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.163 26.545 Td [(celles)-403(qui)-409(son)24(t)-403(vides)-409(on)]TJ -28.831 -1.1988 Td [(obtien)30(t)-337(une)-337(famille)-337(a)30(y)24(an)24(t)-337(la)-331(propri)24(\266)469(et)24(\266)469(e)-337(annonc)24(\266)475(ee.)]TJ ET 512.64 301.71 m 519.84 301.71 l 519.84 310.14 l 512.64 310.14 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 23.28 Td [(On)-337(en)-331(d)24(\266)475(eduit)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 16.545 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 19.834 Td [(8.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8976 0 Td [(Les)-343(fonctions)-319(\266)469(etag)24(\266)475(ees)-343(son)18(t)-343(exactemen)24(t)-343(les)-349(fonctions)-343(mesurables)]TJ -11.657 -1.1988 Td [(\265)499(a)-331(v)48(aleurs)-337(\257nies)-337(et)-337(ne)-331(prenan)24(t)-337(qu'un)-337(nom)24(bre)-337(\257ni)-337(de)-331(v)48(aleurs.)]TJ 0 -1.5602 Td [(Les)-331(fonctions)-307(\266)469(etag)24(\266)475(ees)-337(\(en)-337(particulier)-337(les)-331(fonctions)-337(constan)24(tes\))-337(son)24(t)-337(donc)-337(mesurables.)]TJ /F1 1 Tf 0 -1.8072 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Si)]TJ /F8 1 Tf 1.1145 0 Td [(f)]TJ /F2 1 Tf 0.88554 0 Td [(est)-289(mesurable)-283(et)-289(ne)-283(prend)-289(qu'un)-289(nom)24(bre)-283(\257ni)-289(de)-283(v)54(aleurs)]TJ /F8 1 Tf 23.958 0 Td [(\270)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 58.233 21.633 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.145 15.268 Td [(;)-168(:)-162(:)-168(:)-168(;)-168(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 21.633 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 13.159 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.1747 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.578 18.616 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8614 13.159 Td [(2)]TJ /F3 1 Tf 0.93976 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-319(les)-319(images)-319(r)30(\266)469(ecipro)-24(ques)]TJ /F8 1 Tf 10.566 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.25 18.616 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.608 13.159 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.716 19.418 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.337 13.159 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.647 18.616 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.801 13.159 Td [(\))-319(son)24(t)-313(mesurable)-319(et)-313(l'on)-319(a)]TJ /F8 1 Tf 11.428 0 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(=)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.509 20.616 Td [(i)]TJ /F4 1 Tf 0.40517 0 Td [(=)]TJ /F7 1 Tf 0.87931 0 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 40.157 14.105 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.578 17.142 Td [(i)]TJ /F4 1 Tf 0.40517 0 Td [(=1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.771 13.159 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.612 18.616 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.687 13.159 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.888 18.616 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 87.699 25.813 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 13.159 Td [(,)]TJ -38.886 -2.0783 Td [(par)-331(suite)]TJ /F8 1 Tf 4.1747 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(est)-307(\266)469(etag)24(\266)475(ee.)]TJ -5.1084 -2.7169 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-451(si)]TJ /F8 1 Tf 8.9217 0 Td [(f)]TJ /F2 1 Tf 1.0241 0 Td [(=)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.621 13.754 Td [(i)]TJ /F4 1 Tf 0.40517 0 Td [(=)]TJ /F7 1 Tf 0.87931 0 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 16.482 9.3159 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.69 10.28 Td [(i)]TJ /F4 1 Tf 0.40517 0 Td [(=1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.096 8.3641 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.81 11.754 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.066 8.3641 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.086 11.754 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 40.458 16.222 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.018 8.3641 Td 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[(Z)]TJ /F2 1 Tf 0.92169 0 Td [(d'en)24(tiers)-259(compris)]TJ -31.801 -1.3855 Td [(en)30(tre)-373(0)-373(et)-379(9.)-560(Cette)-349(\266)469(ecriture)-373(signi\257e)-373(que)-373(l'on)-379(a)]TJ /F8 1 Tf 20.602 0 Td [(x)]TJ /F2 1 Tf 0.91566 0 Td [(=)]TJ /F9 1 Tf 1.1205 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.026 44.159 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(Z)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.584 32.063 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.095 45.668 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.602 32.063 Td [(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.224 46.478 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.096 32.063 Td [(.)-560(On)-379(p)-30(eut)-373(m)24(^)475(eme)-373(exclure)-379(les)]TJ -26.765 -2.1506 Td [(familles)-325(telles)-331(que)]TJ /F8 1 Tf 8.0361 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.888 42.59 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.711 29.912 Td [(est)-325(constammen)24(t)-295(\266)469(egal)-319(\265)499(a)-325(9)-319(p)-30(our)-325(les)-325(indices)-331(su\261sammen)24(t)-325(p)-30(etits,)-325(il)-325(y)-325(a)]TJ -9.3795 -1.1988 Td [(alors)-337(une)-337(unique)-307(\266)475(ecriture)-337(de)-337(ce)-337(t)24(yp)-30(e)-337(p)-30(our)-337(c)24(haque)]TJ /F8 1 Tf 21.723 0 Td [(x)]TJ /F2 1 Tf 0.90362 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.431 40.875 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.476 28.714 Td [(est)-343(la)-331(d)24(\266)469(ecimale)-331(d'indice)]TJ /F8 1 Tf 10.512 0 Td [(k)]TJ /F2 1 Tf 0.88554 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(.)]TJ -38.452 -1.7289 Td [(Av)30(ec)-668(les)-662(notations)-668(pr)30(\266)469(ec)24(\266)475(eden)24(tes)-668(p)-30(osons)]TJ /F8 1 Tf 18.596 0 Td [(r)]TJ /F2 1 Tf 1.3012 0 Td [(=)]TJ /F8 1 Tf 1.6084 0 Td [(:)-162(:)-168(:)]TJ /F2 1 Tf 1.3313 0 Td [(0)-168(0)]TJ /F8 1 Tf 1.1687 0 Td [(;)-331(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.612 38.4 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.717 26.985 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.147 38.4 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.488 26.985 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.681 38.4 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(3)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.259 26.985 Td [(:)-162(:)-168(:)]TJ /F2 1 Tf 1.988 0 Td [(=)]TJ /F9 1 Tf 1.6145 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.603 36.892 Td [(k)-25(<)]TJ /F4 1 Tf 1.5259 0 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.488 26.985 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.69 38.4 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.506 26.985 Td [(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.828 39.202 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 26.985 Td [(et)]TJ /F8 1 Tf -38.331 -2.506 Td [(n)]TJ /F2 1 Tf 1.1265 0 Td [(=)]TJ /F8 1 Tf 1.3012 0 Td [(:)-168(:)-168(:)-168(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 13.767 34.814 Td [(3)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.241 24.479 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.405 34.814 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.386 24.479 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.043 34.814 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.361 24.479 Td [(;)]TJ /F2 1 Tf 0.60843 0 Td [(0)-168(0)-168(0)]TJ /F8 1 Tf 2.006 0 Td [(:)-162(:)-168(:)]TJ /F2 1 Tf 1.6928 0 Td [(=)]TJ /F9 1 Tf 1.3133 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.716 33.306 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(>)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.602 24.479 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.81 34.814 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.627 24.479 Td [(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.948 35.616 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.145 24.479 Td [(,)-518(on)-481(a)-481(alors)]TJ /F8 1 Tf 5.8614 0 Td [(x)]TJ /F2 1 Tf 1.1024 0 Td [(=)-524(10)]TJ /F8 1 Tf 2.3012 0 Td [(n)]TJ /F2 1 Tf 0.92169 0 Td [(+)]TJ /F8 1 Tf 1.1024 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 50.034 34.814 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.729 24.479 Td [(+)]TJ /F8 1 Tf 1.1024 0 Td [(r)]TJ /F2 1 Tf 0.96386 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.3434 0 Td [(n)]TJ /F6 1 Tf 1.1265 0 Td [(2)]TJ /F3 1 Tf 1.1928 0 Td [(N)]TJ /F2 1 Tf 1.2048 0 Td [(et)]TJ -38.331 -2.2892 Td [(0)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(r)-313(<)]TJ /F2 1 Tf 1.8133 0 Td [(1.)-445(On)-331(en)-331(d)30(\266)469(eduit)-325(que,)-337(p)-30(our)]TJ /F8 1 Tf 11.819 0 Td [(p)]TJ /F6 1 Tf 0.77711 0 Td [(2)-283(f)]TJ /F2 1 Tf 1.4458 0 Td [(0)]TJ /F8 1 Tf 0.5 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(;)-168(:)-168(:)-168(:)-168(;)]TJ /F2 1 Tf 2.2229 0 Td [(9)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(,)-331(on)-331(a)]TJ /F8 1 Tf 2.8253 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.414 31.538 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.765 22.189 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(p)]TJ /F2 1 Tf 0.83735 0 Td [(si)-331(et)-331(seulemen)30(t)-331(s'il)-337(existe)]TJ -28.325 -1.1988 Td [(un)-337(en)30(tier)]TJ /F8 1 Tf 4.253 0 Td [(n)]TJ /F2 1 Tf 0.93373 0 Td [(tel)-337(que)-337(10)]TJ /F8 1 Tf 4.3072 0 Td [(n)]TJ /F2 1 Tf 0.8253 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(p)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(x)-283(<)]TJ /F2 1 Tf 1.9036 0 Td [(10)]TJ /F8 1 Tf 1 0 Td [(n)]TJ /F2 1 Tf 0.8253 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(p)]TJ /F2 1 Tf 0.72289 0 Td [(+)-228(1.)]TJ -18.608 -1.5843 Td [(Plus)-283(g)30(\266)469(en)30(\266)469(eralemen)24(t,)-295(p)-30(our)]TJ /F8 1 Tf 10.819 0 Td [(k)]TJ /F6 1 Tf 0.83133 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(Z)]TJ /F2 1 Tf 0.66867 0 Td [(,)-289(en)-277(consid)24(\266)475(eran)24(t)-283(l')30(\266)469(ecriture)-283(d)30(\266)469(ecimale)-277(de)-283(10)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.828 28.357 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.331 19.406 Td [(x)]TJ /F2 1 Tf 0.85542 0 Td [(on)-277(conclut)-283(que)]TJ /F8 1 Tf -32.855 -1.1988 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3879 25.84 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6265 18.207 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(p)]TJ /F2 1 Tf 0.8253 0 Td [(si)-331(et)-331(seulemen)30(t)-331(s'il)-331(existe)-331(un)-331(en)30(tier)]TJ /F8 1 Tf 15.404 0 Td [(n)]TJ /F2 1 Tf 0.92771 0 Td [(tel)-331(que)-331(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.698 26.642 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.633 18.207 Td [(\(10)]TJ /F8 1 Tf 1.3855 0 Td [(n)]TJ /F2 1 Tf 0.81325 0 Td [(+)]TJ /F8 1 Tf 0.98795 0 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(x)-283(<)]TJ /F2 1 Tf 1.9096 0 Td [(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.31 26.642 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.44 18.207 Td [(\(10)]TJ /F8 1 Tf 1.3916 0 Td [(n)]TJ /F2 1 Tf 0.80723 0 Td [(+)]TJ /F8 1 Tf 0.98795 0 Td [(p)]TJ /F2 1 Tf 0.71687 0 Td [(+)-204(1\).)]TJ -37.012 -1.5843 Td [(Consid)30(\266)469(erons)-385(alors)-385(une)-385(fonction)]TJ /F8 1 Tf 13.97 0 Td [(f)]TJ /F2 1 Tf 0.98193 0 Td [(mesurable)-385(et)-379(p)-30(ositiv)24(e.)-596(P)30(our)-385(tout)]TJ /F8 1 Tf 14.934 0 Td [(x)]TJ /F6 1 Tf 0.92771 0 Td [(2)]TJ /F8 1 Tf 1.0241 0 Td [(X)]TJ /F2 1 Tf 1.2892 0 Td [(et)-385(tout)]TJ /F8 1 Tf 3.4337 0 Td [(k)]TJ /F6 1 Tf 0.90964 0 Td [(2)]TJ /F3 1 Tf 1.0241 0 Td [(Z)]TJ /F2 1 Tf -38.494 -1.2048 Td [(notons)]TJ /F8 1 Tf 3.1867 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.948 21.849 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.5422 15.418 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(la)-295(d)24(\266)475(ecimale)-295(d'indice)]TJ /F8 1 Tf 9.512 0 Td [(k)]TJ /F2 1 Tf 0.84337 0 Td [(de)]TJ /F8 1 Tf 1.2952 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(si)]TJ /F8 1 Tf 1.6446 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(est)-301(\257ni)-295(et)]TJ /F8 1 Tf 5.012 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.448 21.849 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.952 15.418 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)-277(1)-295(si)]TJ /F8 1 Tf 3.4819 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)-283(+)]TJ /F6 1 Tf 2.5 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)-433(On)]TJ -37.12 -1.3976 Td [(a)-355(donc)]TJ /F8 1 Tf 3.2651 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-313(=)]TJ /F9 1 Tf 1.7892 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.095 18.34 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(Z)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.56 14.021 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.164 19.849 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.584 14.021 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\)10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.233 20.65 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.428 14.021 Td [(.)-506(P)30(our)-361(c)24(haque)]TJ /F8 1 Tf 6.5964 0 Td [(p)]TJ /F6 1 Tf 0.81325 0 Td [(2)-319(f)]TJ /F2 1 Tf 1.4819 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(;)]TJ /F2 1 Tf 0.60843 0 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(;)-337(:)-168(:)-168(:)-162(;)]TJ /F2 1 Tf 2.5602 0 Td [(9)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.85542 0 Td [(notons)]TJ /F8 1 Tf 3.247 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.293 19.849 Td [(p;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.337 14.021 Td [(l'ensem)24(ble)-355(des)]TJ /F8 1 Tf 6.5843 0 Td [(x)]TJ /F2 1 Tf -38.59 -3.259 Td [(p)-30(our)-331(lesquels)]TJ /F8 1 Tf 6.0181 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.991 15.185 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.367 10.762 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(p)]TJ /F2 1 Tf 0.83133 0 Td [(;)-337(on)-337(a)-331(alors)]TJ /F8 1 Tf 5.2349 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.983 15.185 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.145 10.762 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.276 17.185 Td [(9)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 24.825 11.714 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.543 13.728 Td [(p)]TJ /F4 1 Tf 0.58621 0 Td [(=1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.44 10.762 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.362 15.185 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 56.205 21.005 Td [(p;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.524 10.762 Td [(et)-337(donc)]TJ /F8 1 Tf 3.5602 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F9 1 Tf 2.3554 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.207 13.685 Td [(1)]TJ /F17 1 Tf 0.57758 0 Td [(6)]TJ /F7 1 Tf 0.88794 0 Td [(p)]TJ /F17 1 Tf 0.59483 0 Td [(6)]TJ /F4 1 Tf 0.89655 0 Td [(9)]TJ /F7 1 Tf -2.2328 -1.0603 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(Z)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.229 10.762 Td [(p)]TJ /F2 1 Tf 0.50602 0 Td [(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.293 15.987 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.229 10.762 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.802 15.185 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 84.783 21.005 Td [(p;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.476 10.762 Td [(.)]TJ -38.145 -3.2711 Td [(Il)-385(ne)-385(reste)-391(plus)-391(qu')0(\265)500(a)-385(prouv)24(er)-391(que)-391(les)]TJ /F8 1 Tf 16.434 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.216 10.504 Td [(p;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.042 7.4906 Td [(son)24(t)-391(des)-391(parties)-385(mesurables)-397(p)-24(our)-391(conclure)-385(que)]TJ -18.711 -1.2048 Td [(l')30(\266)469(ecriture)-289(pr)24(\266)469(ec)24(\266)475(eden)24(te)-289(de)]TJ /F8 1 Tf 10.404 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(est)-289(du)-295(t)24(yp)-30(e)-295(souhait)24(\266)469(e.)-433(P)24(our)-289(cela)-295(la)-289(remarque)-289(du)-289(d)24(\266)469(ebut)-295(mon)24(tre)]TJ /F5 1 Tf -11.964 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(44)]TJ ET endstream endobj 167 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 166 0 R /ProcSet 2 0 R >> /Contents 165 0 R >> endobj 166 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F16 81 0 R /F17 97 0 R >> endobj 168 0 obj << /Length 22351 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ /F2 1 Tf 0 -2.4036 Td [(que)]TJ /F8 1 Tf 1.7651 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.914 88.823 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1205 62.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(p)]TJ /F2 1 Tf 0.71687 0 Td [(\266)475(equiv)54(aut)-246(\265)499(a)-240(l'existence)-240(de)]TJ /F8 1 Tf 10.681 0 Td [(n)]TJ /F6 1 Tf 0.87952 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(N)]TJ /F2 1 Tf 0.96386 0 Td [(tel)-240(que)-240(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.647 89.633 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.596 62.22 Td [(\(10)]TJ /F8 1 Tf 1.3855 0 Td [(n)]TJ /F2 1 Tf 0.63253 0 Td [(+)]TJ /F8 1 Tf 0.81326 0 Td [(p)]TJ /F2 1 Tf 0.50602 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(<)]TJ /F2 1 Tf 1.0542 0 Td [(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.716 89.633 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.428 62.22 Td [(\(10)]TJ /F8 1 Tf 1.3916 0 Td [(n)]TJ /F2 1 Tf 0.63253 0 Td [(+)]TJ /F8 1 Tf 0.80722 0 Td [(p)]TJ /F2 1 Tf 0.53615 0 Td [(+)-36(1\))]TJ -37.464 -1.1988 Td [(ou,)-403(lorsque)]TJ /F8 1 Tf 5.2169 0 Td [(p)]TJ /F2 1 Tf 0.87349 0 Td [(=)-367(1,)]TJ /F8 1 Tf 2.3253 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-373(=)-367(+)]TJ /F6 1 Tf 2.6807 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)-614(Autremen)24(t)-391(dit)]TJ /F8 1 Tf 7.6145 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.147 87.107 Td [(p;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.886 61.021 Td [(est)-391(la)-391(r)24(\266)475(eunion,)-409(p)-30(our)]TJ /F8 1 Tf 9.1687 0 Td [(n)]TJ /F2 1 Tf 0.98795 0 Td [(v)54(arian)24(t)-391(dans)]TJ /F3 1 Tf -33.711 -1.1988 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(,)-451(des)-433(images)-433(r)30(\266)469(ecipro)-30(ques)]TJ /F8 1 Tf 11.319 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.716 86.193 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 19.042 60.629 Td [(\241)]TJ /F2 1 Tf 0.46386 -0.80723 Td [([10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.741 86.193 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.271 59.822 Td [(\(10)]TJ /F8 1 Tf 1.3916 0 Td [(n)]TJ /F2 1 Tf 0.88554 0 Td [(+)]TJ /F8 1 Tf 1.0663 0 Td [(p)]TJ /F2 1 Tf 0.5 0 Td [(\))-174(;)-331(10)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.043 86.193 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.777 59.822 Td [(\(10)]TJ /F8 1 Tf 1.3916 0 Td [(n)]TJ /F2 1 Tf 0.88554 0 Td [(+)]TJ /F8 1 Tf 1.0602 0 Td [(p)]TJ /F2 1 Tf 0.78916 0 Td [(+)-289(1\)[)]TJ /F9 1 Tf 2.2349 0.80723 Td [(\242)]TJ /F2 1 Tf 0.88554 -0.80723 Td [(et,)-457(lorsque)]TJ /F8 1 Tf 5.0843 0 Td [(p)]TJ /F2 1 Tf 0.93976 0 Td [(=)-439(1,)-451(de)]TJ /F8 1 Tf -35.717 -1.3193 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4828 84.314 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7 58.503 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\).)-433(Lorsque)]TJ /F8 1 Tf 4.8253 0 Td [(f)]TJ /F2 1 Tf 0.88554 0 Td [(est)-295(mesurable)-289(ces)-289(images)-295(r)30(\266)469(ecipro)-30(ques)-289(son)24(t)-295(mesurables)-289(ainsi)-289(donc)-289(que)]TJ -9.5482 -1.1988 Td [(les)]TJ /F8 1 Tf 1.4518 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.776 81.788 Td [(p;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.006 57.304 Td [(qui)-337(en)-331(son)24(t)-337(des)-337(r)24(\266)475(eunions)-337(d)30(\266)469(enom)24(brables.)]TJ -3.6747 -1.8735 Td [(Cela)-355(d)30(\266)469(emon)24(tre)-349(la)-349(premi)24(\265)469(ere)-349(partie)-355(de)-349(l')24(\266)475(enonc)24(\266)469(e.)-493(P)30(our)-355(prouv)30(er)-355(la)-349(derni)24(\265)469(ere)-349(il)-355(su\261t)-349(d')24(\266)469(ecrire)]TJ /F8 1 Tf 38.645 0 Td [(I)]TJ /F2 1 Tf -38.645 -1.1988 Td [(comme)-397(r)24(\266)475(eunion)-403(d'une)-403(suite)-397(croissan)30(te)-403(de)-397(parties)-397(\257nies)]TJ /F8 1 Tf 24.416 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.198 77.392 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.127 54.232 Td [(\(c'est)-403(p)-24(ossible)-403(parce)-403(que)]TJ /F8 1 Tf 11.223 0 Td [(I)]TJ /F2 1 Tf 0.91566 0 Td [(est)]TJ -37.934 -1.4036 Td [(d)30(\266)469(enom)24(brable\),)]TJ /F8 1 Tf 6.6084 0 Td [(f)]TJ /F2 1 Tf 0.94578 0 Td [(est)-355(alors)-355(la)-349(limite)-349(de)-349(la)-349(suite)-349(croissan)24(te)-355(des)-349(sommes)-355(partielles)]TJ /F9 1 Tf 27.386 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.466 73.892 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 82.675 103.07 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.994 52.828 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.009 75.383 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.964 52.828 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.284 75.383 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 88.253 105.16 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 50.641 Td [(qui)-331(son)24(t)-337(des)-337(fonctions)-307(\266)469(etag)24(\266)469(ees)-331(p)-30(ositiv)30(es.)]TJ ET 512.64 604.04 m 519.84 604.04 l 519.84 612.46 l 512.64 612.46 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 39.741 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 47.641 Td [(10.)]TJ /F15 1 Tf 2.3253 0 Td [(Cor)18(ollaire.)]TJ /F12 1 Tf 6.9639 0 Td [(Sur)-463(un)-463(espace)-463(mesurable)]TJ /F8 1 Tf 11.645 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-493(toute)-463(fonction)-469(mesurable)]TJ /F8 1 Tf 12.367 0 Td [(f)]TJ /F12 1 Tf 1.0602 0 Td [(\()0(\265)500(a)]TJ -38.277 -1.1988 Td [(v)54(aleurs)-337(dans)]TJ ET 0.504 w 131.98 565.05 m 140.62 565.05 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 11.042 46.442 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(\))-337(est)-337(limite)-337(simple)-331(d'une)-337(suite)-337(de)-331(fonctions)-313(\266)475(etag)24(\266)469(ees)]TJ /F8 1 Tf 22.398 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.586 66.245 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 35.199 46.442 Td [(.)]TJ -29.867 -1.5843 Td [(En)-331(outre)-343(on)-331(p)-30(eut)-337(exiger)]TJ /F6 1 Tf 10.801 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.181 63.978 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.446 44.858 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F6 1 Tf 1.0542 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F12 1 Tf 0.60843 0 Td [(et)-337(alors)-337(les)]TJ /F8 1 Tf 5.0181 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.276 63.978 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.928 44.858 Td [(s'ann)24(ulen)30(t)-337(en)-337(tout)-337(p)-30(oin)30(t)-337(o)-30(\265)530(u)]TJ /F8 1 Tf 12.289 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(s'ann)30(ule.)]TJ /F1 1 Tf -34.819 -1.8735 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Appliquons)-475(le)-481(r)24(\266)475(esultat)-481(pr)24(\266)469(ec)24(\266)475(eden)24(t)-481(\265)499(a)-475(la)-475(partie)-487(p)-24(ositiv)24(e)-481(et)-481(\265)500(a)-475(la)-475(partie)]TJ -8.8193 -1.1988 Td [(n)30(\266)469(egativ)24(e)-325(de)]TJ /F8 1 Tf 5.259 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)-325(on)-325(obtien)24(t)-325(deux)-325(suites)-325(croissan)24(tes)-331(de)-325(fonctions)-295(\266)469(etag)24(\266)475(ees)-325(p)-30(ositiv)30(es)-331(\()]TJ /F8 1 Tf 29.054 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.405 59.581 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.361 41.786 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.741 59.581 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 41.786 Td [(et)]TJ -38.331 -1.2048 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8793 57.866 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.747 40.581 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.216 57.866 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0181 40.581 Td [(telles)-295(que)]TJ /F8 1 Tf 4.3434 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.819 57.866 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.789 40.581 Td [(=)-283(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.664 57.788 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.783 40.581 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.267 57.866 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.187 40.581 Td [(et)]TJ /F8 1 Tf 1.1265 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.776 57.866 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.759 40.581 Td [(=)-277(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.638 57.788 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.747 40.581 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.112 57.866 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.777 40.581 Td [(.)-433(Alors)]TJ /F8 1 Tf 3.3133 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.095 57.866 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.313 40.581 Td [(\241)]TJ /F8 1 Tf 0.92168 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.259 57.866 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.687 40.581 Td [(est)-301(la)-289(limite)-295(de)-289(la)-295(suite)]TJ -29.355 -1.1988 Td [(des)]TJ /F8 1 Tf 1.759 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.845 56.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4518 39.382 Td [(=)]TJ /F8 1 Tf 1.1084 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.5 56.142 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.922 39.382 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.784 56.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.331 39.382 Td [(qui)-367(son)24(t)-367(des)-367(fonctions)-337(\266)469(etag)24(\266)469(ees.)-536(En)-367(outre,)-373(les)-367(suites)-367(\()]TJ /F8 1 Tf 23.108 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.966 56.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.554 39.382 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.302 56.142 Td [(n)]TJ /F10 1 Tf 0.70689 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.898 39.382 Td [(et)-367(\()]TJ /F8 1 Tf 1.5904 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.06 56.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.518 39.382 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.397 56.142 Td [(n)]TJ /F10 1 Tf 0.70689 0 Td [(2)]TJ /F16 1 Tf 0.77587 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3012 38.183 Td [(\266)469(etan)24(t)-307(croissan)24(tes,)-313(on)-307(a)-307(0)]TJ /F11 1 Tf 10.783 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.345 54.426 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 18.536 38.183 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.733 54.426 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.054 38.183 Td [(et)-307(0)]TJ /F11 1 Tf 1.9157 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.078 54.426 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 25.331 38.183 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.466 54.426 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.554 38.183 Td [(.)-433(P)24(ar)-307(suite)]TJ /F6 1 Tf 4.9337 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.586 54.426 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.801 38.183 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.491 54.426 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.699 38.183 Td [(+)]TJ /F8 1 Tf 0.94579 0 Td [(v)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.56 54.426 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 38.952 38.183 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.948 54.426 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.325 38.183 Td [(+)]TJ /F8 1 Tf 0.94578 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.198 54.426 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.717 38.183 Td [(=)]TJ /F6 1 Tf -38.386 -1.1988 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.61446 0 Td [(et,)-331(lorsque)]TJ /F8 1 Tf 4.8675 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(s'ann)30(ule,)-337(il)-337(en)-331(v)48(a)-331(de)-331(m)24(^)469(eme)-331(p)-30(our)]TJ /F8 1 Tf 14.265 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.172 52.711 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.922 36.985 Td [(.)]TJ ET 512.64 440.82 m 519.84 440.82 l 519.84 449.24 l 512.64 449.24 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 33.967 Td [(Remarque.)]TJ /F2 1 Tf 6.2349 0 Td [(La)-319(stabilit)24(\266)469(e)-319(des)-325(fonctions)-319(mesurables)-325(par)-319(passage)-319(\265)500(a)-319(la)-319(limite)-319(simple)-319(de)-319(suite)]TJ -6.2349 -1.1988 Td [(de)-301(fonctions)-301(et)-301(ce)-307(corollaire)-301(mon)24(tren)24(t)-307(en)-301(fait)-301(que)-307(les)-301(fonctions)-307(mesurables)-301(son)18(t)-301(exactemen)24(t)]TJ 0 -1.2048 Td [(les)-379(limites)-385(simples)-391(des)-379(suites)-385(de)-385(fonctions)-355(\266)469(etag)30(\266)469(ees.)-590(On)-385(en)-379(d)24(\266)475(eduit)-385(un)-379(deuxi)24(\265)475(eme)-355(\266)469(enonc)30(\266)469(e)-385(de)]TJ 0 -1.1988 Td [(stabilit)24(\266)475(e)-337(de)-337(l'ensem)30(ble)-337(des)-337(applications)-331(mesurables.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 22.831 Td [(I)-30(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3373 27.37 Td [(11.)]TJ /F15 1 Tf 2.3253 0 Td [(Pr)18(oposition.)]TJ /F12 1 Tf 6.8735 0 Td [(Soit)]TJ /F2 1 Tf 2.0542 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(un)-337(espace)-337(mesurable.)-445(Alors)-337(:)]TJ /F1 1 Tf -16.97 -1.5904 Td [(\(i\))]TJ /F12 1 Tf 1.9458 0 Td [(Si)]TJ /F8 1 Tf 1.2048 0 Td [(f)]TJ /F12 1 Tf 0.96386 0 Td [(et)]TJ /F8 1 Tf 1.1988 0 Td [(g)]TJ /F12 1 Tf 0.87952 0 Td [(son)24(t)-373(des)-373(fonctions)-367(mesurables)-373(la)-367(somme)]TJ /F8 1 Tf 17.916 0 Td [(f)]TJ /F2 1 Tf 0.84337 0 Td [(+)]TJ /F8 1 Tf 1.0241 0 Td [(g)]TJ /F12 1 Tf 0.87952 0 Td [(est)-373(une)-367(fonction)-373(mesurable)]TJ -27.349 -1.1988 Td [(si)-343(elle)-349(est)-349(d)30(\266)469(e\257nie)-343(\()]TJ /F2 1 Tf 8.2771 0 Td [(i.e.)]TJ /F12 1 Tf 1.747 0 Td [(il)-349(n'y)-337(a)-343(pas)-349(de)-343(forme)-337(ind)24(\266)469(etermin)24(\266)475(ee\),)-349(le)-343(pro)-30(duit)]TJ /F8 1 Tf 20.687 0 Td [(f)-114(g)]TJ /F12 1 Tf 1.4518 0 Td [(est)-349(une)-343(fonction)]TJ -32.163 -1.1988 Td [(mesurable)-331(\(a)24(v)24(ec)-337(la)-337(con)30(v)18(en)30(tion)]TJ /F2 1 Tf 13.428 0 Td [(0)]TJ /F6 1 Tf 0.72289 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(a)]TJ /F6 1 Tf 0.74698 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(0)-283(=)-277(0)]TJ /F12 1 Tf 2.6687 0 Td [(p)-30(our)-337(tout)]TJ /F8 1 Tf 4.5361 0 Td [(a)]TJ /F12 1 Tf 0.5241 0 Td [(\).)]TJ /F1 1 Tf -25.319 -1.7892 Td [(\(ii\))]TJ /F12 1 Tf 1.994 0 Td [(P)30(our)-234(toute)-234(famille)-234(d)30(\266)469(enom)24(brable)]TJ ET 185.18 256.46 m 252 256.46 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 21.313 21.593 Td [(de)-228(fonctions)-234(mesurables)-234(la)-234(somme)]TJ /F9 1 Tf 14.657 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.629 29.193 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.584 21.593 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.483 30.685 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.639 21.593 Td [(est)-234(mesurable)]TJ -33.307 -2.1446 Td [(si)-337(elle)-337(est)-337(d)24(\266)469(e\257nie.)]TJ /F1 1 Tf 0 -1.8675 Td [(D)24(\266)548(emonstration.)]TJ 0.49398 -1.5904 Td [(\(i\))]TJ /F2 1 Tf 2.1024 0 Td [(D'apr)30(\265)469(es)-445(le)-451(corollaire)-445(pr)30(\266)469(ec)24(\266)469(eden)30(t)]TJ /F8 1 Tf 14.036 0 Td [(f)]TJ /F2 1 Tf 1.0422 0 Td [(et)]TJ /F8 1 Tf 1.2771 0 Td [(g)]TJ /F2 1 Tf 0.95783 0 Td [(p)-24(euv)18(en)30(t)-421(^)475(etre)-445(consid)24(\266)475(er)18(\266)475(ees)-445(comme)-445(les)-445(limites)]TJ -19.91 -1.1988 Td [(simples)-259(de)-253(suites)-259(de)-259(fonctions)-228(\266)469(etag)24(\266)475(ees)]TJ /F8 1 Tf 16.211 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.526 20.952 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.831 14.792 Td [(et)]TJ /F8 1 Tf 1.0904 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.914 20.952 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.94 14.792 Td [(.)-421(On)-259(a)-253(alors)]TJ /F8 1 Tf 5.3614 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-66(+)]TJ /F8 1 Tf 1.2952 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)-277(lim)]TJ /F9 1 Tf 3.2771 0.80723 Td [(\241)]TJ /F8 1 Tf 0.46386 -0.80723 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.612 20.952 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.404 14.792 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-66(+)]TJ /F8 1 Tf 1.2952 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.31 20.952 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.687 14.792 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F9 1 Tf 0.39156 0.80723 Td [(\242)]TJ /F2 1 Tf -38.705 -2.1084 Td [(en)-349(tout)-355(p)-30(oin)24(t)-355(o)-30(\265)530(u)-355(le)-361(premier)-355(mem)24(bre)-349(est)-361(d)24(\266)475(e\257ni)-355(et)]TJ /F8 1 Tf 21.687 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.62651 0 Td [(\242)]TJ /F8 1 Tf 0.51205 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56626 0 Td [(\))-313(=)-313(lim)]TJ /F9 1 Tf 3.3434 0.81325 Td [(\241)]TJ /F8 1 Tf 0.46386 -0.81325 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.776 19.09 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.024 13.491 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.62048 0 Td [(\242)]TJ /F8 1 Tf 0.51807 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.241 19.09 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.145 13.491 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.81325 Td [(\242)]TJ /F2 1 Tf 0.81325 -0.81325 Td [(en)-355(tout)]TJ -35.976 -1.1988 Td [(p)-30(oin)30(t)-451(\(si)-445(l'on)-451(a)-445(pris)-451(soin)-451(d'exiger)-451(que)]TJ /F8 1 Tf 17.319 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.121 17.375 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.687 12.292 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-445(s'ann)24(ule)-445(c)24(haque)-451(fois)-445(que)]TJ /F8 1 Tf 12.163 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-445(s'ann)24(ule,)-475(de)]TJ -33.042 -1.2048 Td [(m)24(^)475(eme)-463(p)-30(our)]TJ /F8 1 Tf 5.512 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.207 15.659 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.331 11.087 Td [(et)]TJ /F8 1 Tf 1.2952 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\).)-837(T)78(out)-469(revien)24(t)-469(donc)-463(\265)500(a)-463(v)24(oir)-463(qu'une)-469(somme)-463(ou)-463(un)-469(pro)-30(duit)-463(de)-463(deux)]TJ -8.8133 -1.1988 Td [(fonctions)-289(\266)469(etag)24(\266)475(ees)-319(est)-319(une)-319(fonction)-295(\266)475(etag)24(\266)469(ee.)-439(D'apr)24(\265)469(es)-319(la)-313(d)24(\266)469(e\257nition)-319(des)-319(fonctions)-289(\266)469(etag)30(\266)469(ees,)-319(c'est)]TJ -0.030121 -1.1988 Td [(\266)469(eviden)24(t)-343(p)-30(our)-343(la)-337(somme,)-343(et)-343(p)-30(our)-343(le)-343(pro)-30(duit)-343(tout)-343(revien)24(t)-343(\265)500(a)-343(prouv)30(er)-343(que,)-349(p)-30(our)-343(des)-343(parties)]TJ /F8 1 Tf 38.446 0 Td [(A)]TJ /F2 1 Tf -38.416 -1.1988 Td [(et)]TJ /F8 1 Tf 1.2108 0 Td [(B)]TJ /F2 1 Tf 1.1867 0 Td [(mesurables,)-397(le)-379(pro)-30(duit)-379(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.448 10.504 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.386 7.4906 Td [(\242)]TJ /F2 1 Tf 0.53012 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.31 10.504 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.548 7.4906 Td [(est)-379(une)-379(fonction)-361(\266)475(etag)24(\266)469(ee.)-578(Ce)-385(dernier)-379(p)-30(oin)24(t)-379(r)24(\266)475(esulte)-379(de)]TJ -16.217 -1.2048 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.431 8.7797 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7651 6.2858 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.198 8.7797 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.7952 6.2858 Td [(=)-277(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.897 8.7797 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [(\\)]TJ /F7 1 Tf 0.77586 0 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.241 6.2858 Td [(.)]TJ 12.006 -2.3976 Td [(45)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 170 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 169 0 R /ProcSet 2 0 R >> /Contents 168 0 R >> endobj 169 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F16 81 0 R >> endobj 171 0 obj << /Length 5206 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-12765(I)-36(I)-36(I.)-385(La)-385(notion)-391(de)-385(mesure.)]TJ 0.1747 -2.4518 Td [(\(ii\))]TJ /F2 1 Tf 2.0482 0 Td [(Si)-265(la)-259(somme)]TJ /F9 1 Tf 5.3976 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.69 87.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.566 62.171 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.543 88.754 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.651 62.171 Td [(est)-259(d)24(\266)469(e\257nie)-259(a)24(v)24(ec)]TJ /F8 1 Tf 6.6988 0 Td [(I)]TJ /F2 1 Tf 0.77711 0 Td [(d)24(\266)469(enom)24(brable)-259(on)-265(sait)-259(que)-265(l'on)-259(p)-30(eut)-234(\266)469(ecrire)]TJ /F8 1 Tf 17.536 0 Td [(I)]TJ /F2 1 Tf 0.77711 0 Td [(comme)]TJ -36.108 -2.5 Td [(la)-271(r)30(\266)469(eunion)-271(d'une)-277(suite)-271(croissan)24(te)-271(de)-271(parties)-271(\257nies)]TJ /F8 1 Tf 21.12 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.483 85.176 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.705 59.671 Td [(de)-271(fa)-6(\273)439(con)-271(que)]TJ /F9 1 Tf 5.6506 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.879 83.676 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.964 59.671 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.741 85.176 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.06 59.671 Td [(soit)-271(la)-271(limite)-271(simple)]TJ -30.729 -2.5 Td [(des)-421(sommes)-427(partielles)]TJ /F9 1 Tf 10.127 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.966 80.107 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 33.048 111.75 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.181 57.171 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.293 81.599 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.006 57.171 Td [(.)-710(A)-427(l'aide)-421(de)]TJ /F1 1 Tf 6.3373 0 Td [(\(i\))]TJ /F2 1 Tf 1.6386 0 Td [(on)-421(mon)24(tre)-427(ais)24(\266)469(emen)30(t)-427(par)-421(r)24(\266)469(ecurrence)-421(sur)-427(le)]TJ -20.651 -2.5482 Td [(nom)24(bre)-349(d')24(\266)475(el)18(\266)475(emen)24(ts)-349(de)]TJ /F8 1 Tf 9.8916 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.414 77.961 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.56 54.623 Td [(que)-355(les)-349(sommes)-355(partielles)]TJ /F9 1 Tf 11.512 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.017 76.461 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 58.277 106.66 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.795 54.623 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.336 77.961 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.97 54.623 Td [(son)24(t)-355(mesurables,)-355(par)-355(passage)-349(\265)500(a)]TJ -25.639 -2.5422 Td [(la)-331(limite)-337(simple)-331(la)-337(somme)]TJ /F9 1 Tf 11.518 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.259 72.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.458 52.081 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.112 74.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.614 52.081 Td [(est)-337(aussi)-337(mesurable.)]TJ ET 512.64 621.25 m 519.84 621.25 l 519.84 629.67 l 512.64 629.67 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 48.882 Td [(Remarque.)]TJ /F2 1 Tf 6.1867 0 Td [(Le)-289(r)24(\266)475(esultat)]TJ /F1 1 Tf 4.988 0 Td [(\(ii\))]TJ /F2 1 Tf 1.8193 0 Td [(est)-289(v)48(alable)-289(dans)-283(tous)-289(les)-289(cas)-289(o)-24(\265)524(u)-283(l'on)-289(a)-289(donn)24(\266)469(e)-289(un)-289(sens)-283(\265)500(a)]TJ /F9 1 Tf 23.458 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.948 68.237 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.392 48.882 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.802 69.737 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 48.882 Td [(,)]TJ -38.886 -2.1988 Td [(en)-271(particulier)-277(d)30(\265)469(es)-271(que)-277(les)]TJ /F8 1 Tf 10.91 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.94 66.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.337 46.683 Td [(son)24(t)-277(\265)499(a)-271(v)54(aleurs)-277(p)-30(ositiv)24(es)-271(\()-6(\265)500(a)-271(condition)-277(d'admettre)-277(la)-271(v)48(aleur)-271(+)]TJ /F6 1 Tf 25.488 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\).)]TJ -38.494 -1.2048 Td [(Dans)-337(le)-337(cas)-337(des)-337(s)24(\266)475(eries)-337(\()]TJ /F8 1 Tf 10.078 0 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F3 1 Tf 1.0542 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(\),)-337(c'est)-313(\266)475(egalemen)24(t)-337(v)54(alable)-337(p)-30(our)-337(des)-337(s)24(\266)475(eries)-337(semi-con)24(v)24(ergen)24(tes.)]TJ /F5 1 Tf -12.651 -41.59 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(46)]TJ ET endstream endobj 173 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 172 0 R /ProcSet 2 0 R >> /Contents 171 0 R >> endobj 172 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F13 39 0 R >> endobj 174 0 obj << /Length 14265 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F0 1 Tf 20.592 0 0 20.592 0 0 Tm 4.1329 32.152 Td [(Chapitre)-377(IV.)-1132(Construction)-377(de)-377(l'in)27(t)24(\266)538(egrale)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 50.979 Td [(Dans)-283(ce)-283(c)24(hapitre)-283(nous)-277(supp)-30(osons)-283(donn)24(\266)469(e)-283(un)-283(espace)-283(mesur)30(\266)469(e)-283(\()]TJ /F8 1 Tf 25.56 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.79518 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3253 0 Td [(\).)-433(Cette)-283(donn)24(\266)469(ee)-283(p)-30(ermet)]TJ -28.976 -1.1988 Td [(de)-391(d)30(\266)469(e\257nir)-391(imm)24(\266)469(ediatemen)24(t)-391(l'in)24(t)24(\266)469(egrale)-391(des)-397(fonctions)-391(indicatrices)-391(des)-391(parties)-397(mesurables)-391(en)]TJ 0 -1.6566 Td 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0 Tm 20.807 13.479 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.586 19.073 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 43.952 26.463 Td [(j)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.735 13.479 Td [(alors)-337(on)-337(a)]TJ /F9 1 Tf 6.6205 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.026 17.581 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.97 13.479 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.095 19.073 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.94 13.479 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.457 19.073 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.295 13.479 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.422 17.581 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.627 13.479 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.517 19.073 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.614 13.479 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.017 19.073 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.06 13.479 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F1 1 Tf -38.452 -2.8554 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-337(preuv)30(e)-343(v)54(a)-331(se)-337(faire)-337(en)-337(plusieurs)-307(\266)469(etap)-30(es.)]TJ /F1 1 Tf -8.006 -1.5663 Td [(1.)]TJ /F2 1 Tf 1.5602 0 Td [(Supp)-30(osons)]TJ /F8 1 Tf 4.9277 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(J)]TJ /F2 1 Tf 0.98193 0 Td [(\257nis.)]TJ ET 92.088 106.56 m 211.61 106.56 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 7.4906 Td [(Appliquons)-301(le)-313(lemme)-307(de)-301(d)24(\266)469(ecomp)-30(osition)-307(\(cf.)]TJ /F1 1 Tf 19.139 0 Td [(I)-36(I)-36(I.5.7.)]TJ /F2 1 Tf 3.4819 0 Td [(\))-307(\265)500(a)-307(la)-301(famille)-307(form)24(\266)469(ee)-307(par)-307(les)-301(parties)]TJ /F8 1 Tf 15.458 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 10.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 6.2858 Td [(et)]TJ /F8 1 Tf 1.1988 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.431 8.7797 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7108 6.2858 Td [(.)-548(On)-367(disp)-30(ose)-367(donc)-367(d'une)-373(famille)-367(\257nie)-373(de)-367(parties)]TJ /F8 1 Tf 21.398 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.681 8.7797 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.681 6.2858 Td [(qui)-367(son)24(t)-373(mesurables,)-379(non)-373(vides,)]TJ -6.4337 -2.3976 Td [(47)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 176 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 175 0 R /ProcSet 2 0 R >> /Contents 174 0 R >> endobj 175 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 177 0 obj << /Length 32867 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(deux)-319(\265)500(a)-325(deux)-319(disjoin)24(tes,)-325(recouvren)24(t)]TJ /F8 1 Tf 15.247 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(et)-325(telles)-331(que,)-325(p)-30(our)-325(c)24(haque)]TJ /F8 1 Tf 11.518 0 Td [(i)]TJ /F2 1 Tf 0.66868 0 Td [(dans)]TJ /F8 1 Tf 2.3253 0 Td [(I)]TJ /F2 1 Tf 0.84337 0 Td [(\(resp.)]TJ /F8 1 Tf 2.8916 0 Td [(j)]TJ /F2 1 Tf 0.79518 0 Td [(dans)]TJ /F8 1 Tf 2.3253 0 Td [(J)]TJ /F2 1 Tf 0.6506 0 Td [(\),)]TJ /F8 1 Tf -38.494 -1.1988 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 87.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7952 61.021 Td [(\(resp.)]TJ /F8 1 Tf 3.0301 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.147 87.107 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.006 61.021 Td [(\))-385(est)-385(la)-379(r)30(\266)469(eunion)-385(de)-379(certains)-379(des)]TJ /F8 1 Tf 14.127 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.991 87.107 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.343 61.021 Td [(,)-391(en)-379(particulier,)-391(p)-30(our)-379(c)24(haque)]TJ /F8 1 Tf 13.012 0 Td [(k)]TJ /F2 1 Tf 0.55421 0 Td [(,)]TJ /F8 1 Tf 0.66867 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.147 87.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.042 61.021 Td [(\(resp.)]TJ /F8 1 Tf -36.711 -1.1988 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7155 85.392 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.512 59.822 Td [(\))-337(con)30(tien)24(t)]TJ /F8 1 Tf 4.5542 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.862 85.392 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.608 59.822 Td [(ou)-337(ne)-331(le)-337(rencon)24(tre)-337(pas.)]TJ -7.2771 -2.1566 Td [(P)30(our)-506(c)24(haque)]TJ /F8 1 Tf 6.1024 0 Td [(k)]TJ /F2 1 Tf 0.55422 0 Td [(,)-542(en)-500(m)24(ultiplian)24(t)-506(par)-506(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.707 82.314 Td [(C)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 45.458 114.82 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.717 57.665 Td [(la)-506(relation)]TJ /F9 1 Tf 5.1205 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.422 80.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.446 57.665 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.491 82.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.422 57.665 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.767 82.314 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 65.169 114.84 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 33.506 57.665 Td [(6)]TJ /F9 1 Tf 1.3373 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.879 80.814 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.458 57.665 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.983 82.314 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.44 57.665 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.388 82.314 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 77.217 114.84 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.03 57.665 Td [(,)-548(on)-500(obtien)24(t)]TJ /F9 1 Tf -33.699 -1.6265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7759 77.116 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 55.087 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.853 78.616 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.9157 55.087 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.129 78.616 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 18.157 109.67 Td [(i)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.448 78.616 Td [(\\)]TJ /F7 1 Tf 0.76724 0 Td [(C)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 21.012 109.66 Td [(k)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 11.44 55.087 Td [(6)]TJ /F9 1 Tf 1.2229 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.147 77.116 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.277 55.087 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.241 78.616 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.265 55.087 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.647 78.616 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 32.867 109.67 Td [(j)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.052 78.616 Td [(\\)]TJ /F7 1 Tf 0.76724 0 Td [(C)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 35.831 109.66 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.404 55.087 Td [(.)-746(Cep)-30(endan)24(t)]TJ /F8 1 Tf 6.1867 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.259 78.616 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.964 55.087 Td [(\\)]TJ /F8 1 Tf 0.95783 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.543 78.616 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.56 55.087 Td [(ainsi)-433(que)]TJ /F8 1 Tf 4.4036 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.259 78.616 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.434 55.087 Td [(\\)]TJ /F8 1 Tf 0.95783 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.672 78.616 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.036 55.087 Td [(est)-439(soit)-433(vide)-439(soit)]TJ -31.735 -2.6867 Td [(\266)469(egal)-361(\265)500(a)]TJ /F8 1 Tf 2.9819 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.879 74.78 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.494 52.4 Td [(.)-536(La)-373(relation)-361(se)-367(r)24(\266)469(e)24(\266)475(ecrit)-367(donc)]TJ /F9 1 Tf 12.783 1.1145 Td [(\263)]TJ 0.8494 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.974 73.28 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 48.446 102.2 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.825 52.4 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.44 74.78 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 25.795 53.515 Td [(\264)]TJ /F2 1 Tf 0.60241 -1.1145 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.578 74.78 Td [(C)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 55.06 104.29 Td [(k)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 28.349 52.4 Td [(6)]TJ /F9 1 Tf 1.1084 1.1145 Td [(\263)]TJ 0.93976 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.25 73.28 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(J)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 63.157 102.2 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.181 52.4 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.862 74.78 Td [(j)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 33.169 53.515 Td [(\264)]TJ /F2 1 Tf 0.59639 -1.1145 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.121 74.78 Td [(C)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 69.795 104.29 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.753 52.4 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.4217 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.828 74.78 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.47 52.4 Td [(\(resp.)]TJ /F8 1 Tf 2.994 0 Td [(J)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.129 74.78 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.512 52.4 Td [(\))-367(est)]TJ -37.181 -2.2771 Td [(l'ensem)30(ble)-361(des)]TJ /F8 1 Tf 6.5964 0 Td [(i)]TJ /F2 1 Tf 0.70482 0 Td [(\(resp.)]TJ /F8 1 Tf 2.9759 0 Td [(j)]TJ /F2 1 Tf 0.46988 0 Td [(\))-361(tels)-361(que)]TJ /F8 1 Tf 4.506 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.526 71.512 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.03 50.123 Td [(\(resp.)]TJ /F8 1 Tf 2.9759 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.871 71.512 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.187 50.123 Td [(\))-361(con)24(tien)24(t)]TJ /F8 1 Tf 4.253 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.578 71.512 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.645 50.123 Td [(.)-524(Comme)]TJ /F8 1 Tf 4.5 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.75 71.512 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.711 50.123 Td [(est)-367(non)-361(vide)-367(on)]TJ -32.38 -1.5422 Td [(en)-331(d)24(\266)475(eduit)]TJ /F9 1 Tf 4.5301 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.991 67.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 21.904 94.56 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.56 48.581 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.457 69.314 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 12.807 48.581 Td [(6)]TJ /F9 1 Tf 1.2289 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.836 67.814 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 30.434 94.56 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.819 48.581 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.448 69.314 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.145 48.581 Td [(et)-331(donc)]TJ /F9 1 Tf 3.5542 1.1145 Td [(\263)]TJ 0.8494 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.716 67.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 45.277 94.56 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.247 48.581 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.181 69.314 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 24.217 49.695 Td [(\264)]TJ /F8 1 Tf 0.59639 -1.1145 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.353 69.314 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.289 48.581 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.1145 Td [(\263)]TJ 0.93976 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.612 67.814 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(J)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 62.265 94.56 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.735 48.581 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.224 69.314 Td [(j)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 32.723 49.695 Td [(\264)]TJ /F8 1 Tf 0.59638 -1.1145 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.517 69.314 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.795 48.581 Td [(\).)]TJ -30.464 -3.2952 Td [(La)-283(relation)-283(pr)24(\266)469(ec)24(\266)475(eden)24(te)-283(p)-30(eut)-289(encore)-283(s')24(\266)469(ecrire)]TJ /F9 1 Tf 18.62 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.431 63.099 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.566 45.286 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.5 64.59 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.705 45.286 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.103 64.59 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.175 45.286 Td [(\\)]TJ /F8 1 Tf 0.78915 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.905 64.59 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.169 45.286 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.095 63.099 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.506 45.286 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.19 64.59 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.663 45.286 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.931 64.59 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.229 45.286 Td [(\\)]TJ /F8 1 Tf 0.78915 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.853 64.59 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.223 45.286 Td [(\),)-295(puisque)]TJ /F8 1 Tf -34.892 -2.2831 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 61.323 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6386 43.003 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.793 61.323 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0663 43.003 Td [(\(resp.)]TJ /F8 1 Tf 2.8976 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.207 61.323 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.367 43.003 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.422 61.323 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.464 43.003 Td [(\))-337(est)-313(\266)475(egal)-337(\265)500(a)]TJ /F8 1 Tf 5.1747 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.56 61.323 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.181 43.003 Td [(p)-30(our)]TJ /F8 1 Tf 2.3614 0 Td [(i)]TJ /F6 1 Tf 0.62651 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38 61.323 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.38 43.003 Td [(\(resp.)]TJ /F8 1 Tf 2.4518 0 Td [(j)]TJ /F6 1 Tf 0.74699 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(J)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.905 61.323 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.572 43.003 Td [(\))-337(et)-331(vide)-343(sinon.)]TJ -27.241 -1.9157 Td [(Comme)-457(les)]TJ /F8 1 Tf 5.3614 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.328 58.59 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.355 41.087 Td [(formen)24(t)-457(un)-457(recouvremen)18(t)-457(de)]TJ /F8 1 Tf 13.229 0 Td [(X)]TJ /F2 1 Tf 1.3614 0 Td [(par)-457(des)-457(parties)-463(deux)-451(\265)499(a)-457(deux)-457(disjoin)30(tes,)]TJ -21.614 -1.1988 Td [(c)24(haque)]TJ /F8 1 Tf 3.3133 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.44 56.866 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.036 39.888 Td [(\(resp.)]TJ /F8 1 Tf 2.8916 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.586 56.866 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.108 39.888 Td [(\))-313(est)-319(la)-307(r)24(\266)475(eunion)-319(disjoin)30(te)-313(des)]TJ /F8 1 Tf 12.608 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.31 56.866 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.982 39.888 Td [(\\)]TJ /F8 1 Tf 0.84337 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.267 56.866 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.343 39.888 Td [(\(resp.)]TJ /F8 1 Tf 2.8916 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.647 56.866 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.59 39.888 Td [(\\)]TJ /F8 1 Tf 0.84337 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.733 56.866 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.639 39.888 Td [(\),)-319(par)-313(suite)]TJ /F8 1 Tf 5.1145 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.638 56.866 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 39.888 Td [(\))]TJ -38.777 -1.1988 Td [(\(resp.)]TJ /F8 1 Tf 2.8855 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.655 55.15 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.663 38.689 Td [(\)\))-301(est)-301(la)-295(somme)-301(des)]TJ /F8 1 Tf 8.6747 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.56 55.15 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 21.837 38.689 Td [(\\)]TJ /F8 1 Tf 0.81325 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.44 55.15 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.855 38.689 Td [(\))-301(\(resp.)]TJ /F8 1 Tf 3.5723 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.155 55.15 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.024 38.689 Td [(\\)]TJ /F8 1 Tf 0.81928 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.164 55.15 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.048 38.689 Td [(\)\).)-433(En)-301(somman)24(t)-301(les)-301(relations)]TJ -26.717 -1.4398 Td [(obten)24(us)-337(p)-30(our)-337(tous)-337(les)]TJ /F8 1 Tf 9.6867 0 Td [(k)]TJ /F2 1 Tf 0.88554 0 Td [(on)-337(obtien)24(t)-337(donc)]TJ /F9 1 Tf 7.1988 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.216 51.599 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.717 37.25 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.284 53.09 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.855 37.25 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.888 53.09 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.205 37.25 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.853 51.599 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.542 37.25 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.948 53.09 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.699 37.25 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.69 53.09 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.145 37.25 Td [(\).)]TJ /F1 1 Tf -29 -2.9398 Td [(2.)]TJ /F2 1 Tf 1.5602 0 Td [(Supp)-30(osons)]TJ /F8 1 Tf 4.9277 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(\257ni)-337(et)]TJ /F8 1 Tf 2.8916 0 Td [(J)]TJ /F2 1 Tf 0.98193 0 Td [(d)24(\266)475(enom)24(brable)-337(quelconque.)]TJ ET 0.504 w 92.088 408.45 m 339.48 408.45 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 32.394 Td [(On)-439(p)-24(eut)-415(\266)475(ecrire)]TJ /F8 1 Tf 7.006 0 Td [(J)]TJ /F2 1 Tf 1.0904 0 Td [(comme)-439(la)-433(r)24(\266)475(eunion)-439(d'une)-439(suite)-439(croissan)24(te)-439(de)-433(parties)-439(\257nies)]TJ /F8 1 Tf 25.94 0 Td [(J)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.129 46.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.47 32.394 Td [(.)-746(P)24(osons)]TJ -35.139 -1.3434 Td [(alors)]TJ /F8 1 Tf 2.5 0 Td [(f)]TJ /F2 1 Tf 1.0422 0 Td [(=)]TJ /F9 1 Tf 1.2229 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.603 42.719 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.705 31.051 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.672 44.219 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.681 31.051 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.948 44.219 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 27.687 61.596 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.645 31.051 Td [(,)]TJ /F8 1 Tf 1.1747 0 Td [(g)]TJ /F2 1 Tf 0.95783 0 Td [(=)]TJ /F9 1 Tf 1.2229 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.776 42.719 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.608 31.051 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.871 44.219 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.596 31.051 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.284 44.219 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 43.53 61.596 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.62 31.051 Td [(et,)-463(p)-30(our)-433(c)18(haque)]TJ /F8 1 Tf 7.4759 0 Td [(n)]TJ /F2 1 Tf 0.59638 0 Td [(,)]TJ /F8 1 Tf 0.74097 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.664 44.219 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.898 31.051 Td [(=)]TJ /F9 1 Tf 1.4277 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.836 42.719 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 70.952 59.511 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.133 31.051 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.517 44.219 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.12 31.051 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.922 44.219 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 76.578 61.596 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.711 31.051 Td [(.)-746(On)-439(a)-433(donc)]TJ /F8 1 Tf -33.38 -2.2289 Td [(f)]TJ /F11 1 Tf 0.87349 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.79518 0 Td [(=)-277(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.026 40.961 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.042 28.822 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.483 41.03 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.066 28.822 Td [(.)]TJ -6.7349 -1.9157 Td [(Fixons)]TJ /F8 1 Tf 3.247 0 Td [(\270)]TJ /F2 1 Tf 0.92169 0 Td [(dans)-337(]0;)-168(1[)-343(et)-337(p)-30(osons)]TJ /F8 1 Tf 9.1265 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.836 38.288 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.283 26.906 Td [(=)]TJ /F6 1 Tf 1.0663 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.85542 0 Td [(2)]TJ /F8 1 Tf 0.95181 0 Td [(X)]TJ /F6 1 Tf 1.1928 0 Td [(j)]TJ /F8 1 Tf 0.56627 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.052 38.288 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.44 26.906 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.6747 0 Td [(>)]TJ /F8 1 Tf 1.0663 0 Td [(\270f)]TJ /F2 1 Tf 1.1807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)-457(Comme)-343(la)-337(suite)-343(des)]TJ /F8 1 Tf 9.6627 0 Td [(J)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.655 38.288 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 26.906 Td [(est)]TJ -37.934 -1.1988 Td [(croissan)30(te,)-307(il)-289(en)-289(v)48(a)-289(de)-289(m)24(^)469(eme)-289(p)-30(our)-289(la)-295(suite)-289(des)]TJ /F8 1 Tf 19.807 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.655 36.573 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.452 25.707 Td [(et)-289(donc)-295(celle)-295(des)]TJ /F8 1 Tf 7.3374 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.534 36.573 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.163 25.707 Td [(.)-427(P)24(ar)-289(ailleurs,)-301(lorsque)]TJ /F8 1 Tf -29.831 -1.1988 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56626 0 Td [(\))-301(=)-295(0,)]TJ /F8 1 Tf 2.8855 0 Td [(x)]TJ /F2 1 Tf 0.91566 0 Td [(appartien)24(t)-349(\265)500(a)-343(tous)-343(les)]TJ /F8 1 Tf 9.3675 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.879 34.857 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.771 24.509 Td [(et,)-349(lorsque)]TJ /F8 1 Tf 4.8916 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.68675 0 Td [(>)]TJ /F2 1 Tf 1.0723 0 Td [(0,)-349(on)-343(a)]TJ /F8 1 Tf 3.3675 0 Td [(\270f)]TJ /F2 1 Tf 1.1807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.68072 0 Td [(<)-301(f)]TJ /F2 1 Tf 1.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-301(=)-295(lim)]TJ /F8 1 Tf 3.3133 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.966 34.857 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.145 24.509 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ -38.777 -1.1988 Td [(donc)]TJ /F8 1 Tf 2.5301 0 Td [(x)]TJ /F6 1 Tf 1.0843 0 Td [(2)]TJ /F8 1 Tf 1.1867 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.681 33.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.982 23.31 Td [(p)-30(our)]TJ /F8 1 Tf 2.506 0 Td [(n)]TJ /F2 1 Tf 1.0723 0 Td [(su\261sammen)24(t)-481(grand.)-873(Ainsi)]TJ /F8 1 Tf 12.572 0 Td [(X)]TJ /F2 1 Tf 1.3855 0 Td [(est)-475(la)-481(r)30(\266)469(eunion)-475(croissan)24(te)-481(des)]TJ /F8 1 Tf 13.325 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.5 33.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 23.31 Td [(.)]TJ -38.886 -1.2048 Td [(Remarquons)-409(en\257n)-415(que)-415(les)]TJ /F8 1 Tf 11.843 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.767 31.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.958 22.105 Td [(son)18(t)-409(mesurables)-415(puisqu'on)-415(p)-30(eut)-415(les)-415(d)24(\266)475(e\257nir)-415(par)-415(la)-409(relation)]TJ /F8 1 Tf -13.627 -1.1988 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3103 29.702 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.3494 20.906 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(\270f)]TJ /F2 1 Tf 1.1867 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(0.)]TJ -7.4578 -2.1145 Td [(Sur)-313(c)24(haque)]TJ /F8 1 Tf 5.1205 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.147 26.676 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.133 18.792 Td [(la)-307(relation)]TJ /F8 1 Tf 4.7349 0 Td [(\270f)]TJ /F11 1 Tf 1.4578 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.414 26.676 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.711 18.792 Td [(s')24(\266)469(ecrit)]TJ /F9 1 Tf 2.9277 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.983 25.176 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.253 18.792 Td [(\270\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.888 26.676 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.807 18.792 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.164 26.676 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 55.94 37.078 Td [(i)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.483 26.676 Td [(\\)]TJ /F7 1 Tf 0.76724 0 Td [(X)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 58.964 37.078 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 30.301 18.792 Td [(6)]TJ /F9 1 Tf 1.259 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.871 25.176 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 65.422 34.993 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.367 18.792 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.56 26.676 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.355 18.792 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.966 26.676 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 71.048 37.078 Td [(j)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.371 26.676 Td [(\\)]TJ /F7 1 Tf 0.76724 0 Td [(X)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 74.181 37.078 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.633 18.792 Td [(,)-313(ce)-313(qui,)-313(d'apr)24(\265)469(es)]TJ -32.301 -2.5843 Td [(le)-530(cas)-530(pr)24(\266)469(ec)24(\266)475(eden)24(t,)-578(donne)]TJ /F9 1 Tf 11.307 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.966 21.487 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.253 16.207 Td [(\270\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.871 22.978 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.807 16.207 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.233 22.978 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.512 16.207 Td [(\\)]TJ /F8 1 Tf 1.0181 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.862 22.978 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.904 16.207 Td [(\))]TJ /F11 1 Tf 0.99398 0 Td [(6)]TJ /F9 1 Tf 1.5783 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.034 21.487 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 57.253 29.825 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.283 16.207 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.716 22.978 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.271 16.207 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.224 22.978 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.072 16.207 Td [(\\)]TJ /F8 1 Tf 1.0181 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.974 22.978 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.464 16.207 Td [(\))-530(et)-530(donc)]TJ /F14 1 Tf 4.8614 0 Td [(a)-542(fortiori)]TJ /F9 1 Tf -34.994 -1.6325 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7759 17.788 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 13.623 Td [(\270\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.681 19.288 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5 13.623 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.052 19.288 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.072 13.623 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.302 19.288 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.337 13.623 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.569 17.788 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.669 13.623 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.664 19.288 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.657 13.623 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.164 19.288 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.102 13.623 Td [(\).)]TJ -14.771 -2.994 Td [(Si)-325(l'on)-331(fait)-331(tendre)]TJ /F8 1 Tf 8.006 0 Td [(n)]TJ /F2 1 Tf 0.92771 0 Td [(v)24(ers)-331(+)]TJ /F6 1 Tf 2.8373 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-331(la)-325(propri)24(\266)469(et)24(\266)475(e)-337(de)-325(Borel)-331(assure)-325(p)-30(our)-331(c)24(haque)]TJ /F8 1 Tf 18.669 0 Td [(i)]TJ /F2 1 Tf 0.66867 0 Td [(que)]TJ /F8 1 Tf 1.8554 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.121 14.995 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.855 10.629 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.336 14.995 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 10.629 Td [(\))]TJ -38.777 -1.4036 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.733 12.995 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.024 9.2256 Td [(\).)-451(Comme)]TJ /F8 1 Tf 4.7831 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(est)-337(\257ni)-337(on)-337(en)-331(tire)]TJ /F9 1 Tf 7.8434 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.647 11.495 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.114 9.2256 Td [(\270\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.552 12.995 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.669 9.2256 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.914 12.995 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.024 9.2256 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.879 11.495 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.355 9.2256 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.974 12.995 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.343 9.2256 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.474 12.995 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.789 9.2256 Td [(\).)]TJ -32.458 -2.9398 Td [(Il)-331(ne)-337(reste)-337(plus)-331(qu')0(\265)499(a)-337(faire)-337(tendre)]TJ /F8 1 Tf 14.386 0 Td [(\270)]TJ /F2 1 Tf 0.91566 0 Td [(v)24(ers)-337(1)-331(p)-30(our)-337(obtenir)-337(la)-331(relation)-337(annonc)24(\266)475(ee.)]TJ /F5 1 Tf -15.301 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(48)]TJ ET endstream endobj 179 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 178 0 R /ProcSet 2 0 R >> /Contents 177 0 R >> endobj 178 0 obj << /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F13 39 0 R /F14 43 0 R >> endobj 180 0 obj << /Length 15155 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ 0.81325 -2.4036 Td [(3.)]TJ /F2 1 Tf 1.5602 0 Td [(Cas)-337(g)30(\266)469(en)30(\266)469(eral.)]TJ ET 0.504 w 92.088 742.03 m 156.02 742.03 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 60.491 Td [(P)30(our)-493(toute)]TJ /F8 1 Tf 5.3614 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.043 87.15 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.982 60.491 Td [(partie)-500(\257nie)-493(de)]TJ /F8 1 Tf 6.8675 0 Td [(I)]TJ /F2 1 Tf 1.012 0 Td [(on)-493(a)-463(\266)469(evidemmen)30(t)]TJ /F9 1 Tf 8.3374 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.319 84.849 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 58.807 118.99 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.843 60.491 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.621 86.349 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.813 60.491 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.897 86.349 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 63.952 120.48 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 32.886 60.491 Td [(6)]TJ /F9 1 Tf 1.3193 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.966 84.849 Td [(j)]TJ /F10 1 Tf 0.53449 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.819 60.491 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.069 86.349 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.807 60.491 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.474 86.349 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 75.94 120.48 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.392 60.491 Td [(.)-921(Le)-493(r)24(\266)475(esultat)]TJ -33.06 -2.5843 Td [(pr)30(\266)469(ec)24(\266)469(eden)30(t)-313(assure)-307(donc)]TJ /F9 1 Tf 9.8855 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.733 81.159 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 32.843 113.82 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.855 57.906 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.043 82.65 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.831 57.906 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.405 82.65 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.181 57.906 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.362 81.159 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.518 57.906 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.466 82.65 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.506 57.906 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.966 82.65 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.952 57.906 Td [(\).)-439(Cela)-283(\266)469(etan)24(t)-313(vrai)-307(p)-30(our)-313(toute)-307(partie)-313(\257nie)]TJ -21.62 -2.5783 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-337(on)-331(en)-337(d)30(\266)469(eduit)]TJ /F9 1 Tf 6.4518 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.655 77.461 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.241 55.328 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.724 78.961 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.217 55.328 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.095 78.961 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.566 55.328 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.052 77.461 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.904 55.328 Td [(\257)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.147 78.961 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.886 55.328 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.655 78.961 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.337 55.328 Td [(\).)]TJ ET 512.64 660.06 m 519.84 660.06 l 519.84 668.48 l 512.64 668.48 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 52.159 Td [(Nous)-331(p)-30(ouv)18(ons)-337(donc)-337(donner)-337(la)-331(d)24(\266)469(e\257nition)-331(suiv)48(an)30(te)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 41.022 Td [(IV.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 49.177 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83133 0.084339 Td [(\266)]TJ -0.0060234 -0.084339 Td [(efinition.)]TJ /F12 1 Tf 5.3193 0 Td [(Si)]TJ /F8 1 Tf 1.1807 0 Td [(f)]TJ /F12 1 Tf 0.94578 0 Td [(est)-355(une)-349(fonction)]TJ /F6 1 Tf 7.3614 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurable)-349(p)-30(ositiv)24(e,)-355(l')]TJ /F1 1 Tf 9.8012 0 Td [(in)30(t)24(\266)548(egrale)]TJ /F12 1 Tf 4.7048 0 Td [(de)]TJ /F8 1 Tf 1.3494 0 Td [(f)]TJ /F12 1 Tf 0.94579 0 Td [(est)]TJ -37.934 -1.1988 Td [(par)-331(d)24(\266)469(e\257nition)-337(:)]TJ /F9 1 Tf 14.361 -0.29518 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)-271(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.776 64.573 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.807 46.322 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.845 66.073 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.777 46.322 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.207 66.073 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.127 46.322 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(,)]TJ -24.518 -2.8795 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3554 0 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.483 60.452 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.229 43.442 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.552 61.952 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.199 43.442 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.836 61.952 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 24.735 86.379 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 13.03 43.442 Td [(est)-307(une)-271(\266)469(ecriture)-301(de)]TJ /F8 1 Tf 8.3253 0 Td [(f)]TJ /F12 1 Tf 0.89157 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1626 0 Td [(I)]TJ /F12 1 Tf 0.81928 0 Td [(d)24(\266)475(enom)24(brable,)-313(les)-301(co)-30(e\261cien)24(ts)]TJ /F8 1 Tf 12.614 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.078 61.952 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 39.12 43.442 Td [(r)30(\266)469(eels)-301(p)-30(ositifs)]TJ -33.789 -2.1988 Td [(\(\257nis\))-331(et)-337(les)]TJ /F8 1 Tf 5.512 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.595 58.806 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 12.259 41.244 Td [(des)-337(parties)]TJ /F6 1 Tf 5.0181 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurables)-331(de)]TJ /F8 1 Tf 6.7952 0 Td [(X)]TJ /F12 1 Tf 0.90362 0 Td [(.)]TJ /F1 1 Tf -20.446 -2.4217 Td [(Remarques.)]TJ 0.81325 -1.5783 Td [(1.)]TJ /F2 1 Tf 1.5602 0 Td [(Cette)-337(in)24(t)24(\266)475(egrale)-343(appartien)24(t)-337(\265)500(a)-331([0;)-168(+)]TJ /F6 1 Tf 14.512 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(].)]TJ /F1 1 Tf -17.072 -1.5783 Td [(2.)]TJ /F2 1 Tf 1.8554 0 Td [(Dans)-487(la)-481(notation)-487(de)-481(l'in)24(t)24(\266)475(egrale,)-518(on)-487(p)-24(eut)-487(c)24(hoisir)-481(une)-487(lettre,)-518(par)-481(exemple)]TJ /F8 1 Tf 33.096 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-518(p)-30(our)]TJ -36.337 -1.6566 Td [(repr)30(\266)469(esen)24(ter)-337(une)-331(v)48(ariable)-331(dans)]TJ /F8 1 Tf 13.139 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-331(on)-337(p)-30(ourra)-331(alors)-343(noter)-337(l'in)24(t)24(\266)475(egrale)]TJ /F9 1 Tf 14.916 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)]TJ -34.434 -2.7048 Td [(Dans)-325(le)-325(cas)-325(de)-325(la)-325(mesure)-325(de)-325(Leb)-30(esgue)-325(on)-325(utilise)-325(m)24(^)469(eme)-319(la)-325(notation)]TJ /F9 1 Tf 28.982 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0964 0 Td [(.)-439(P)24(ar)-319(abus)-325(on)]TJ -33.193 -1.6566 Td [(dit)-337(parfois)-337(que)]TJ /F8 1 Tf 6.6747 0 Td [(dx)]TJ /F2 1 Tf 1.4277 0 Td [(est)-337(la)-337(mesure)-337(de)-337(Leb)-30(esgue.)]TJ -8.1024 -1.5783 Td [(Dans)-403(cette)-397(derni)24(\265)469(ere)-403(notation)]TJ /F8 1 Tf 13.096 0 Td [(x)]TJ /F2 1 Tf 0.96988 0 Td [(est)-403(une)-403(v)54(ariable)-403(m)24(uette,)-415(c'est-)0(\265)500(a-dire)-403(que)-403(la)-397(lettre)]TJ /F8 1 Tf 22.157 0 Td [(x)]TJ /F2 1 Tf 0.96988 0 Td [(p)-30(eut)]TJ -37.223 -1.2048 Td [(^)469(etre)-331(remplac)24(\266)469(ee)-337(par)-337(toute)-331(autre)-337(lettre)-343(non)-337(utilis)24(\266)475(ee)-337(aupara)24(v)48(an)24(t.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 18.736 Td [(IV.2.)-236(Propri)20(\266)537(et)25(\266)537(es)-381(de)-376(l'in)30(t)20(\266)537(egrale)-376(des)-376(fonctions)-376(mesurables)-376(p)-35(ositiv)30(es.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 20.882 Td [(Rapp)-30(elons)-331(que)]TJ /F6 1 Tf 6.7289 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.974 29.668 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.922 20.882 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-325(d)24(\266)469(esigne)-325(l'ensem)24(ble)-325(des)-325(fonctions)-331(mesurables)-331(de)]TJ /F8 1 Tf 21.434 0 Td [(X)]TJ /F2 1 Tf 1.2349 0 Td [(dans)]TJ ET 480.74 259.48 m 489.38 259.48 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 40.223 20.882 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 58.595 29.668 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.608 20.882 Td [(.)-439(V)78(oici)]TJ -36.277 -1.2048 Td [(les)-337(propri)24(\266)475(et)24(\266)469(es)-331(fondamen)18(tales)-331(de)-337(l'in)24(t)24(\266)475(egrale)-337(que)-343(nous)-331(v)24(enons)-337(de)-337(d)30(\266)469(e\257nir)-337(sur)]TJ /F6 1 Tf 32.283 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 55.552 27.952 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.482 19.677 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 13.927 Td [(IV.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 16.695 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084335 Td [(\266)]TJ 0 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ -0.0060244 -0.084335 Td [(eme.)]TJ /F12 1 Tf 2.9518 0 Td [(Soit)]TJ /F2 1 Tf 2.1446 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.80121 0 Td [(;)-162(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))]TJ /F12 1 Tf 0.81325 0 Td [(un)-421(espace)-421(mesur)24(\266)475(e)-427(et)]TJ /F6 1 Tf 9.4759 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.112 23.676 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.283 16.695 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.81326 0 Td [(l'ensem)30(ble)-421(des)-427(ap-)]TJ -31.06 -1.1988 Td [(plications)-421(mesurables)-421(de)]TJ /F8 1 Tf 11.217 0 Td [(X)]TJ /F12 1 Tf 1.3253 0 Td [(dans)]TJ /F2 1 Tf 2.4217 0 Td [([0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.28313 0 Td [(.)-692(Alors)-415(nous)-421(v)24(enons)-421(de)-415(d)24(\266)475(e\257nir)-421(une)-415(application)]TJ /F8 1 Tf -18.247 -1.7108 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F12 1 Tf 2.5 0 Td [(de)]TJ /F6 1 Tf 1.3313 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.345 19.512 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.181 13.786 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(dans)]TJ /F2 1 Tf 2.3434 0 Td [([0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.60843 0 Td [(qui)]TJ /F1 1 Tf -16.006 -2.7048 Td [(0.)]TJ /F12 1 Tf 1.5602 0 Td [(prolonge)]TJ /F8 1 Tf 4.0602 0 Td [(m)]TJ /F12 1 Tf 1.2108 0 Td [(:)]TJ /F6 1 Tf 1.7229 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(T)]TJ /F9 1 Tf 2.9096 1.3614 Td [(Z)]TJ /F2 1 Tf 1.006 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.707 15.642 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.578 11.081 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ /F1 1 Tf -20.904 -2.7048 Td [(1.)]TJ /F12 1 Tf 1.3434 0 Td [(est)]TJ /F1 1 Tf 1.4518 0 Td [(p)-30(ositiv)30(emen)30(t)-259(homog)18(\265)548(ene)]TJ /F12 1 Tf 11.928 0 Td [(:)]TJ /F6 1 Tf 1.6687 0 Td [(8)]TJ /F8 1 Tf 0.56024 0 Td [(\270)]TJ /F6 1 Tf 0.86144 0 Td [(2)]TJ /F2 1 Tf 0.94579 0 Td [([0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F6 1 Tf 0.5 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-283(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.767 11.78 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.645 8.3762 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F9 1 Tf 1.5542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\270f)-277(dm)]TJ /F2 1 Tf 3.0241 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf -37 -2.0904 Td [(\(a)24(v)24(ec)-337(la)-331(con)24(v)24(en)30(tion)-337(0)]TJ /F6 1 Tf 9.4217 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(\(+)]TJ /F6 1 Tf 1.1627 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(\))-277(=)-283(\(+)]TJ /F6 1 Tf 2.8855 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(0)-277(=)-277(0\))]TJ /F12 1 Tf 3.0542 0 Td [(;)]TJ /F2 1 Tf -1.2289 -2.3976 Td [(49)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 182 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 181 0 R /ProcSet 2 0 R >> /Contents 180 0 R >> endobj 181 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F18 107 0 R >> endobj 183 0 obj << /Length 21843 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ 0.81325 -2.7651 Td [(2.)]TJ /F12 1 Tf 1.5602 0 Td [(est)]TJ /F1 1 Tf 1.5602 0 Td [(croissan)24(te)]TJ /F12 1 Tf 5.2651 0 Td [(:)]TJ /F6 1 Tf 1.2771 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-283(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.741 88.306 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.048 61.858 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F6 1 Tf 0.72289 0 Td [(8)]TJ /F8 1 Tf 0.56024 0 Td [(g)]TJ /F6 1 Tf 0.78916 0 Td [(2)-283(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.578 88.306 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.229 61.858 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F8 1 Tf 1.3855 0 Td [(f)]TJ /F11 1 Tf 0.87952 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F6 1 Tf 0.78916 0 Td [(\))]TJ /F9 1 Tf 1.2831 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-271(dm)]TJ /F11 1 Tf 2.4398 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F12 1 Tf 2.0783 0 Td [(;)]TJ /F1 1 Tf -34.343 -2.1627 Td [(3.)]TJ /F12 1 Tf 1.9518 0 Td [(est)]TJ /F1 1 Tf 1.759 0 Td [(d)24(\266)548(enom)30(brablemen)30(t)-608(additiv)30(e)]TJ /F12 1 Tf 14.175 0 Td [(:)-1837(p)-30(our)-530(toute)-530(famille)]TJ /F2 1 Tf 10.934 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.293 85.211 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.175 59.695 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.319 85.211 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.373 59.695 Td [(d')24(\266)475(el)18(\266)475(emen)24(ts)-530(de)]TJ /F6 1 Tf -33.042 -1.7108 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 9.3448 82.762 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1928 57.985 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.1928 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(d)24(\266)469(enom)24(brable)]TJ /F9 1 Tf 6.9277 1.3615 Td [(Z)]TJ 1 -0.40964 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.897 82.762 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.789 57.985 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ 1.6145 0.40964 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.957 82.762 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.12 57.985 Td [(dm)]TJ /F12 1 Tf 1.7349 0 Td [(;)]TJ /F1 1 Tf -24.711 -2.1024 Td [(4.)]TJ /F12 1 Tf 1.5602 0 Td [(a)-331(la)]TJ /F1 1 Tf 1.9458 0 Td [(propri)24(\266)548(et)18(\266)548(e)-385(de)-385(Bepp)-36(o)-385(Levi)]TJ /F12 1 Tf 12.651 0 Td [(:)]TJ -14.952 -1.6566 Td [(p)-30(our)-337(toute)-337(suite)-331(croissan)24(te)-337(d')24(\266)475(el)18(\266)475(emen)24(ts)]TJ /F8 1 Tf 16.898 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.405 77.383 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.614 54.226 Td [(de)]TJ /F6 1 Tf 1.3373 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.284 77.383 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.819 54.226 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F9 1 Tf 1.3855 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.216 77.383 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.705 54.226 Td [(dm)]TJ /F6 1 Tf 1.6747 0 Td [(!)]TJ /F9 1 Tf 1.2831 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.879 77.306 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.596 54.226 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.362 77.383 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.795 54.226 Td [(dm)]TJ /F12 1 Tf 1.3976 0 Td [(.)]TJ /F1 1 Tf -36.861 -2.4337 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6506 Td [(Le)-337(p)-30(oin)30(t)]TJ /F1 1 Tf 4.012 0 Td [(0)]TJ /F2 1 Tf 0.90964 0 Td [(r)24(\266)475(esulte)-337(imm)24(\266)469(ediatemen)30(t)-337(de)-337(la)-331(d)24(\266)475(e\257nition)-337(de)-337(l'in)24(t)24(\266)475(egrale.)]TJ -4.9217 -2.1205 Td [(Le)-313(p)-30(oin)24(t)]TJ /F1 1 Tf 3.9699 0 Td [(1)]TJ /F2 1 Tf 0.89157 0 Td [(est)-319(imm)24(\266)469(ediat)-313(lorsque)]TJ /F8 1 Tf 9.3735 0 Td [(\270)]TJ /F2 1 Tf 0.89157 0 Td [(est)-319(\257ni,)-325(en)-313(e\256et,)-313(si)]TJ /F8 1 Tf 8.2831 0 Td [(f)]TJ /F2 1 Tf 0.8735 0 Td [(=)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.112 68.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.259 48.021 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.397 68.504 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 68.843 95.548 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.102 48.021 Td [(on)-313(a)]TJ /F8 1 Tf 2.1807 0 Td [(\270f)]TJ /F2 1 Tf 1.4578 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\270\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.009 68.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.964 48.021 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.284 68.504 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 88.253 95.548 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 46.009 Td [(par)-397(suite)]TJ /F9 1 Tf 4.3012 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(\270f)-277(dm)]TJ /F2 1 Tf 3.1265 0 Td [(=)]TJ /F9 1 Tf 1.1627 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\270\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.405 65.625 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.09 46.009 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.776 65.625 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.44 46.009 Td [(\))-385(=)]TJ /F8 1 Tf 1.9277 0 Td [(\270)]TJ /F9 1 Tf 0.75301 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.31 65.625 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.705 46.009 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.672 65.625 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.054 46.009 Td [(\))-385(=)]TJ /F8 1 Tf 1.9337 0 Td [(\270)]TJ /F9 1 Tf 0.74699 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)-632(Nous)-397(traiterons)-397(le)-403(cas)]TJ /F8 1 Tf -28.566 -1.6566 Td [(\270)]TJ /F2 1 Tf 0.86145 0 Td [(=)-277(+)]TJ /F6 1 Tf 1.8313 0 Td [(1)]TJ /F2 1 Tf 1.3373 0 Td [(plus)-331(loin.)]TJ -4.0301 -1.9699 Td [(L'additivit)24(\266)475(e)-301(d)24(\266)475(enom)24(brable)-295(r)24(\266)475(esulte)-295(des)-301(propri)24(\266)475(et)24(\266)469(es)-295(des)-295(familles)-301(sommables.)-433(Plus)-295(pr)24(\266)469(ecis)30(\266)469(emen)30(t,)]TJ 0 -1.3494 Td [(en)-307(\266)469(ecriv)54(an)24(t)-343(c)24(haque)]TJ /F8 1 Tf 8.4639 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.448 58.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.958 41.033 Td [(sous)-343(la)-337(forme)]TJ /F9 1 Tf 6.3795 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.155 56.995 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(K)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 45.458 79.463 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.211 41.033 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.129 58.504 Td [(i;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.861 41.033 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.388 58.504 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 52.06 81.56 Td [(i;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.012 41.033 Td [(,)-337(on)-343(a)]TJ /F9 1 Tf 2.8494 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.888 57.012 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.476 41.033 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.741 58.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.584 41.033 Td [(=)]TJ /F9 1 Tf 1.0663 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.379 58.504 Td [(i;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.916 41.033 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.629 58.504 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 76.157 81.56 Td [(i;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.398 41.033 Td [(o)-30(\265)530(u)-343(le)-343(m)24(ulti-)]TJ -34.066 -2.1988 Td [(indice)]TJ /F8 1 Tf 3.0361 0 Td [(i;)-168(k)]TJ /F2 1 Tf 1.8193 0 Td [(parcourt)-481(l'ensem)30(ble)-481(d)24(\266)469(enom)24(brable)-481(des)]TJ /F8 1 Tf 17.114 0 Td [(i;)-168(k)]TJ /F2 1 Tf 1.8193 0 Td [(tels)]TJ /F8 1 Tf 1.988 0 Td [(i)]TJ /F6 1 Tf 0.86747 0 Td [(2)]TJ /F8 1 Tf 1.1867 0 Td [(I)]TJ /F2 1 Tf 1 0 Td [(et)]TJ /F8 1 Tf 1.3133 0 Td [(k)]TJ /F6 1 Tf 1.0783 0 Td [(2)]TJ /F8 1 Tf 1.1867 0 Td [(K)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.224 55.357 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.922 38.834 Td [(.)-885(Il)-481(s'ensuit)]TJ /F9 1 Tf -33.59 -0.29518 Td [(Z)]TJ 1 -0.40964 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.069 52.987 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.9337 37.177 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.914 52.987 Td [(i;k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.928 37.177 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.25 52.987 Td [(i;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.958 37.177 Td [(\))-283(=)]TJ /F9 1 Tf 1.7289 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.319 51.495 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 21.295 38.292 Td [(\263)]TJ 1.0301 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.569 51.487 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(K)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 47.422 71.752 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.199 37.177 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.543 52.987 Td [(i;k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.849 37.177 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.879 52.987 Td [(i;k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.88 37.177 Td [(\))]TJ /F9 1 Tf 0.39157 1.1145 Td [(\264)]TJ /F2 1 Tf 0.87349 -1.1145 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ 1.6145 0.40964 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.086 52.987 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.801 37.177 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(.)]TJ -30.873 -2.9699 Td [(Nous)-283(allons)-289(d)30(\266)469(eduire)-283(les)-289(p)-30(oin)24(ts)]TJ /F1 1 Tf 13.145 0 Td [(2)]TJ /F2 1 Tf 0.86144 0 Td [(et)]TJ /F1 1 Tf 1.1145 0 Td [(4)]TJ /F2 1 Tf 0.86144 0 Td [(de)-283(cette)-289(propri)24(\266)469(et)24(\266)475(e)-289(d'additivit)24(\266)469(e.)-433(P)30(our)-289(cela)-289(nous)-283(aurons)]TJ -15.982 -1.1988 Td [(b)-30(esoin)-337(d'un)-337(lemme)-337(auxiliaire.)]TJ /F15 1 Tf 0 -1.9759 Td [(Lemme.)]TJ /F12 1 Tf 4.0904 0 Td [(Soien)24(t)]TJ /F8 1 Tf 2.994 0 Td [(f)]TJ /F12 1 Tf 0.89157 0 Td [(et)]TJ /F8 1 Tf 1.1265 0 Td [(g)]TJ /F12 1 Tf 0.80723 0 Td [(des)-301(fonctions)-295(mesurables)-301(p)-30(ositiv)30(es.)-433(On)-301(supp)-30(ose)]TJ /F8 1 Tf 20.825 0 Td [(f)]TJ /F11 1 Tf 0.87349 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(g)]TJ /F12 1 Tf 0.80723 0 Td [(alors)-295(il)-301(existe)]TJ /F8 1 Tf -33.476 -1.2048 Td [(h)]TJ /F12 1 Tf 0.90964 0 Td [(mesurable)-331(p)-30(ositiv)24(e)-337(telle)-337(que)]TJ /F8 1 Tf 12.464 0 Td [(g)]TJ /F2 1 Tf 0.78916 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.8253 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(h)]TJ /F12 1 Tf 0.57229 0 Td [(.)]TJ /F1 1 Tf -17.614 -1.9699 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(L'id)30(\266)469(ee)-337(naturelle)-337(est)-337(de)-337(p)-30(oser)]TJ /F8 1 Tf 12.681 0 Td [(h)]TJ /F2 1 Tf 0.57229 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\),)-343(cep)-30(endan)24(t)-337(cela)-343(p)-30(eut)]TJ -29.404 -1.2048 Td [(en)30(tra)108(^)385(\263ner)-421(des)-421(formes)-421(ind)24(\266)475(etermin)24(\266)469(ees.)-698(Lorsque)-421(cela)-421(se)-421(pro)-30(duit)-415(on)-421(a)]TJ /F8 1 Tf 29.705 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))-427(=)-421(+)]TJ /F6 1 Tf 2.7892 0 Td [(1)]TJ /F2 1 Tf 1.4157 0 Td [(\(et)-427(donc)]TJ /F8 1 Tf -35.464 -1.1988 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))-493(=)-481(+)]TJ /F6 1 Tf 2.9217 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\).)-819(P)24(osons)-457(alors)]TJ /F8 1 Tf 7.4639 0 Td [(A)]TJ /F2 1 Tf 1.2349 0 Td [(=)]TJ /F8 1 Tf 1.2651 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.457 37.021 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.355 25.454 Td [(\()]TJ /F3 1 Tf 0.39157 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\),)-493(il)-457(s'agit)-463(d'une)-457(partie)-463(mesurable)-463(et)-457(l'on)-463(p)-30(eut)]TJ -18.139 -1.1988 Td [(d)30(\266)469(e\257nir)-373(\(sans)-379(probl)24(\265)469(eme)-373(de)-373(formes)-373(ind)24(\266)469(etermin)30(\266)469(ees\))-373(la)-373(fonction)-379(mesurable)]TJ /F8 1 Tf 31.349 0 Td [(h)]TJ /F2 1 Tf 0.92169 0 Td [(=)]TJ /F8 1 Tf 1.1205 0 Td [(g)]TJ /F6 1 Tf 0.75903 0 Td [(\241)]TJ /F8 1 Tf 1.0301 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.629 34.495 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.319 24.256 Td [(.)-566(On)]TJ -36.988 -1.2048 Td [(v)54(\266)469(eri\257e)-331(ais)24(\266)469(emen)24(t)-337(que)]TJ /F8 1 Tf 8.9578 0 Td [(h)]TJ /F2 1 Tf 0.90964 0 Td [(a)-331(les)-337(propri)24(\266)469(et)30(\266)469(es)-337(demand)30(\266)469(ees.)]TJ -9.8675 -2.2952 Td [(Ce)-361(lemme)-361(mon)18(tre)-361(en)-361(particulier)-355(que)-367(p)-30(our)]TJ /F8 1 Tf 18.675 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(et)]TJ /F8 1 Tf 1.1928 0 Td [(g)]TJ /F2 1 Tf 0.8735 0 Td [(dans)]TJ /F6 1 Tf 2.3675 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.784 29.487 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.259 20.756 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(\),)-373(si)]TJ /F8 1 Tf 2.0663 0 Td [(f)]TJ /F11 1 Tf 0.92168 0 Td [(6)]TJ /F8 1 Tf 1.0964 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(,)-367(il)-355(existe)]TJ /F8 1 Tf 4.3976 0 Td [(h)]TJ /F2 1 Tf 0.93373 0 Td [(dans)]TJ /F6 1 Tf -37.157 -1.7169 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 9.3448 27.038 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1928 19.039 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-343(tel)-349(que)]TJ /F8 1 Tf 4.0602 0 Td [(g)]TJ /F2 1 Tf 0.81325 0 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(f)]TJ /F2 1 Tf 0.8253 0 Td [(+)]TJ /F8 1 Tf 1.012 0 Td [(h)]TJ /F2 1 Tf 0.92169 0 Td [(et)-343(l'additivit)24(\266)475(e)-349(de)-349(l'in)24(t)24(\266)475(egrale)-349(assure)]TJ /F9 1 Tf 15.301 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(g)-204(dm)]TJ /F2 1 Tf 2.3735 0 Td [(=)]TJ /F9 1 Tf 1.0783 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)-277(dm)]TJ /F2 1 Tf 2.3916 0 Td [(+)]TJ /F9 1 Tf 1.006 1.3675 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3675 Td [(h)-168(dm)]TJ /F2 1 Tf -37.024 -2.3193 Td [(donc)]TJ /F9 1 Tf 2.3855 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)-271(dm)]TJ /F11 1 Tf 2.4398 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F2 1 Tf 2.4157 0 Td [(puisque)]TJ /F9 1 Tf 3.6446 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)-168(dm)]TJ /F2 1 Tf 2.4759 0 Td [(est)-337(p)-30(ositif.)-445(Cela)-313(\266)469(etablit)-337(le)-337(p)-30(oin)24(t)]TJ /F1 1 Tf 14.127 0 Td [(2.)]TJ /F2 1 Tf 0.89759 0 Td [(.)]TJ -32.446 -2.759 Td [(Prouv)30(ons)-391(le)-391(p)-30(oin)24(t)]TJ /F1 1 Tf 8.2349 0 Td [(4)]TJ /F2 1 Tf 0.57831 0 Td [(.)-596(Si)-391(l'on)-385(a)-385(une)-385(suite)-391(croissan)30(te)-391(de)-385(fonctions)-385(mesurables)-391(p)-30(ositiv)30(es)]TJ /F8 1 Tf 29.036 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.5 19.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 13.961 Td [(,)]TJ -38.886 -1.1988 Td [(p)-30(our)-349(c)24(haque)]TJ /F8 1 Tf 5.7289 0 Td [(n)]TJ /F11 1 Tf 0.89759 0 Td [(>)]TJ /F2 1 Tf 1.0843 0 Td [(1)-343(le)-355(lemme)-349(pr)24(\266)475(ec)24(\266)469(eden)24(t)-349(p)-30(ermet)-349(de)-349(d)24(\266)475(e\257nir)-349(une)-349(fonction)-355(mesurable)-349(p)-30(ositiv)30(e)]TJ /F8 1 Tf -7.7108 -1.3434 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4483 16.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.747 11.418 Td [(telle)-301(que)]TJ /F8 1 Tf 3.9578 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.026 16.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.018 11.418 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.414 16.125 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.283 11.418 Td [(+)]TJ /F8 1 Tf 0.92771 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.026 16.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.337 11.418 Td [(.)-433(En)-295(p)-30(osan)24(t)]TJ /F8 1 Tf 5.4337 0 Td [(h)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 33.414 16.125 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.078 11.418 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.664 16.125 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.367 11.418 Td [(on)-301(a)]TJ /F8 1 Tf 2.1506 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.517 16.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.831 11.418 Td [(=)]TJ /F9 1 Tf 1.1145 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.207 14.616 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.62 11.418 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.5 16.125 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.982 11.418 Td [(et)-301(donc)-301(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.612 16.056 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.404 11.418 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.086 16.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.717 11.418 Td [(=)]TJ /F9 1 Tf 1.0783 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.776 14.616 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F16 1 Tf 0.76725 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.428 11.418 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.966 16.125 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 8.5207 Td [(P)30(ar)-427(suite)]TJ /F9 1 Tf 4.4639 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.147 11.978 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.994 8.5207 Td [(dm)]TJ /F2 1 Tf 1.8373 0 Td [(=)]TJ /F9 1 Tf 1.2771 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.534 10.469 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 16.777 9.8822 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.267 11.978 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.012 8.5207 Td [(dm)]TJ /F2 1 Tf 2.259 0 Td [(et)]TJ /F9 1 Tf 1.6928 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.276 11.9 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.898 8.5207 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.759 11.978 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.096 8.5207 Td [(dm)]TJ /F2 1 Tf 1.8374 0 Td [(=)]TJ /F9 1 Tf 1.2349 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.147 10.469 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 31.801 9.8822 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.767 11.978 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.036 8.5207 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(.)-728(La)-433(somme)-433(globale)]TJ -30.139 -2.2349 Td [(\266)469(etan)24(t)-337(la)-331(limite)-337(des)-337(sommes)-337(partielles,)-337(la)-331(propri)18(\266)475(et)24(\266)469(e)-337(de)-331(Bepp)-36(o)-331(Levi)-337(en)-331(d)24(\266)469(ecoule.)]TJ /F5 1 Tf 0.030121 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(50)]TJ ET endstream endobj 185 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 184 0 R /ProcSet 2 0 R >> /Contents 183 0 R >> endobj 184 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F16 81 0 R /F17 97 0 R >> endobj 186 0 obj << /Length 14347 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(Remarquons)-349(alors)-361(que,)-361(p)-24(our)]TJ /F8 1 Tf 12.753 0 Td [(a)]TJ /F6 1 Tf 0.83735 0 Td [(2)]TJ ET 0.504 w 237.82 753.55 m 246.46 753.55 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 19.898 62.22 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 29.509 88.823 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.283 62.22 Td [(,)-361(\(+)]TJ /F6 1 Tf 1.8012 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(:a)]TJ /F2 1 Tf 1.1566 0 Td [(est)-361(la)-349(limite)-355(de)-349(la)-355(suite)-355(croissan)30(te)-355(des)]TJ /F8 1 Tf 16.825 0 Td [(n:a)]TJ /F2 1 Tf 1.759 0 Td [(.)]TJ -38.886 -1.7108 Td [(A)-409(partir)-409(de)-409(la)-415(relation)]TJ /F9 1 Tf 10.422 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\270f)-271(dm)]TJ /F2 1 Tf 3.1506 0 Td [(=)]TJ /F8 1 Tf 1.1807 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5723 0 Td [(prouv)54(\266)469(ee)-409(p)-30(our)]TJ /F8 1 Tf 6.247 0 Td [(\270)]TJ /F2 1 Tf 0.98795 0 Td [(\257ni,)-433(la)-409(propri)18(\266)475(et)24(\266)469(e)-409(de)-415(Bepp)-30(o)]TJ -27.313 -1.6566 Td [(Levi)-355(p)-30(ermet)-355(donc)-361(de)-355(prouv)30(er)-361(la)-355(m)24(^)469(eme)-355(relation)-355(p)-24(our)]TJ /F8 1 Tf 23.259 0 Td [(\270)]TJ /F2 1 Tf 0.89759 0 Td [(=)-313(+)]TJ /F6 1 Tf 1.8675 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)-512(Cela)-355(ac)18(h)30(\265)469(ev)24(e)-361(de)-355(d)30(\266)469(emon)24(trer)]TJ -27.024 -1.1988 Td [(le)-337(p)-30(oin)24(t)]TJ /F1 1 Tf 3.6687 0 Td [(2)]TJ /F2 1 Tf 0.90964 0 Td [(et)-331(donc)-337(le)-337(th)18(\266)475(eor)18(\265)475(eme.)]TJ ET 512.64 687.85 m 519.84 687.85 l 519.84 696.27 l 512.64 696.27 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 54.81 Td [(L')24(\266)469(enonc)30(\266)469(e)-493(suiv)48(an)30(t)-493(mon)24(tre)-493(que)-500(l'homog)30(\266)469(en)30(\266)469(eit)24(\266)475(e)-500(p)-30(ositiv)30(e)-493(et)-493(l'additivit)24(\266)475(e)-500(d)30(\266)469(enom)24(brable)-493(carac-)]TJ 0 -1.2048 Td [(t)24(\266)475(erisen)24(t)-337(les)-337(in)24(t)24(\266)475(egrales.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 42.178 Td [(IV.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 50.563 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.3614 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.1867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.88554 0 Td [(un)-487(espace)-493(mesurable)-493(et)]TJ /F8 1 Tf 11.09 0 Td [(\301)]TJ /F12 1 Tf 1.0843 0 Td [(une)-493(application)-493(de)]TJ /F6 1 Tf -30.789 -1.1988 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 9.3448 70.426 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1928 49.364 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.77108 0 Td [(dans)]TJ ET 139.25 599.9 m 147.89 599.9 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 11.651 49.364 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.707 70.426 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 13.422 49.364 Td [(qui)-379(est)-391(p)-30(ositiv)30(emen)24(t)-385(homog)24(\265)469(ene)-385(et)-385(d)30(\266)469(enom)24(brablemen)30(t)-385(additiv)24(e)]TJ /F2 1 Tf 27.283 0 Td [(\(cf.)-590(pro-)]TJ -35.373 -1.2048 Td [(pri)24(\266)475(et)24(\266)469(es)]TJ /F1 1 Tf 3.2349 0 Td [(1)]TJ /F2 1 Tf 0.91566 0 Td [(et)]TJ /F1 1 Tf 1.1687 0 Td [(3.)]TJ /F2 1 Tf 1.3554 0 Td [(du)-337(th)18(\266)475(eor)18(\265)475(eme)-337(pr)24(\266)469(ec)24(\266)475(eden)24(t\))]TJ /F12 1 Tf 10.38 0 Td [(.)-457(Alors)-343(l'application)]TJ /F8 1 Tf 9.1205 0 Td [(A)]TJ /F6 1 Tf 1.0361 0 Td [(7!)]TJ /F8 1 Tf 1.2831 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.621 68.702 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.024 48.159 Td [(\))]TJ /F12 1 Tf 0.72892 0 Td [(est)-343(une)-337(mesure)]TJ /F8 1 Tf 6.8614 0 Td [(m)]TJ /F12 1 Tf -38.283 -1.1988 Td [(sur)]TJ /F2 1 Tf 1.6747 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)-445(P)30(our)-337(cette)-337(mesure,)-337(l'in)24(t)24(\266)469(egrale)-337(d'une)-337(fonction)-343(mesurable)-331(p)-30(ositiv)24(e)]TJ /F8 1 Tf 28.946 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)]TJ /F8 1 Tf 1.5663 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -37.958 -1.9217 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(P)30(our)-222(mon)24(trer)-228(que)]TJ /F8 1 Tf 7.7771 0 Td [(m)]TJ /F2 1 Tf 1.0964 0 Td [(est)-228(une)-222(mesure)-228(on)-222(doit)-222(prouv)24(er)-228(l'additivit)30(\266)469(e)-222(d)24(\266)469(enom-)]TJ -17.693 -1.1988 Td [(brable)-319(de)]TJ /F8 1 Tf 4.3554 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\),)-319(mais)-319(elle)-319(d)24(\266)475(ecoule)-319(de)-319(l'additivit)30(\266)469(e)-319(d)30(\266)469(enom)24(brable)-319(de)]TJ /F8 1 Tf 22.205 0 Td [(\301)]TJ /F2 1 Tf 0.90964 0 Td [(puisque,)-319(p)-30(our)-319(des)]TJ /F8 1 Tf -31.512 -1.4036 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 60.512 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.747 42.436 Td [(deux)-331(\265)499(a)-337(deux)-331(disjoin)24(ts)-331(de)-331(r)24(\266)469(eunion)]TJ /F8 1 Tf 14.578 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-337(on)-331(a)-337(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.448 60.512 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.398 42.436 Td [(=)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F2 1 Tf 1.6084 -0.95181 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.397 60.512 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 60.458 84.379 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.596 42.436 Td [(.)]TJ -25.265 -1.759 Td [(P)30(ar)-240(d)24(\266)475(e\257nition)-246(l'in)24(t)24(\266)475(egrale)-246(p)-30(our)-240(cette)-240(mesure)-246(co)108(\304)385(\263ncide)-240(a)30(v)24(ec)]TJ /F8 1 Tf 24.458 0 Td [(\301)]TJ /F2 1 Tf 0.83133 0 Td [(sur)-246(les)-240(fonctions)-240(caract)24(\266)469(eristiques)]TJ -25.289 -1.1988 Td [(de)-403(parties)]TJ /F6 1 Tf 4.759 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurables.)-650(L'homog)24(\266)469(en)24(\266)475(eit)24(\266)469(e)-403(p)-30(ositiv)24(e)-403(et)-409(l'additivit)30(\266)469(e)-403(d)24(\266)469(enom)24(brable)-403(de)]TJ /F8 1 Tf 30.367 0 Td [(\301)]TJ /F2 1 Tf 1 0 Td [(et)-403(de)]TJ -36.928 -1.3434 Td [(l'in)24(t)24(\266)475(egrale)-409(en)30(tra)108(^)385(\263nen)24(t)-403(alors)-403(la)-403(co)108(\304)385(\263ncidence)-403(sur)-403(toutes)-403(les)-403(sommes)]TJ /F9 1 Tf 28.687 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.836 52.857 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.633 38.135 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.905 54.357 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.608 38.135 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.457 54.357 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.09 38.135 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.259 0 Td [(I)]TJ /F2 1 Tf 0.92168 0 Td [(d)24(\266)475(enom-)]TJ -35.94 -2.1446 Td [(brable,)]TJ /F8 1 Tf 3.3795 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.379 51.288 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.012 35.991 Td [(2)]TJ /F3 1 Tf 1 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.793 51.288 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.765 35.991 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.06 51.288 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.416 35.991 Td [(mesurable,)-379(c'est-)0(\265)500(a-dire)-373(toutes)-367(les)-373(fonctions)]TJ /F6 1 Tf 19.024 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(-mesurables)-367(p)-24(ositiv)24(es)]TJ -29.91 -1.2048 Td [(d'apr)24(\265)469(es)-307(le)-313(th)24(\266)469(eor)24(\265)469(eme)]TJ /F1 1 Tf 8.7711 0 Td [(I)-30(I)-36(I.5.9.)]TJ /F2 1 Tf 3.4819 0 Td [(.)-433(L')18(\266)475(egalit)24(\266)469(e)-313(de)-307(l'in)24(t)24(\266)469(egrale)-313(et)-307(de)]TJ /F8 1 Tf 13.133 0 Td [(\301)]TJ /F2 1 Tf 0.90362 0 Td [(sur)]TJ /F6 1 Tf 1.6506 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.336 49.564 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.139 34.786 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-313(en)-307(d)30(\266)469(ecoule.)]TJ ET 513.22 414.61 m 520.42 414.61 l 520.42 423.03 l 513.22 423.03 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 32.25 Td [(Remarque.)]TJ /F2 1 Tf 6.8494 0 Td [(L'additivit)24(\266)469(e)-487(d)30(\266)469(enom)24(brable)-487(en)30(tra)108(^)385(\263ne)-481(en)-481(particulier)-487(l'additivit)30(\266)469(e)-487(simple)-481(\(i.e.)]TJ /F8 1 Tf -6.8494 -1.1988 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.86747 0 Td [(+)]TJ /F8 1 Tf 1.0482 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\))-397(=)]TJ /F8 1 Tf 1.9578 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-271(+)]TJ /F8 1 Tf 1.7048 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\)\).)-662(Nous)-409(a)30(v)24(ons)-409(vu)-409(qu'elle)-415(impliquait)-409(aussi)-403(la)-409(propri)24(\266)469(et)30(\266)469(e)-409(de)-409(Bepp)-30(o)]TJ -10.163 -1.1988 Td [(Levi.)]TJ 0 -1.6145 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-433(l'additivit)30(\266)469(e)-409(simple)-409(p)-30(ermet)-409(de)-409(prouv)30(er,)-433(par)-409(r)30(\266)469(ecurrence,)-427(l'additivit)24(\266)475(e)-415(\257nie)]TJ 0 -1.4458 Td [(\(i.e.)]TJ /F8 1 Tf 2.3193 0 Td [(\301)]TJ /F9 1 Tf 0.59639 1.1084 Td [(\263)]TJ 0.76506 -0.15662 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.052 36.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.627 26.792 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.905 38.125 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 11.446 27.9 Td [(\264)]TJ /F2 1 Tf 0.99397 -1.1084 Td [(=)]TJ /F9 1 Tf 1.1687 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.629 36.625 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.223 26.792 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.897 38.125 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.03 26.792 Td [(\))-409(lorsque)]TJ /F8 1 Tf 4.2892 0 Td [(I)]TJ /F2 1 Tf 0.92169 0 Td [(est)-409(\257ni\),)-421(et)-409(la)-403(propri)24(\266)475(et)24(\266)469(e)-409(de)-403(Bepp)-30(o)-403(Levi)-409(p)-24(ermet)-409(de)]TJ -16.91 -2.1446 Td [(passer)-319(\265)500(a)-319(l'additivit)24(\266)475(e)-325(d)24(\266)469(enom)24(brable)-319(en)-325(consid)30(\266)469(eran)30(t)-325(les)-325(sommes)-319(d)24(\266)469(enom)24(brables)-319(comme)-319(limites)]TJ 0 -1.1988 Td [(des)-337(suites)-337(croissan)24(tes)-337(form)24(\266)475(ees)-337(par)-331(des)-337(sommes)-337(partielles.)]TJ 0 -1.6145 Td [(Ainsi,)-289(dans)-277(l')24(\266)469(enonc)30(\266)469(e)-283(pr)30(\266)469(ec)24(\266)475(eden)24(t,)-295(on)-277(p)-30(eut)-277(remplacer)-283(l'additivit)30(\266)469(e)-277(d)24(\266)469(enom)24(brable)-277(par)-277(l'additivit)24(\266)475(e)]TJ 0 -1.2048 Td [(simple)-331(join)24(te)-337(\265)500(a)-331(la)-337(propri)24(\266)475(et)24(\266)469(e)-337(de)-331(Bepp)-36(o)-331(Levi.)]TJ /F1 1 Tf 0 -2.2289 Td [(Exemples.)]TJ /F2 1 Tf 0 -2.0663 Td [(Lorsque)]TJ /F8 1 Tf 3.7711 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(est)-343(la)-331(mesure)-337(de)-337(Dirac)-337(au)-337(p)-30(oin)24(t)]TJ /F8 1 Tf 14.12 0 Td [(a)]TJ /F2 1 Tf 0.86747 0 Td [(on)-331(a)]TJ /F9 1 Tf 2.2229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ -28.205 -2.741 Td [(De)-337(m)24(^)475(eme,)-337(lorsque)]TJ /F8 1 Tf 8.1386 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(est)-337(la)-337(mesure)-337(de)-331(d)24(\266)475(ecompte,)-343(on)-331(a)]TJ /F9 1 Tf 14.41 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.216 17.745 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.199 13.593 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(\))-337(.)]TJ -31.199 -2.9096 Td [(Il)-433(su\261t)-433(p)-30(our)-433(v)24(oir)-433(cela)-439(d'observ)24(er)-439(que)]TJ /F8 1 Tf 17.193 0 Td [(f)]TJ /F6 1 Tf 1.0422 0 Td [(7!)]TJ /F8 1 Tf 1.4458 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.52409 0 Td [(\))-439(et)]TJ /F8 1 Tf 2.0904 0 Td [(f)]TJ /F6 1 Tf 1.0422 0 Td [(7!)]TJ /F9 1 Tf 1.4398 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(\))-433(son)18(t)-433(p)-30(ositiv)30(emen)24(t)-439(ho-)]TJ -28.711 -1.3494 Td [(mog)24(\265)475(enes)-337(et)-337(d)30(\266)469(enom)24(brablemen)30(t)-337(additiv)24(es)-331(puis)-337(d'appliquer)-331(la)-337(prop)-30(osition)-337(pr)24(\266)469(ec)24(\266)475(eden)24(te.)]TJ 18.916 -5.4458 Td [(51)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 188 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 187 0 R /ProcSet 2 0 R >> /Contents 186 0 R >> endobj 187 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 189 0 obj << /Length 16878 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(IV.3.)-236(In)30(t)20(\266)537(egrale)-376(sup)-10(\266)537(erieure)-376(des)-376(fonctions)-376(p)-35(ositiv)25(es.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 60.629 Td [(Nous)-295(allons)-271(\266)475(etendre)-301(la)-295(fonctionnelle)-295(d'in)18(t)24(\266)475(egration)-301(\265)500(a)-295(l'ensem)24(ble)-295(de)-295(toutes)-301(les)-301(fonctions)-295(p)-30(osi-)]TJ 0 -1.1988 Td [(tiv)24(es.)-451(Cep)-24(endan)24(t)-337(cela)-343(se)-337(fera)-337(au)-337(prix)-337(de)-337(la)-331(p)-30(erte)-337(de)-337(l'additivit)30(\266)469(e,)-343(qui)-331(sera)-337(remplac)24(\266)469(ee)-337(par)-337(la)]TJ 0 -1.1988 Td [(sous-additivit)24(\266)475(e.)-614(P)30(ours)-367(\266)475(eviter)-391(toute)-385(confusion)-391(cette)-391(fonctionnelle)-385(prolong)24(\266)469(ee)-391(ne)-385(s'app)-30(ellera)]TJ 0 -1.7349 Td [(plus)-331(in)24(t)24(\266)469(egrale)-337(mais)]TJ /F1 1 Tf 8.5422 0 Td [(in)30(t)18(\266)554(egrale)-385(sup)-12(\266)548(erieure)]TJ /F2 1 Tf 10.331 0 Td [(et)-331(l'on)-337(notera)]TJ /F9 1 Tf 6.2229 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.983 82.443 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.223 56.497 Td [(au)-337(lieu)-337(de)]TJ /F9 1 Tf 4.6145 1.3615 Td [(Z)]TJ /F2 1 Tf 1 -1.3615 Td [(.)]TJ -32.506 -2.1024 Td [(Rapp)-30(elons)-409(que)]TJ /F6 1 Tf 6.8855 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.517 77.625 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.602 54.394 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-409(d)24(\266)469(esigne)-409(l'ensem)30(ble)-409(de)-403(toutes)-409(les)-409(applications)-409(de)]TJ /F8 1 Tf 22.313 0 Td [(X)]TJ /F2 1 Tf 1.3133 0 Td [(dans)-409([0;)-168(+)]TJ /F6 1 Tf 4.4157 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(].)]TJ -38.608 -1.2048 Td [(On)-337(p)-24(ose)-337(alors)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 41.867 Td [(IV.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 50.189 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83133 0.084335 Td [(\266)]TJ -0.0060234 -0.084335 Td [(efinition.)]TJ /F12 1 Tf 5.3614 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4578 0 Td [(f)]TJ /F6 1 Tf 0.92169 0 Td [(2)-331(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.491 71.607 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.976 50.189 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(,)-373(on)-361(app)-30(elle)]TJ /F1 1 Tf 5.512 0 Td [(in)30(t)24(\266)548(egrale)-415(sup)-12(\266)548(erieure)]TJ /F12 1 Tf 10.386 0 Td [(de)]TJ /F8 1 Tf 1.3614 0 Td [(f)]TJ /F12 1 Tf 0.59639 0 Td [(,)-373(not)24(\266)469(ee)]TJ /F9 1 Tf -36.181 -0.42771 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 70.857 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 48.4 Td [(f)-277(dm)]TJ /F12 1 Tf 2.1627 0 Td [(,)-307(la)-295(b)-30(orne)-295(inf)30(\266)469(erieure)-295(des)]TJ /F9 1 Tf 10.5 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(h)-168(dm)]TJ /F12 1 Tf 2.4337 0 Td [(o)-30(\265)524(u)]TJ /F8 1 Tf 1.3554 0 Td [(h)]TJ /F12 1 Tf 0.86747 0 Td [(parcourt)-295(l'ensem)24(ble)-295(des)-271(\266)475(el)24(\266)469(emen)24(ts)-295(de)]TJ /F6 1 Tf 15.837 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 60.31 69.047 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.807 48.4 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf -38.777 -1.7169 Td [(qui)-331(ma)-60(joren)24(t)]TJ /F8 1 Tf 5.9759 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(\(cet)-337(ensem)24(ble)-337(p)-24(oss)24(\265)475(ede)-343(en)-331(particulier)-337(la)-331(constan)24(te)]TJ /F2 1 Tf 21.675 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F12 1 Tf 1 0 Td [(\).)]TJ /F1 1 Tf -30.361 -2.4578 Td [(Remarque.)]TJ /F2 1 Tf 6.2711 0 Td [(Comme)-337(p)-30(our)-343(l'in)24(t)24(\266)475(egrale,)-343(si)-337(l'on)-343(note)]TJ /F8 1 Tf 16.157 0 Td [(x)]TJ /F2 1 Tf 0.90964 0 Td [(la)-337(v)54(ariable)-343(dans)]TJ /F8 1 Tf 7.2169 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(on)-337(utilisera)-343(aussi)]TJ -31.801 -1.7349 Td [(la)-331(notation)]TJ /F9 1 Tf 5.1145 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.379 62.4 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.904 42.491 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(et)-331(m)24(^)469(eme)]TJ /F9 1 Tf 4.7771 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.707 62.4 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.62 42.491 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.4217 0 Td [(s'il)-343(s'agit)-331(de)-337(la)-331(mesure)-343(de)-331(Leb)-36(esgue.)]TJ -20.825 -2.6265 Td [(L')24(\266)469(enonc)30(\266)469(e)-337(suiv)54(an)24(t)-337(regroup)-30(e)-337(les)-337(principales)-337(propri)24(\266)469(et)24(\266)475(es)-337(de)-331(cette)-337(in)24(t)24(\266)469(egrale)-337(sup)0(\266)469(erieure)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 30.751 Td [(IV.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 36.864 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084335 Td [(\266)]TJ 0 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ -0.0060244 -0.084335 Td [(eme.)]TJ /F12 1 Tf 2.7831 0 Td [(Soit)]TJ /F2 1 Tf 2.0904 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F8 1 Tf 0.79518 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3253 0 Td [(\))]TJ /F12 1 Tf 0.75301 0 Td [(un)-373(espace)-367(mesur)30(\266)469(e,)]TJ /F6 1 Tf 8.3494 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.397 52.538 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.392 36.864 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90963 0 Td [(\))]TJ /F12 1 Tf 0.75301 0 Td [(l'ensem)24(ble)-367(des)-367(applica-)]TJ -29.108 -1.7892 Td [(tions)-331(de)]TJ /F8 1 Tf 3.7771 0 Td [(X)]TJ /F12 1 Tf 1.2349 0 Td [(dans)]TJ /F2 1 Tf 2.3373 0 Td [([0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(.)-445(Alors)-331(l'in)24(t)24(\266)475(egrale)-337(sup)0(\266)469(erieure)-331(est)-337(une)-331(application)]TJ /F8 1 Tf 21.428 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.017 51.788 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41 35.075 Td [(f)-277(dm)]TJ /F12 1 Tf 2.494 0 Td [(de)]TJ /F6 1 Tf -38.163 -1.7169 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6552 47.521 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7108 33.358 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(dans)]TJ /F2 1 Tf 2.3373 0 Td [([0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.61446 0 Td [(qui)-331(:)]TJ /F1 1 Tf -8.5422 -2.1807 Td [(0.)]TJ /F12 1 Tf 1.5602 0 Td [(prolonge)-337(l'in)24(t)24(\266)469(egrale)-337(:)]TJ /F6 1 Tf 10.428 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-277(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 31.069 44.409 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.373 31.177 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F9 1 Tf 1.5602 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.534 46.211 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.687 31.177 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F12 1 Tf 2.494 0 Td [(;)]TJ /F1 1 Tf -27.536 -2.7892 Td [(1.)]TJ /F12 1 Tf 1.3373 0 Td [(est)]TJ /F1 1 Tf 1.4518 0 Td [(p)-30(ositiv)30(emen)30(t)-259(homog)18(\265)548(ene)]TJ /F12 1 Tf 11.928 0 Td [(:)]TJ /F6 1 Tf 1.6627 0 Td [(8)]TJ /F8 1 Tf 0.56024 0 Td [(\270)]TJ /F6 1 Tf 0.86145 0 Td [(2)]TJ /F2 1 Tf 0.94578 0 Td [([0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F6 1 Tf 0.5 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.8735 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.06 40.409 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.151 28.388 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F9 1 Tf 1.5542 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.526 42.219 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.464 28.388 Td [(\270f)-277(dm)]TJ /F2 1 Tf 3.0241 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.526 42.219 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.753 28.388 Td [(f)-277(dm)]TJ /F2 1 Tf -37.422 -2.1024 Td [(\(a)24(v)24(ec)-337(la)-331(con)24(v)24(en)30(tion)-337(0)]TJ /F6 1 Tf 9.4217 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(\(+)]TJ /F6 1 Tf 1.1627 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(\))-277(=)-283(\(+)]TJ /F6 1 Tf 2.8855 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(0)-277(=)-277(0\))]TJ /F12 1 Tf 3.0542 0 Td [(;)]TJ /F1 1 Tf -19.331 -2.1807 Td [(2.)]TJ /F12 1 Tf 1.5602 0 Td [(est)]TJ /F1 1 Tf 1.5602 0 Td [(croissan)24(te)]TJ /F12 1 Tf 5.2651 0 Td [(:)]TJ /F6 1 Tf 1.2771 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.052 34.28 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.566 24.105 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(\))]TJ /F6 1 Tf 0.72289 0 Td [(8)]TJ /F8 1 Tf 0.56024 0 Td [(g)]TJ /F6 1 Tf 0.78916 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.198 34.28 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.259 24.105 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F8 1 Tf 1.3855 0 Td [(f)]TJ /F11 1 Tf 0.87952 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F6 1 Tf 0.78916 0 Td [(\))]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.147 36.09 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.41 24.105 Td [(f)-277(dm)]TJ /F11 1 Tf 2.4398 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.241 36.09 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.367 24.105 Td [(g)-204(dm)]TJ /F12 1 Tf 2.0783 0 Td [(;)]TJ /F1 1 Tf -34.301 -2.0422 Td [(3.)]TJ /F12 1 Tf 1.5602 0 Td [(est)]TJ /F1 1 Tf 1.5602 0 Td [(sous-additiv)30(e)]TJ /F12 1 Tf 6.8675 0 Td [(:)]TJ /F6 1 Tf -5.3554 -1.5422 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.845 29.15 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.536 20.521 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F6 1 Tf 0.72289 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(g)]TJ /F6 1 Tf 0.78916 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28 29.15 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.229 20.521 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F9 1 Tf 1.5542 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.466 30.961 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.373 20.521 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.914 30.961 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.849 20.521 Td [(f)-271(dm)]TJ /F2 1 Tf 2.3855 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.853 30.961 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.693 20.521 Td [(g)-204(dm)]TJ /F12 1 Tf 2.4157 0 Td [(;)]TJ /F1 1 Tf -33.283 -2.1024 Td [(3'.)]TJ /F12 1 Tf 2.0422 0 Td [(et)-415(m)30(^)469(eme)]TJ /F1 1 Tf 4.2169 0 Td [(d)24(\266)548(enom)30(brablemen)30(t)-475(sous-additiv)30(e)]TJ /F12 1 Tf 16.428 0 Td [(:)-1608(p)-24(our)-421(toute)-409(famille)]TJ /F2 1 Tf 10.355 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.871 26.142 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.078 18.418 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.905 26.142 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 42.163 18.418 Td [(d')30(\266)469(el)24(\266)469(e-)]TJ -36.831 -1.7892 Td [(men)24(ts)-331(de)]TJ /F6 1 Tf 4.259 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.75 23.581 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.97 16.629 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1988 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(d)24(\266)475(enom)24(brable)]TJ /F9 1 Tf 6.9217 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.284 25.392 Td [(\244)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 24.422 17.581 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.957 23.581 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.024 16.629 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ 1.6145 0.40964 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.319 25.392 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.825 16.629 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.681 23.581 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.819 16.629 Td [(dm)]TJ /F12 1 Tf 1.7289 0 Td [(;)]TJ /F1 1 Tf -29.404 -2.0482 Td [(4.)]TJ /F12 1 Tf 1.5602 0 Td [(a)-331(la)]TJ /F1 1 Tf 1.9458 0 Td [(propri)24(\266)548(et)18(\266)548(e)-385(de)-385(Bepp)-36(o)-385(Levi)]TJ /F12 1 Tf 12.651 0 Td [(:)]TJ -15.169 -1.7349 Td [(p)-30(our)-337(toute)-331(suite)-337(croissan)24(te)-337(d')30(\266)469(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 16.898 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.086 18.168 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.398 12.846 Td [(de)]TJ /F6 1 Tf 1.3374 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.284 18.168 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.114 12.846 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F9 1 Tf 1.3916 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.509 19.978 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.265 12.846 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.871 18.168 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.464 12.846 Td [(dm)]TJ /F6 1 Tf 1.6747 0 Td [(!)]TJ /F9 1 Tf 1.2831 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.552 19.978 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.88 12.846 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.198 18.099 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.813 12.846 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.672 18.168 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.012 12.846 Td [(dm)]TJ /F12 1 Tf 1.4036 0 Td [(.)]TJ -37.084 -2.0241 Td [(En)-331(outre)-343(:)]TJ /F1 1 Tf 0.81325 -1.5904 Td [(5.)]TJ /F12 1 Tf 1.5602 0 Td [(p)-30(our)-337(toute)]TJ /F8 1 Tf 4.9759 0 Td [(f)]TJ /F12 1 Tf 0.93374 0 Td [(dans)]TJ /F6 1 Tf 2.3373 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 23.862 12.995 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.337 9.2316 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(il)-337(existe)-337(une)-331(fonction)]TJ /F8 1 Tf 9.4578 0 Td [(g)]TJ /F12 1 Tf 0.84337 0 Td [(mesurable)-337(telle)-337(que)]TJ /F8 1 Tf -11.542 -1.7952 Td [(f)]TJ /F11 1 Tf 0.87349 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(g)]TJ /F12 1 Tf 1.512 0 Td [(et)]TJ /F9 1 Tf 1.8313 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.914 12.237 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.855 7.4364 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(g)-198(dm)]TJ /F12 1 Tf 2.0783 0 Td [(.)]TJ /F5 1 Tf -26.102 -3.5482 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(52)]TJ ET endstream endobj 191 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 190 0 R /ProcSet 2 0 R >> /Contents 189 0 R >> endobj 190 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 192 0 obj << /Length 19530 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ 0 -2.4036 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -2.0301 Td [(0.)]TJ /F2 1 Tf 1.6145 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4578 0 Td [(f)]TJ /F6 1 Tf 0.92169 0 Td [(2)-325(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.078 85.918 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.994 60.189 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\),)-373(dans)-361(la)-361(d)24(\266)475(e\257nition)-361(de)]TJ /F9 1 Tf 10.373 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.155 87.728 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.127 60.189 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1626 0 Td [(,)-367(la)-361(croissance)-367(de)-361(l'in)24(t)24(\266)475(egrale)-367(assure)]TJ /F9 1 Tf -23.958 -1.0361 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)-168(dm)]TJ /F11 1 Tf 2.4217 0 Td [(>)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.494 0 Td [(et)-337(on)-337(p)-30(eut)-337(prendre)]TJ /F8 1 Tf 8.5422 0 Td [(h)]TJ /F2 1 Tf 0.85542 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)-445(On)-337(en)-331(d)24(\266)475(eduit)]TJ /F9 1 Tf 6.8374 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.06 84.297 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.645 57.792 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4458 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)]TJ /F1 1 Tf -33.163 -2.2048 Td [(1.)]TJ /F2 1 Tf 1.4458 0 Td [(P)30(our)-277(0)]TJ /F8 1 Tf 3.1506 0 Td [(<)-283(\270)-283(<)]TJ /F2 1 Tf 2.9759 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-289(les)-277(ma)-60(joran)30(ts)-277(de)]TJ /F8 1 Tf 7.9096 0 Td [(\270f)]TJ /F2 1 Tf 1.4518 0 Td [(qui)-277(appartiennen)24(t)-283(\265)500(a)]TJ /F6 1 Tf 8.7771 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.853 79.331 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.5 55.587 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-283(son)24(t)-277(exactemen)24(t)]TJ -31.464 -1.1988 Td [(les)]TJ /F8 1 Tf 1.6145 0 Td [(\270h)]TJ /F2 1 Tf 1.6566 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.5241 0 Td [(h)]TJ /F2 1 Tf 1.0783 0 Td [(ma)-54(joran)24(t)]TJ /F8 1 Tf 4.5 0 Td [(f)]TJ /F2 1 Tf 1.0904 0 Td [(dans)]TJ /F6 1 Tf 2.506 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.767 77.616 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.169 54.388 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\).)-939(De)-500(l'homog)30(\266)469(en)30(\266)469(eit)24(\266)475(e)-506(p)-30(ositiv)30(e)-500(de)-500(l'in)24(t)24(\266)469(egrale)-500(on)]TJ -18.133 -1.7892 Td [(conclut)-457(donc)]TJ /F9 1 Tf 6.1325 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.836 76.866 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.922 52.599 Td [(\270f)-277(dm)]TJ /F2 1 Tf 3.2289 0 Td [(=)]TJ /F8 1 Tf 1.253 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.414 76.866 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.614 52.599 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1687 0 Td [(.)-801(Cette)-433(\266)469(egalit)24(\266)475(e)-463(est)-457(encore)-457(v)54(alable)-457(p)-30(our)]TJ /F8 1 Tf 18.102 0 Td [(\270)]TJ /F2 1 Tf 1.0602 0 Td [(=)-481(0)-451(car)]TJ /F9 1 Tf -35.614 -1.0422 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 73.426 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 50.195 Td [(0)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.9096 0 Td [(=)]TJ /F9 1 Tf 1.2952 1.3615 Td [(Z)]TJ /F2 1 Tf 1 -1.3615 Td [(0)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F2 1 Tf 1.9096 0 Td [(=)-518(0.)-867(Comme)-475(p)-30(our)-475(l'in)24(t)24(\266)469(egrale,)-512(le)-481(r)30(\266)469(esultat)-475(p)-30(eut)-451(^)469(etre)-445(\266)469(etendu)-475(au)-475(cas)]TJ /F8 1 Tf -8.9096 -1.6566 Td [(\270)]TJ /F2 1 Tf 0.86145 0 Td [(=)-277(+)]TJ /F6 1 Tf 1.8313 0 Td [(1)]TJ /F2 1 Tf 1.3373 0 Td [(gr)0(^)500(ace)-337(\265)500(a)-331(la)-337(propri)24(\266)469(et)30(\266)469(e)-337(de)-337(Bepp)-30(o)-337(Levi)-331(que)-337(nous)-337(prouv)24(erons)-337(ult)24(\266)469(erieuremen)30(t.)]TJ /F1 1 Tf -3.2169 -2.1446 Td [(2.)]TJ /F2 1 Tf 1.7771 0 Td [(Lorsque)]TJ /F8 1 Tf 3.8795 0 Td [(f)]TJ /F11 1 Tf 1.0602 0 Td [(6)]TJ /F8 1 Tf 1.2349 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(,)-469(p)-30(our)]TJ /F8 1 Tf 3.2229 0 Td [(h)]TJ /F6 1 Tf 1.0301 0 Td [(2)-463(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.328 66.176 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.855 46.394 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-445(ma)-60(joran)30(t)]TJ /F8 1 Tf 5.2771 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(,)]TJ /F8 1 Tf 0.74699 0 Td [(h)]TJ /F2 1 Tf 1.0181 0 Td [(ma)-60(jore)-445(aussi)]TJ /F8 1 Tf 6.0422 0 Td [(f)]TJ /F2 1 Tf 1.0361 0 Td [(d'o)-30(\265)524(u)]TJ /F9 1 Tf 2.3313 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)-174(dm)]TJ /F11 1 Tf 2.6024 0 Td [(>)]TJ /F9 1 Tf -38.386 -1.0422 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 64.547 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 43.991 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5 0 Td [(et)-331(donc,)-337(en)-337(prenan)24(t)-337(la)-331(b)-30(orne)-337(inf)30(\266)469(erieure,)]TJ /F9 1 Tf 17.488 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.75 64.547 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.241 43.991 Td [(g)-204(dm)]TJ /F11 1 Tf 2.3554 0 Td [(>)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.724 64.547 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.114 43.991 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)]TJ /F1 1 Tf -29.452 -2.1446 Td [(3'.)]TJ /F2 1 Tf 1.9398 0 Td [(Comme)]TJ /F8 1 Tf 3.6988 0 Td [(I)]TJ /F2 1 Tf 0.88554 0 Td [(est)-367(d)24(\266)469(enom)24(brable)-367(on)-367(p)-24(eut)-373(trouv)24(er)-367(une)-367(famille)-367(de)-361(r)24(\266)475(eels)-367(strictemen)18(t)-367(p)-30(ositifs)]TJ /F8 1 Tf -7.0181 -1.3434 Td [(")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.2931 57.745 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5301 40.503 Td [(telle)-403(que)]TJ /F8 1 Tf 4.1627 0 Td [(")]TJ /F2 1 Tf 0.85542 0 Td [(=)]TJ /F9 1 Tf 1.1687 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.172 57.745 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.53 40.503 Td [(soit)-403(\257ni)-403(\(par)-403(exemple,)-421(en)-397(se)-403(ramenan)24(t)-403(\265)500(a)]TJ /F8 1 Tf 18.301 0 Td [(I)]TJ /F2 1 Tf 0.90964 0 Td [(=)]TJ /F3 1 Tf 1.1627 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(,)-421(on)-403(p)-24(eut)-409(prendre)]TJ /F8 1 Tf -31.295 -1.5 Td [(")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.2931 55.607 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7229 39.003 Td [(=)-385(2)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12 56.409 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.5542 39.003 Td [(\).)-638(Fixons)-397(alors)]TJ /F8 1 Tf 7.0663 0 Td [(\270)]TJ /F2 1 Tf 0.98193 0 Td [(strictemen)18(t)-397(p)-30(ositif.)-632(P)24(our)-397(c)24(haque)]TJ /F8 1 Tf 14.861 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(,)-415(comme)]TJ /F8 1 Tf 4.1446 0 Td [(\270")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.379 55.607 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.717 39.003 Td [(>)]TJ /F2 1 Tf 1.1627 0 Td [(0,)-415(on)-397(p)-30(eut)]TJ -34.548 -1.7892 Td [(trouv)24(er)]TJ /F8 1 Tf 3.5422 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.517 53.038 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.114 37.214 Td [(2)-337(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.629 53.038 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.982 37.214 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-367(telle)-373(que)]TJ /F8 1 Tf 4.8554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.078 53.038 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 20.289 37.214 Td [(6)]TJ /F8 1 Tf 1.1084 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.448 53.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.675 37.214 Td [(et)]TJ /F9 1 Tf 1.1988 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.422 53.038 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.952 37.214 Td [(dm)]TJ /F11 1 Tf 1.7349 0 Td [(6)]TJ /F9 1 Tf 1.1084 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.647 54.849 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.259 37.214 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44 53.038 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.247 37.214 Td [(dm)]TJ /F2 1 Tf 1.6446 0 Td [(+)]TJ /F8 1 Tf 1.0241 0 Td [(\270")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.034 53.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.663 37.214 Td [(\(par)-373(d)24(\266)469(e\257nition)-367(de)-367(la)]TJ -30.331 -2.3976 Td [(b)-30(orne)-415(inf)24(\266)475(erieure)-421(si)]TJ /F9 1 Tf 8.488 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.207 51.418 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.283 34.816 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.569 49.607 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.271 34.816 Td [(dm)]TJ /F2 1 Tf 1.8193 0 Td [(est)-421(\257ni,)-439(en)-421(prenan)24(t)]TJ /F8 1 Tf 8.9518 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.517 49.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.367 34.816 Td [(=)-415(+)]TJ /F6 1 Tf 1.9759 0 Td [(1)]TJ /F2 1 Tf 1.4157 0 Td [(sinon\).)-698(Alors)]TJ /F8 1 Tf 6.3795 0 Td [(h)]TJ /F2 1 Tf 0.99398 0 Td [(=)]TJ /F9 1 Tf 1.1928 0.94579 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94579 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.845 49.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 34.816 Td [(est)]TJ -37.934 -2.012 Td [(mesurable)-331(p)-30(ositiv)24(e,)-337(ma)-60(jore)]TJ /F9 1 Tf 12.078 0.94578 Td [(X)]TJ /F8 1 Tf 1.6084 -0.94578 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.922 46.728 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.175 32.804 Td [(et)-337(l'on)-337(a)]TJ /F9 1 Tf -13.62 -0.72892 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.81 45.538 Td [(\244)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 8.012 31.659 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.474 43.737 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.614 30.707 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F9 1 Tf 1.0602 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(h)-168(dm)]TJ /F2 1 Tf 2.4217 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ 1.6145 0.41566 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.069 43.737 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.512 30.707 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ 1.6145 0.41566 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.44 45.538 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.319 30.707 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.793 43.737 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.307 30.707 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1.006 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\270")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.069 43.737 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.205 30.707 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ 1.6145 0.41566 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.198 45.538 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.331 30.707 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.552 43.737 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.319 30.707 Td [(dm)]TJ /F2 1 Tf 1.6265 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(\270")]TJ /F2 1 Tf 1.0482 0 Td [(.)]TJ -37.663 -2.1446 Td [(En)-331(faisan)18(t)-331(tendre)]TJ /F8 1 Tf 7.9157 0 Td [(\270)]TJ /F2 1 Tf 0.91566 0 Td [(v)24(ers)-337(0)-331(on)-337(obtien)30(t)-337(la)-337(relation)-331(annonc)24(\266)469(ee.)]TJ -8.8313 -2.0301 Td [(L')24(\266)469(enonc)30(\266)469(e)]TJ /F1 1 Tf 4.1807 0 Td [(3.)]TJ /F2 1 Tf 1.3373 0 Td [(est)-343(un)-337(cas)-331(particulier)-337(de)]TJ /F1 1 Tf 10.855 0 Td [(3')]TJ /F2 1 Tf 0.89759 0 Td [(.)]TJ /F1 1 Tf -16.458 -2.2229 Td [(5.)]TJ /F2 1 Tf 1.6566 0 Td [(P)30(ar)-385(d)24(\266)475(e\257nition)-385(de)]TJ /F9 1 Tf 7.8072 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.767 36.383 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.066 24.31 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5482 0 Td [(on)-379(p)-30(eut)-385(trouv)24(er)-385(une)-385(suite)-385(d')24(\266)475(el)18(\266)475(emen)24(ts)]TJ /F8 1 Tf 16.699 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.793 34.573 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.819 24.31 Td [(de)]TJ /F6 1 Tf 1.3795 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.819 34.573 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.066 24.31 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-385(qui)]TJ -37.03 -2.4036 Td [(ma)-54(joren)18(t)]TJ /F8 1 Tf 4.4277 0 Td [(f)]TJ /F2 1 Tf 1.0783 0 Td [(et)-481(tels)-475(que)]TJ /F9 1 Tf 5.3072 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.353 31.133 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.428 21.906 Td [(dm)]TJ /F2 1 Tf 1.8795 0 Td [(tende)-487(v)24(ers)]TJ /F9 1 Tf 5.0843 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.767 32.943 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.849 21.906 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)-885(P)30(osons)]TJ /F8 1 Tf 4.6386 0 Td [(g)]TJ /F2 1 Tf 1.0361 0 Td [(=)-518(inf)]TJ /F8 1 Tf 2.6807 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.302 31.133 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.488 21.906 Td [(,)-518(c'est)-481(encore)]TJ -33.157 -2.4036 Td [(une)-391(fonction)-391(mesurable)-391(p)-30(ositiv)30(e)-397(qui)-385(ma)-60(jore)]TJ /F8 1 Tf 19.578 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(et)-391(on)-391(a)-361(\266)475(evidemmen)24(t)]TJ /F9 1 Tf 9.2289 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.698 29.504 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.584 19.503 Td [(f)-277(dm)]TJ /F11 1 Tf 2.5361 0 Td [(6)]TJ /F9 1 Tf 1.1446 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F11 1 Tf 2.4518 0 Td [(6)]TJ /F9 1 Tf -38.386 -1.0361 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.8879 24.262 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6205 17.105 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-445(Ceci)-337(a)30(y)24(an)24(t)-337(lieu)-337(p)-30(our)-331(tout)]TJ /F8 1 Tf 12.09 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)-331(en)-337(passan)24(t)-337(\265)500(a)-331(la)-337(limite)-331(il)-337(vien)24(t)]TJ /F9 1 Tf 13.711 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.121 26.073 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.88 17.105 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4458 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F2 1 Tf 2.0783 0 Td [(.)]TJ -38.127 -2.8916 Td [(On)-391(conclut)-403(donc)-391(que)-397(la)-397(b)-30(orne)-391(inf)24(\266)475(erieure)-397(qui)-397(sert)-397(\265)499(a)-391(d)24(\266)469(e\257nir)]TJ /F9 1 Tf 26.018 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.293 21.935 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.813 14.214 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5542 0 Td [(est)-397(attein)24(te,)-415(c'est)-397(un)]TJ -30.036 -1.6386 Td [(minim)30(um.)]TJ /F1 1 Tf 0.81325 -1.6867 Td [(4.)]TJ /F2 1 Tf 1.3675 0 Td [(La)-246(croissance)-240(de)-240(la)-240(suite)-234(des)]TJ /F8 1 Tf 12.145 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.828 15.366 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.928 10.888 Td [(et)-240(la)-234(croissance)-246(de)-240(l'in)24(t)24(\266)475(egrale)-246(sup)0(\266)475(erieure)-246(assuren)24(t)-240(que)-240(la)]TJ -15.596 -1.7349 Td [(suite)-259(des)]TJ /F9 1 Tf 3.9699 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.75 14.693 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.765 9.1533 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.103 12.883 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.964 9.1533 Td [(dm)]TJ /F2 1 Tf 1.6566 0 Td [(est)-265(croissan)24(te)-259(donc)-265(a)-253(une)-265(limite)-259(\(dans)]TJ ET 0.504 w 357.77 119.3 m 366.41 119.3 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 29.934 9.1533 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.871 12.883 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.325 9.1533 Td [(\).)-421(En)-259(outre,)-277(p)-30(our)-265(c)24(haque)]TJ /F8 1 Tf 10.964 0 Td [(n)]TJ /F2 1 Tf 0.59638 0 Td [(,)-277(on)]TJ -37.554 -2.4036 Td [(a)]TJ /F8 1 Tf 0.86747 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.569 9.4435 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 7.5723 6.7497 Td [(6)]TJ /F2 1 Tf 1.1145 0 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.422 9.3745 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.62 6.7497 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.897 9.4435 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.024 6.7497 Td [(d'o)-30(\265)524(u)]TJ /F9 1 Tf 2.259 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.871 11.254 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.747 6.7497 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.233 9.4435 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.946 6.7497 Td [(dm)]TJ /F11 1 Tf 1.7349 0 Td [(6)]TJ /F9 1 Tf 1.1145 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.759 11.254 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.259 6.7497 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.414 9.3745 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.193 6.7497 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.888 9.4435 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.392 6.7497 Td [(dm)]TJ /F2 1 Tf 1.7711 0 Td [(et)-373(donc)-367(lim)]TJ /F9 1 Tf 5.1867 1.1084 Td [(\263)]TJ 0.75904 0.25301 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.388 11.254 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.572 6.7497 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.741 9.4435 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.771 6.7497 Td [(dm)]TJ /F9 1 Tf 1.4036 1.1084 Td [(\264)]TJ /F11 1 Tf 0.93373 -1.1084 Td [(6)]TJ /F9 1 Tf 1.1145 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.129 11.254 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.681 6.7497 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.776 9.3745 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.614 6.7497 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.259 9.4435 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.819 6.7497 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)]TJ -19.97 -2.8614 Td [(53)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 194 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 193 0 R /ProcSet 2 0 R >> /Contents 192 0 R >> endobj 193 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R >> endobj 195 0 obj << /Length 21506 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.8434 Td [(Il)-331(s'agit)-337(donc)-337(de)-331(prouv)24(er)-337(l'in)24(\266)469(egalit)24(\266)469(e)-337(in)30(v)18(erse)]TJ /F9 1 Tf 18.916 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.129 90.004 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.705 61.78 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.776 88.125 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.645 61.78 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.259 88.193 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.843 61.78 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(lim)]TJ /F9 1 Tf 1.5542 1.1084 Td [(\263)]TJ 0.76506 0.25301 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.94 90.004 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.355 61.78 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.302 88.193 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.56 61.78 Td [(dm)]TJ /F9 1 Tf 1.3976 1.1084 Td [(\264)]TJ /F2 1 Tf 0.59638 -1.1084 Td [(.)]TJ -33.223 -2.1627 Td [(En)-445(utilisan)24(t)-451(le)-451(r)30(\266)469(esultat)-451(pr)24(\266)475(ec)24(\266)469(eden)30(t)-451(on)-451(p)-30(eut)-451(in)30(tro)-36(duire)-445(une)-451(suite)-445(de)-451(fonctions)-445(mesurables)]TJ /F8 1 Tf 0 -1.7349 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3103 82.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7349 57.882 Td [(telles)-385(que)]TJ /F8 1 Tf 4.5181 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.81 82.616 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 12.651 57.882 Td [(6)]TJ /F8 1 Tf 1.1325 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.414 82.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.187 57.882 Td [(et)]TJ /F9 1 Tf 1.2169 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.586 82.616 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.596 57.882 Td [(dm)]TJ /F2 1 Tf 1.753 0 Td [(=)]TJ /F9 1 Tf 1.1386 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.181 84.426 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.952 57.882 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.543 82.616 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.151 57.882 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-590(P)24(osons)]TJ /F8 1 Tf 4.253 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.466 82.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.277 57.882 Td [(=)-391(sup)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.388 81.625 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.145 57.882 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.543 82.616 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.114 57.882 Td [(,)-391(on)-385(a)-385(ainsi)-379(une)-385(suite)]TJ -29.783 -1.8735 Td [(croissan)30(te)-228(de)-228(fonctions)-228(mesurables)-228(p)-30(ositiv)30(es)-228(qui)-222(a)-228(donc)-222(une)-228(limite)-228(mesurable)]TJ /F8 1 Tf 32.596 0 Td [(h)]TJ /F2 1 Tf 0.57831 0 Td [(.)-409(La)-228(propri)24(\266)469(et)24(\266)475(e)]TJ -33.175 -1.6566 Td [(de)-313(Bepp)-30(o)-313(Levi)-313(p)-30(our)-313(les)-319(fonctions)-313(mesurables)-313(p)-30(ositiv)24(es)-313(assure)-313(alors)]TJ /F9 1 Tf 29.163 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(h)-168(dm)]TJ /F2 1 Tf 2.4157 0 Td [(=)-283(lim)]TJ /F9 1 Tf 2.6145 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.259 77.564 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.819 54.352 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)]TJ -38.886 -2.4036 Td [(P)30(ar)-319(ailleurs)]TJ /F8 1 Tf 5.3012 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.043 74.125 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 12.03 51.948 Td [(>)]TJ /F8 1 Tf 1.0602 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.414 74.125 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 14.386 51.948 Td [(>)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.802 74.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.795 51.948 Td [(assure)]TJ /F8 1 Tf 3.006 0 Td [(h)]TJ /F11 1 Tf 0.8494 0 Td [(>)]TJ /F2 1 Tf 1.0602 0 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.052 74.056 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.645 51.948 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.534 74.125 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.994 51.948 Td [(et)-325(donc)]TJ /F9 1 Tf 3.5301 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.25 75.935 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.982 51.948 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.897 74.056 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.916 51.948 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.379 74.125 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.12 51.948 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)-168(dm)]TJ /F2 1 Tf 2.1386 0 Td [(.)-439(La)-325(relation)]TJ -33.663 -2.3976 Td [(sera)-307(donc)-313(prouv)54(\266)469(ee)-313(si)-307(l'on)-313(mon)24(tre)-313(p)-30(our)-313(tout)]TJ /F8 1 Tf 18.886 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)]TJ /F9 1 Tf 0.59036 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.621 70.693 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.699 49.551 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.983 72.504 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.892 49.551 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.336 70.693 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.096 49.551 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-433(Raisonnons)-313(p)-30(our)-307(cela)]TJ -29.163 -1.6566 Td [(par)-331(r)24(\266)469(ecurrence.)]TJ 0 -1.7048 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)-277(0)-331(on)-337(a)]TJ /F8 1 Tf 4.1084 0 Td [(h)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.078 65.883 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.06 46.189 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 22.31 65.883 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.373 46.189 Td [(et)-337(la)-331(relation)-337(vien)24(t)-337(de)-337(la)-331(d)24(\266)475(e\257nition)-337(de)]TJ /F8 1 Tf 16.681 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.983 65.883 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.976 46.189 Td [(.)]TJ -28.645 -2.241 Td [(Supp)-24(osons)]TJ /F9 1 Tf 5.006 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.052 62.676 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.627 43.948 Td [(dm)]TJ /F2 1 Tf 1.8072 0 Td [(=)]TJ /F9 1 Tf 1.1807 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.776 64.487 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.078 43.948 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.138 62.676 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.277 43.948 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-680(In)30(tro)-36(duisons)-415(la)-409(fonction)]TJ /F8 1 Tf 11.928 0 Td [(k)]TJ /F2 1 Tf 0.96386 0 Td [(=)-403(inf)-78(\()]TJ /F8 1 Tf 2.7892 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.422 62.676 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.476 43.948 Td [(;)-168(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.517 62.676 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.958 43.948 Td [(\).)-680(C'est)-415(une)]TJ -33.627 -1.6566 Td [(fonction)-385(mesurable)-379(p)-30(ositiv)30(e.)-590(P)30(ar)-379(ailleurs,)-397(de)]TJ /F8 1 Tf 20.012 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.086 60.306 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 26.819 42.292 Td [(>)]TJ /F8 1 Tf 1.1325 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.681 60.306 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 29.325 42.292 Td [(>)]TJ /F8 1 Tf 1.1325 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.284 60.306 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.873 42.292 Td [(et)]TJ /F8 1 Tf 1.2108 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.026 60.306 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(+1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 35.476 42.292 Td [(>)]TJ /F8 1 Tf 1.1325 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.086 60.306 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(+1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 39.012 42.292 Td [(>)]TJ /F8 1 Tf 1.1325 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.147 60.306 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.554 42.292 Td [(on)-385(tire)]TJ /F8 1 Tf -36.223 -1.7771 Td [(k)]TJ /F11 1 Tf 0.8494 0 Td [(>)]TJ /F8 1 Tf 1.0723 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.086 57.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2892 40.515 Td [(.)-481(En\257n)-349(on)-343(a)]TJ /F8 1 Tf 5.6988 0 Td [(k)]TJ /F11 1 Tf 0.8494 0 Td [(6)]TJ /F8 1 Tf 1.0723 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.595 57.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.03 40.515 Td [(.)-481(On)-343(en)-349(d)30(\266)469(eduit)]TJ /F9 1 Tf 6.9036 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.681 59.573 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.398 40.515 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.043 57.762 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.596 40.515 Td [(dm)]TJ /F11 1 Tf 1.6988 0 Td [(6)]TJ /F9 1 Tf 1.0723 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(k)-204(dm)]TJ /F11 1 Tf 2.4157 0 Td [(6)]TJ /F9 1 Tf 1.0783 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.707 57.762 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.151 40.515 Td [(dm)]TJ /F2 1 Tf 1.6928 0 Td [(=)]TJ /F9 1 Tf 1.0783 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.129 59.573 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.38 40.515 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.483 57.762 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.578 40.515 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(,)-349(ce)]TJ -37.651 -1.6566 Td [(qui)-331(prouv)24(e)-337(l')24(\266)475(egalit)24(\266)469(e)-337(de)-337(ces)-331(termes.)]TJ 0 -1.7048 Td [(Comme)-349(l'on)-349(a)]TJ /F8 1 Tf 6.4819 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.733 52.952 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.247 37.153 Td [(=)-301(sup)0(\()]TJ /F8 1 Tf 2.9699 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.466 52.952 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.343 37.153 Td [(;)-162(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.569 52.952 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.819 37.153 Td [(\))-349(,)-349(on)-349(en)-343(tire)]TJ /F8 1 Tf 5.9578 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.147 52.952 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.145 37.153 Td [(+)]TJ /F8 1 Tf 1.006 0 Td [(k)]TJ /F2 1 Tf 0.85542 0 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.733 52.952 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.434 37.153 Td [(+)]TJ /F8 1 Tf 1.006 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.966 52.952 Td [(n)]TJ /F4 1 Tf 0.71552 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.819 37.153 Td [(et)-349(donc)-349(\(puisqu'il)]TJ -31.488 -1.7108 Td [(s'agit)-445(de)-445(fonctions)-445(mesurables\))]TJ /F9 1 Tf 14.223 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.241 50.504 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.855 35.442 Td [(dm)]TJ /F2 1 Tf 1.6928 0 Td [(+)]TJ /F9 1 Tf 1.0783 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(k)-198(dm)]TJ /F2 1 Tf 2.5783 0 Td [(=)]TJ /F9 1 Tf 1.241 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.828 50.504 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.735 35.442 Td [(dm)]TJ /F2 1 Tf 1.6928 0 Td [(+)]TJ /F9 1 Tf 1.0783 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.922 50.504 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.705 35.442 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(.)-777(Comme)]TJ -34.777 -1.6566 Td [(nous)-397(v)24(enons)-397(de)-397(v)24(oir)-397(que)-397(le)-403(dernier)-397(terme)-403(du)-397(premier)-397(mem)24(bre)-397(est)-373(\266)469(egal)-391(au)-397(premier)-403(terme)]TJ 0 -1.7349 Td [(du)-487(second)-487(mem)24(bre)-481(on)-487(en)-481(d)24(\266)475(eduit)-487(bien)]TJ /F9 1 Tf 17.542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.983 45.65 Td [(n)]TJ /F4 1 Tf 0.71552 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.175 32.051 Td [(dm)]TJ /F2 1 Tf 1.9277 0 Td [(=)]TJ /F9 1 Tf 1.3072 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.198 45.65 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.608 32.051 Td [(dm)]TJ /F2 1 Tf 1.9277 0 Td [(=)]TJ /F9 1 Tf 1.3072 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.724 47.461 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.307 32.051 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.086 45.65 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.518 32.051 Td [(dm)]TJ /F2 1 Tf 1.8855 0 Td [(lorsque)]TJ /F9 1 Tf -36.072 -1.0422 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.8879 42.211 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6205 29.647 Td [(dm)]TJ /F2 1 Tf 1.8373 0 Td [(=)]TJ /F9 1 Tf 1.2169 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.707 44.021 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.133 29.647 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.06 42.211 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.331 29.647 Td [(dm)]TJ /F2 1 Tf 1.8313 0 Td [(est)-433(\257ni.)-740(Lorsque)-427(ces)-433(quan)30(tit)24(\266)469(es)-427(son)18(t)-427(in\257nies)-433(l')24(\266)475(egalit)18(\266)475(e)-433(a)-433(encore)-433(lieu)]TJ -9.8313 -2.3193 Td [(parce)-469(que)-475(les)-469(minorations)]TJ /F8 1 Tf 12.03 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.664 38.892 Td [(n)]TJ /F4 1 Tf 0.71552 0 Td [(+1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 19.994 27.328 Td [(>)]TJ /F8 1 Tf 1.2831 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.267 38.892 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.861 27.328 Td [(et)]TJ /F8 1 Tf 1.3012 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.276 38.892 Td [(n)]TJ /F4 1 Tf 0.71552 0 Td [(+1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 26.711 27.328 Td [(>)]TJ /F8 1 Tf 1.2831 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.759 38.892 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.494 27.328 Td [(mon)24(tren)24(t)-469(alors)-475(que)]TJ /F9 1 Tf 9 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.345 38.892 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.795 27.328 Td [(dm)]TJ /F2 1 Tf 1.8675 0 Td [(et)]TJ /F9 1 Tf -38.331 -1.0422 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 37.262 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 24.924 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.414 35.452 Td [(n)]TJ /F4 1 Tf 0.71552 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.006 24.924 Td [(dm)]TJ /F2 1 Tf 1.7289 0 Td [(son)18(t)-337(in\257nis.)]TJ ET 512.64 296.67 m 519.84 296.67 l 519.84 305.1 l 512.64 305.1 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 21.352 Td [(On)-439(p)-30(eut)-439(alors)-445(donner,)-463(p)-30(our)-439(l'in)24(t)24(\266)469(egrale)-445(sup)0(\266)475(erieure,)-463(un)-415(\266)469(enonc)30(\266)469(e)-439(analogue)-445(\265)500(a)-433(la)-439(prop)-30(osition)]TJ /F1 1 Tf 0 -1.1988 Td [(IV.2.2.)]TJ /F2 1 Tf 3.747 0 Td [(p)-30(our)-337(les)-337(in)30(t)24(\266)469(egrales.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 14.158 Td [(IV.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 16.973 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.3614 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.1867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.88554 0 Td [(un)-487(espace)-493(mesurable)-493(et)]TJ /F8 1 Tf 11.09 0 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[(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.129 20.633 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.633 14.569 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)-433(On)-301(supp)-30(ose)-295(en)-301(outre)-301(que)-301(toute)]TJ /F8 1 Tf 14.343 0 Td [(f)]TJ /F12 1 Tf 0.89759 0 Td [(dans)]TJ /F6 1 Tf 2.3012 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 46.621 20.633 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.247 14.569 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.68675 0 Td [(admette)-301(un)-295(ma)-60(joran)24(t)]TJ /F8 1 Tf -29.898 -1.1988 Td [(g)]TJ /F12 1 Tf 0.84337 0 Td [(dans)]TJ /F6 1 Tf 2.3434 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 13.905 18.918 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.38 13.37 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(tel)-331(que)]TJ /F8 1 Tf 3.3072 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ 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0.92771 0 Td [(est)]TJ /F8 1 Tf 1.5602 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -19.572 -2.0542 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-265(prop)-36(osition)]TJ /F1 1 Tf 6.5783 0 Td [(IV.2.2.)]TJ /F2 1 Tf 3.6807 0 Td [(prouv)30(e)-271(d)24(\266)475(ej)-6(\265)500(a)-265(la)-259(premi)18(\265)475(ere)-271(partie)-265(de)-265(l')24(\266)475(enonc)24(\266)469(e)-265(a)24(v)24(ec)]TJ -19.078 -1.6566 Td [(l')30(\266)469(egalit)24(\266)469(e)]TJ /F9 1 Tf 3.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)-168(dm)]TJ /F2 1 Tf 2.4217 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(p)-30(our)]TJ /F8 1 Tf 3.0904 0 Td [(h)]TJ /F6 1 Tf 0.85542 0 Td [(2)-283(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.31 9.4435 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.843 6.7497 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\).)]TJ /F5 1 Tf -17.807 -2.8614 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(54)]TJ ET endstream endobj 197 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 196 0 R /ProcSet 2 0 R >> /Contents 195 0 R >> endobj 196 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F17 97 0 R >> endobj 198 0 obj << /Length 14443 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.7651 Td [(Alors,)-343(p)-30(our)]TJ /F8 1 Tf 5.3072 0 Td [(f)]TJ /F6 1 Tf 0.88554 0 Td [(2)-295(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.888 88.306 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.861 61.858 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\),)-349(si)]TJ /F8 1 Tf 2.0241 0 Td [(h)]TJ /F2 1 Tf 0.91566 0 Td [(est)-343(un)-343(ma)-60(joran)24(t)-343(mesurable)-343(de)]TJ /F8 1 Tf 13.446 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(on)-343(a)]TJ /F9 1 Tf 2.2349 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(h)-168(dm)]TJ /F2 1 Tf 2.4337 0 Td [(=)]TJ /F8 1 Tf 1.0663 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F2 1 Tf 0.57832 0 Td [(\))]TJ /F11 1 Tf 0.68072 0 Td [(>)]TJ /F8 1 Tf 1.0663 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ -38.777 -1.7108 Td [(\(puisque)]TJ /F8 1 Tf 4.0663 0 Td [(\301)]TJ /F2 1 Tf 0.95783 0 Td [(est)-373(croissan)30(te\))-373(et)-367(l'h)24(yp)-30(oth)18(\265)475(ese)-367(assure)-367(que)-367(l'on)-367(a)-367(l')30(\266)469(egalit)24(\266)469(e)-367(lorsque)]TJ /F8 1 Tf 28.096 0 Td [(h)]TJ /F2 1 Tf 0.90964 0 Td [(=)]TJ /F8 1 Tf 1.1024 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(.)-536(On)-367(en)]TJ -35.651 -1.7892 Td [(d)30(\266)469(eduit)-331(bien)]TJ /F9 1 Tf 5.2771 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.612 85.107 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.072 58.358 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)]TJ ET 512.64 696.27 m 519.84 696.27 l 519.84 704.7 l 512.64 704.7 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 44.309 Td [(IV.4.)-236(Propri)20(\266)537(et)25(\266)537(es)-381(et)-376(applications)-381(de)-376(l'in)30(t)20(\266)537(egrale)-376(sup)-10(\266)537(erieure.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 51.388 Td [(La)-301(propri)24(\266)469(et)24(\266)475(e)-307(de)-295(Bepp)-36(o)-301(Levi)-301(p)-24(ermet)-307(d')30(\266)469(ev)54(aluer)-301(l'in)24(t)24(\266)469(egrale)-301(sup)0(\266)469(erieure)-301(de)-301(la)-301(limite)-301(d'une)-301(suite)]TJ 0 -1.2048 Td [(croissan)30(te)-337(de)-325(fonctions)-337(p)-24(ositiv)24(es,)-331(on)-331(p)-30(eut)-331(en)-331(d)24(\266)469(eduire)-331(un)-331(r)30(\266)469(esultat)-337(imp)-30(ortan)24(t)-337(concernan)30(t)-337(les)]TJ 0 -1.1988 Td [(suites)-337(quelconques)-337(de)-331(fonctions)-337(p)-30(ositiv)30(es.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 37.565 Td [(IV.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 45.033 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084335 Td [(\266)]TJ 0 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ -0.0060244 -0.084335 Td [(eme.)]TJ /F12 1 Tf 2.8012 0 Td [(\()]TJ /F1 1 Tf 0.39157 0 Td [(de)-433(F)96(atou)]TJ /F12 1 Tf 4.4458 0 Td [(\))-373(Soit)]TJ /F2 1 Tf 2.8554 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.80121 0 Td [(;)-162(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))]TJ /F12 1 Tf 0.76506 0 Td [(un)-373(espace)-373(mesur)24(\266)475(e.)-572(Alors,)-385(p)-30(our)-373(toute)]TJ -22.958 -1.1988 Td [(suite)-331(de)-337(fonctions)]TJ /F8 1 Tf 7.988 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.759 62.512 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.687 43.834 Td [(p)-30(ositiv)30(es)-337(on)-331(a)-337(:)]TJ /F9 1 Tf 2.253 -0.37349 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.672 61.84 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.398 42.099 Td [(lim)-174(inf)]TJ /F8 1 Tf 2.9398 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.241 60.03 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.542 42.099 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(lim)-174(inf)]TJ /F9 1 Tf 2.9458 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.81 61.84 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.675 42.099 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.164 60.03 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.873 42.099 Td [(dm)]TJ /F12 1 Tf 1.7349 0 Td [(;)]TJ /F1 1 Tf -27.277 -2.5542 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Av)30(ec)-313(les)-313(notations)-307(de)-313(l')30(\266)469(enonc)24(\266)475(e,)-319(p)-30(our)-313(tout)-307(en)24(tier)]TJ /F8 1 Tf 20.916 0 Td [(n)]TJ /F2 1 Tf 0.90361 0 Td [(p)-30(osons)]TJ /F8 1 Tf 3.241 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.793 56.375 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.512 39.545 Td [(=)-433(inf)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.483 55.659 Td [(p)]TJ /F17 1 Tf 0.58621 0 Td [(>)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 39.545 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.612 56.375 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 39.545 Td [(.)]TJ -38.886 -1.6807 Td [(Les)]TJ /F8 1 Tf 1.8976 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.026 53.969 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6807 37.864 Td [(formen)24(t)-433(une)-439(suite)-433(croissan)24(te)-433(de)-433(fonctions)-433(p)-30(ositiv)30(es)-439(don)24(t)-433(la)-433(limite)-433(est)-439(par)-433(d)24(\266)475(e\257ni-)]TJ 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[(Mais,)-379(p)-24(our)-373(tout)]TJ /F8 1 Tf 7.3494 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(,)-379(si)]TJ /F8 1 Tf 1.6988 0 Td [(p)]TJ /F11 1 Tf 0.84337 0 Td [(>)]TJ /F8 1 Tf 1.1145 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(,)-379(on)-373(a)]TJ /F8 1 Tf 2.9518 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.991 48.047 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 21.843 33.726 Td [(6)]TJ /F8 1 Tf 1.1145 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.552 48.047 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.277 33.726 Td [(ce)-373(qui)-367(implique)]TJ /F9 1 Tf 7.1084 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.345 49.857 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.849 33.726 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.69 48.047 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.036 33.726 Td [(dm)]TJ /F11 1 Tf 1.7349 0 Td [(6)]TJ /F9 1 Tf 1.1145 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.224 49.857 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.349 33.726 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.578 48.047 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.47 33.726 Td [(dm)]TJ /F2 1 Tf 1.7651 0 Td [(et)-373(donc)]TJ /F9 1 Tf -35.904 -1.0361 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 46.418 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 31.322 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.397 44.616 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.9759 31.322 Td [(dm)]TJ /F11 1 Tf 1.759 0 Td [(6)]TJ /F2 1 Tf 1.2892 0 Td [(inf)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.552 43.9 Td [(p)]TJ /F17 1 Tf 0.58621 0 Td [(>)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 12.566 32.689 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.414 46.418 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.024 31.322 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.776 44.616 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.145 31.322 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-590(Lorsque)]TJ /F8 1 Tf 4.6867 0 Td [(n)]TJ /F2 1 Tf 0.98193 0 Td [(tend)-385(v)24(ers)-379(+)]TJ /F6 1 Tf 5.2108 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-391(le)-391(second)-379(mem)24(bre)-385(tend,)-391(par)-379(d)24(\266)475(e\257ni-)]TJ -23.09 -2.4217 Td [(tion,)-367(v)18(ers)-361(lim)-174(inf)]TJ /F9 1 Tf 7.4036 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.655 42.952 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.199 28.9 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.017 41.142 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.398 28.9 Td [(dm)]TJ /F2 1 Tf 1.7651 0 Td [(et)-367(nous)-361(a)30(v)24(ons)-373(vu)-367(que)-367(le)-367(premier)-367(tendait)-367(v)24(ers)]TJ /F9 1 Tf 20.054 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.69 42.952 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.675 28.9 Td [(lim)-174(inf)]TJ /F8 1 Tf 2.9398 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.259 41.142 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.819 28.9 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)]TJ -38.886 -1.6566 Td [(L'in)30(\266)469(egalit)24(\266)469(e)-337(annonc)24(\266)469(ee)-337(en)-331(r)24(\266)475(esulte.)]TJ ET 512.64 324.39 m 519.84 324.39 l 519.84 332.82 l 512.64 332.82 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 25.147 Td [(Remarque.)]TJ 0.81325 -1.7349 Td [(1.)]TJ /F2 1 Tf 1.7108 0 Td [(Cet)-385(\266)469(enonc)30(\266)469(e)-415(sera)-409(essen)24(tiellemen)24(t)-409(utilis)18(\266)475(e)-415(p)-30(our)-409(des)-415(fonctions)-409(mesurables)-415(et)-409(sera)-415(donc)]TJ -2.5542 -1.1988 Td [(\266)469(ecrit)-343(p)-30(our)-343(des)-343(in)24(t)24(\266)469(egrales)-343(et)-343(non)-343(des)-349(in)30(t)24(\266)469(egrales)-343(sup)0(\266)469(erieures,)-349(mais)-337(il)-343(est)-349(b)-30(on)-343(de)-343(noter)-349(que)-343(la)]TJ 0.030121 -1.1988 Td [(mesurabilit)30(\266)469(e)-337(n'in)24(tervien)24(t)-337(pas)-337(dans)-337(sa)-337(preuv)30(e.)]TJ /F1 1 Tf 0.81325 -1.7349 Td [(2.)]TJ /F2 1 Tf 1.8133 0 Td [(Dans)-463(le)-463(th)24(\266)469(eor)24(\265)469(eme)-457(de)-457(F)78(atou)-463(on)-463(ne)-457(p)-30(eut)-463(pas)-463(remplacer)-463(l'in)24(\266)475(egalit)18(\266)475(e)-463(par)-463(une)-433(\266)475(egalit)18(\266)475(e,)]TJ -2.6265 -1.1988 Td [(m)24(^)475(eme)-379(dans)-373(des)-379(situations)-379(apparemmen)30(t)-379(tr)24(\265)469(es)-379(r)30(\266)469(eguli)30(\265)469(eres,)-391(comme)-373(le)-385(mon)24(tren)24(t)-379(les)-379(exemples)]TJ 0 -1.2048 Td [(suiv)54(an)24(ts)-331(:)]TJ /F1 1 Tf 0.49398 -1.7289 Td [(\(i\))]TJ /F8 1 Tf 1.8795 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(est)-337(la)-337(mesure)-337(de)-337(comptage)-331(sur)]TJ /F3 1 Tf 13.584 0 Td [(N)]TJ /F2 1 Tf 1.0602 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.086 21.461 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.036 15.147 Td [(=)-277(1)295(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.578 21.418 Td [(f)]TJ /F7 1 Tf 0.5862 0 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(g)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.349 15.147 Td [(;)]TJ /F1 1 Tf -23.843 -1.7349 Td [(\(ii\))]TJ /F8 1 Tf 2.1988 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(est)-337(la)-337(mesure)-337(de)-337(Leb)-30(esgue)-337(sur)]TJ /F3 1 Tf 13.41 0 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.836 18.987 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.861 13.412 Td [(=)-277(\(1)]TJ /F8 1 Tf 1.9458 0 Td [(=)]TJ /F2 1 Tf 0.5 0 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(1)301(I)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.474 18.935 Td [([)]TJ /F10 1 Tf 0.32759 0 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(;)]TJ /F7 1 Tf 0.32759 0 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(])]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.759 13.412 Td [(;)]TJ /F1 1 Tf -28.572 -1.7289 Td [(\(iii\))]TJ /F8 1 Tf 2.5181 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(est)-337(la)-337(mesure)-337(de)-337(Leb)-30(esgue)-337(sur)]TJ /F3 1 Tf 13.41 0 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.836 16.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.861 11.683 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F6 1 Tf 0.8253 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(1)301(I)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.216 16.461 Td [(]0;)-198(1)]TJ /F7 1 Tf 1.9828 0 Td [(=n)]TJ /F4 1 Tf 1.3017 0 Td [([)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.705 11.683 Td [(.)]TJ -26.373 -1.7349 Td [(Dans)-337(les)-337(trois)-343(cas)-337(les)-337(fonctions)]TJ /F8 1 Tf 13.681 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.905 14.03 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.386 9.9485 Td [(tenden)30(t)-337(simplemen)24(t)-343(v)24(ers)-337(la)-337(constan)24(te)-337(0)-337(mais)-337(l'in)24(t)24(\266)475(egrale)]TJ -15.054 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.233 12.306 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.0301 8.7497 Td [(reste)-337(constan)24(te)-307(\266)469(egale)-337(\265)500(a)-337(1.)]TJ -2.6988 -2.4639 Td [(Le)-337(th)24(\266)469(eor)24(\265)475(eme)-337(de)-331(F)78(atou)-337(est)-337(souv)24(en)24(t)-337(utilis)24(\266)475(e)-343(sous)-331(la)-337(forme)-331(plus)-337(faible)-331(suiv)48(an)30(te)-337(:)]TJ 18.916 -2.3976 Td [(55)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 200 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 199 0 R /ProcSet 2 0 R >> /Contents 198 0 R >> endobj 199 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F17 97 0 R >> endobj 201 0 obj << /Length 18149 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(IV.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 62.22 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Cor)24(ollaire.)]TJ /F12 1 Tf 6.6265 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.0542 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-162(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))]TJ /F12 1 Tf 0.74699 0 Td [(un)-361(espace)-355(mesur)24(\266)475(e)-367(et)-355(une)-361(suite)-355(de)-361(fonctions)]TJ /F8 1 Tf 19.193 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 88.823 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 60.43 Td [(v)54(\266)469(eri\257an)24(t)-319(la)-325(ma)-54(joration)]TJ /F9 1 Tf 10.133 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.56 88.073 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.928 60.43 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.319 86.262 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.241 60.43 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F12 1 Tf 0.75301 0 Td [(.)-439(Si)-319(la)-325(suite)-319(admet)-319(une)-325(limite)-319(simple)]TJ /F8 1 Tf 16.193 0 Td [(f)]TJ /F12 1 Tf 0.60241 0 Td [(,)-319(cette)-325(limite)]TJ -33.633 -2.4036 Td [(v)54(\266)469(eri\257e)-307(\266)475(egalemen)24(t)]TJ /F9 1 Tf 7.6446 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20 84.633 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.434 58.027 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F12 1 Tf 0.75301 0 Td [(.)]TJ /F1 1 Tf -13.91 -2.5 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(On)-349(a)]TJ /F6 1 Tf 2.5301 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.58434 0 Td [(=)-301(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.491 79.176 Td [(s)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 21.151 55.527 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.371 79.245 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.464 55.527 Td [(j)]TJ /F2 1 Tf 0.57831 0 Td [(=)-307(lim)-168(inf)]TJ /F6 1 Tf 4.0241 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.828 79.245 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.38 55.527 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-487(Le)-355(th)24(\266)469(eor)24(\265)469(eme)-343(de)-349(F)78(atou)-355(assure)-349(donc)]TJ /F9 1 Tf -23.325 -0.42771 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 78.495 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 53.738 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(lim)-168(inf)]TJ /F9 1 Tf 2.9398 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.155 78.495 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.247 53.738 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.914 76.685 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.554 53.738 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-433(L'h)24(yp)-30(oth)18(\265)475(ese)-295(de)-295(ma)-60(joration)-301(en)30(tra)108(^)385(\263ne)-295(imm)24(\266)469(ediatemen)30(t)-301(que)-295(le)]TJ -13.066 -1.6566 Td [(deuxi)30(\265)469(eme)-331(mem)24(bre)-337(est)-337(ma)-60(jor)24(\266)469(e)-337(par)]TJ /F8 1 Tf 14.934 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(.)]TJ ET 512.64 621.25 m 519.84 621.25 l 519.84 629.67 l 512.64 629.67 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 49.762 Td [(La)-409(notion)-403(d'in)24(t)24(\266)469(egrale)-409(sup)0(\266)469(erieure)-409(p)-30(ermet)-403(\265)499(a)-409(son)-403(tour)-415(d')30(\266)469(etendre)-409(la)-403(mesure)-409(aux)-409(parties)-409(non)]TJ 0 -1.1988 Td [(mesurables)-337(en)-331(d)24(\266)469(e\257nissan)24(t)-337(la)-331(mesure)-343(ext)30(\266)469(erieure.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 37.912 Td [(IV.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 45.448 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83133 0.084335 Td [(\266)]TJ -0.0060234 -0.084335 Td [(efinition.)]TJ /F12 1 Tf 5.247 0 Td [(Sur)-277(un)-271(espace)-271(mesur)30(\266)469(e)]TJ /F2 1 Tf 9.5422 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))]TJ /F12 1 Tf 0.66265 0 Td [(on)-271(d)30(\266)469(e\257nit)-271(la)]TJ /F1 1 Tf 5.4217 0 Td [(mesure)-307(ext)18(\266)548(erieure)]TJ /F8 1 Tf -30.205 -1.7892 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 63.073 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6687 43.659 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(d'une)-337(partie)]TJ /F8 1 Tf 5.6145 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(par)]TJ /F8 1 Tf 2.7831 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.034 63.073 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.349 43.659 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.224 64.073 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.675 43.659 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.681 62.262 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.054 43.659 Td [(dm)]TJ /F12 1 Tf 1.3976 0 Td [(.)]TJ /F2 1 Tf -21.12 -2.7711 Td [(Les)-349(propri)24(\266)469(et)24(\266)475(es)-355(de)-355(l'in)24(t)24(\266)475(egrale)-355(sup)0(\266)469(erieure)-355(fournissen)24(t)-355(imm)18(\266)475(ediatemen)24(t)-355(des)-355(propri)24(\266)475(et)24(\266)469(es)-349(p)-30(our)-355(la)]TJ 0 -1.1988 Td [(mesure)-337(ext)30(\266)469(erieure.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 30.51 Td [(IV.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 36.575 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8795 0 Td [(Sur)-337(un)-343(espace)-337(mesur)30(\266)469(e)]TJ /F2 1 Tf 9.8133 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3253 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(la)-337(mesure)-343(ext)24(\266)475(erieure)-343(a)-337(les)-337(pro-)]TJ -25.922 -1.1988 Td [(pri)24(\266)475(et)24(\266)469(es)-331(suiv)48(an)30(tes)-343(:)]TJ /F1 1 Tf 0.81325 -1.6627 Td [(0.)]TJ /F8 1 Tf 1.5602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.284 48.84 Td [(\244)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3795 33.714 Td [(co)108(\304)391(\263ncide)-337(a)24(v)24(ec)]TJ /F8 1 Tf 6.0301 0 Td [(m)]TJ /F12 1 Tf 1.2108 0 Td [(p)-30(our)-331(les)-337(parties)-337(mesurables)-337(;)]TJ /F1 1 Tf -10.476 -1.6566 Td [(1.)-668(croissance)]TJ /F12 1 Tf 6.9157 0 Td [(:)-1006(p)-30(our)]TJ /F8 1 Tf 3.6446 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(B)]TJ /F12 1 Tf 1.1446 0 Td [(parties)-331(de)]TJ /F8 1 Tf 4.6205 0 Td [(X)-746(A)]TJ /F6 1 Tf 2.6024 0 Td [(\275)]TJ /F8 1 Tf 1.0542 0 Td [(B)]TJ /F6 1 Tf 1.0904 0 Td [(\))]TJ /F8 1 Tf 1.2771 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.259 46.461 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.084 32.057 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.267 46.461 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.283 32.057 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ /F1 1 Tf -32.06 -1.6627 Td [(2.)-668(sous-additivit)18(\266)548(e)]TJ /F12 1 Tf 9.2229 0 Td [(:)-1006(p)-24(our)-337(toute)-337(famille)-337(d)30(\266)469(enom)24(brable)]TJ ET 0.504 w 297.36 361.65 m 364.18 361.65 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 30.801 30.394 Td [(de)-331(parties)]TJ /F8 1 Tf 4.6205 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.767 43.288 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.837 30.394 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(on)-331(a)]TJ /F8 1 Tf -20.12 -1.3494 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.871 42.159 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.633 29.045 Td [(\()]TJ /F9 1 Tf 0.39157 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.983 41.349 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.386 29.045 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.491 42.159 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.06 29.045 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.784 41.349 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.53 29.045 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ /F1 1 Tf -24.108 -1.8012 Td [(3.)-668(propri)24(\266)548(et)18(\266)548(e)-385(de)-385(Borel)]TJ /F12 1 Tf 11.163 0 Td [(:)-1006(p)-24(our)-337(toute)-337(suite)-337(croissan)30(te)-337(de)-331(parties)]TJ /F8 1 Tf 17.873 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.422 38.771 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.813 27.244 Td [(on)-331(a)]TJ /F8 1 Tf -17.759 -1.3434 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.526 37.65 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.392 25.9 Td [(\()]TJ /F9 1 Tf 0.38554 0.94579 Td [([)]TJ /F8 1 Tf 1.2831 -0.94579 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.638 36.849 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.355 25.9 Td [(\))-277(=)-283(sup)]TJ /F8 1 Tf 3.3916 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.534 37.65 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.09 25.9 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.828 36.849 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.771 25.9 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ /F1 1 Tf -24.349 -1.8614 Td [(4.)]TJ /F12 1 Tf 1.5361 0 Td [(p)-30(our)-325(toute)-319(partie)]TJ /F8 1 Tf 7.8373 0 Td [(A)]TJ /F12 1 Tf 0.74699 0 Td [(,)]TJ /F8 1 Tf 0.60241 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.397 34.995 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.205 24.039 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F12 1 Tf 0.74699 0 Td [(\))-325(est)-331(la)-319(b)-30(orne)-325(inf)30(\266)469(erieure)-325(des)-325(mesures)]TJ /F8 1 Tf 16.072 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.71084 0 Td [(lorsque)]TJ /F8 1 Tf 3.4157 0 Td [(U)]TJ /F12 1 Tf 1.1084 0 Td [(par-)]TJ -37.38 -1.1988 Td [(court)-295(l'ensem)30(ble)-289(des)-295(parties)-289(mesurables)-295(qui)-289(con)24(tiennen)30(t)]TJ /F8 1 Tf 24.349 0 Td [(A)]TJ /F12 1 Tf 0.75301 0 Td [(.)-427(En)-289(outre)-295(cette)-295(b)-30(orne)-289(inf)24(\266)475(erieure)]TJ -25.102 -1.2048 Td [(est)-337(attein)24(te,)-337(il)-331(existe)-337(une)-337(partie)-337(mesurable)]TJ /F8 1 Tf 18.753 0 Td [(U)]TJ /F12 1 Tf 1.1265 0 Td [(qui)-331(con)24(tien)24(t)]TJ /F8 1 Tf 5.5301 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(et)-331(telle)-337(que)]TJ /F8 1 Tf 5.1928 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.647 31.556 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.139 21.635 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)]TJ /F1 1 Tf -38.337 -1.988 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Les)-457(propri)18(\266)475(et)24(\266)469(es)-463(ci-dessus)-463(son)24(t)-463(des)-463(cons)24(\266)469(equences)-463(imm)18(\266)475(ediates)-463(des)-463(pro-)]TJ -8.8193 -1.1988 Td [(pri)24(\266)475(et)24(\266)469(es)-403(similaires)-403(de)-403(l'in)24(t)24(\266)475(egrale)-409(sup)0(\266)469(erieure.)-644(Le)-409(seul)-403(p)-30(oin)24(t)-403(non)-373(\266)469(eviden)24(t)-409(est)-403(la)-403(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 38.271 0 Td [(4.)]TJ /F2 1 Tf -38.271 -1.1988 Td [(Compte)-313(ten)24(u)-313(de)-313(la)-313(croissance)-313(de)-313(la)-313(mesure)-319(ext)30(\266)469(erieure)-319(il)-313(su\261t)-313(en)-313(fait)-313(de)-313(mon)24(trer)-319(que,)-319(p)-30(our)]TJ /F8 1 Tf 0 -1.2048 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(partie)-337(de)]TJ /F8 1 Tf 4.2229 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-331(il)-337(existe)]TJ /F8 1 Tf 4.3133 0 Td [(U)]TJ /F2 1 Tf 1.1265 0 Td [(partie)-337(mesurable)-337(con)30(tenan)18(t)]TJ /F8 1 Tf 12.241 0 Td [(A)]TJ /F2 1 Tf 1.0783 0 Td [(telle)-337(que)]TJ /F8 1 Tf 4.0301 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.81 23.556 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.464 16.045 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)]TJ -35.271 -1.6566 Td [(P)30(our)-457(cela)-463(utilisons)-457(la)-457(propri)24(\266)469(et)30(\266)469(e)]TJ /F1 1 Tf 14.398 0 Td [(5.)]TJ /F2 1 Tf 1.7108 0 Td [(des)-457(in)24(t)24(\266)475(egrales)-457(sup)0(\266)469(erieures,)-487(il)-457(existe)-457(donc)]TJ /F8 1 Tf 18.554 0 Td [(g)]TJ /F2 1 Tf 0.96988 0 Td [(fonction)]TJ -35.633 -1.7349 Td [(mesurable)-427(p)-30(ositiv)30(e)-433(telle)-427(que)-427(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.802 17.892 Td [(A)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 19.813 12.653 Td [(6)]TJ /F8 1 Tf 1.2108 0 Td [(g)]TJ /F2 1 Tf 0.93976 0 Td [(et)]TJ /F9 1 Tf 1.259 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F2 1 Tf 2.5121 0 Td [(=)]TJ /F9 1 Tf 1.2108 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.422 19.702 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.404 12.653 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.879 17.892 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.783 12.653 Td [(dm)]TJ /F2 1 Tf 1.8313 0 Td [(=)]TJ /F8 1 Tf 1.2108 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.664 18.702 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.163 12.653 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)-722(Notons)-427(alors)]TJ /F8 1 Tf 7.3976 0 Td [(U)]TJ /F2 1 Tf -38.373 -1.7108 Td [(l'ensem)30(ble)-379(des)]TJ /F8 1 Tf 6.6325 0 Td [(x)]TJ /F2 1 Tf 0.94578 0 Td [(de)]TJ /F8 1 Tf 1.3795 0 Td [(X)]TJ /F2 1 Tf 1.2892 0 Td [(tels)-379(que)]TJ /F8 1 Tf 3.7892 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.74097 0 Td [(>)]TJ /F2 1 Tf 1.1325 0 Td [(1.)-578(Il)-379(s'agit)-379(d'une)-385(partie)-385(mesurable)-379(\(puisque)]TJ /F8 1 Tf 19.657 0 Td [(g)]TJ /F2 1 Tf 0.89157 0 Td [(est)]TJ -37.934 -1.7892 Td [(mesurable\))-397(et)-397(on)-397(a)-391(clairemen)24(t)-397(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.009 12.883 Td [(A)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 20.602 9.1533 Td [(6)]TJ /F2 1 Tf 1.1566 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.948 12.883 Td [(U)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 23.373 9.1533 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(,)-409(par)-397(suite)]TJ /F9 1 Tf 4.988 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.405 14.693 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.488 9.1533 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.862 12.883 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.867 9.1533 Td [(dm)]TJ /F11 1 Tf 1.7771 0 Td [(6)]TJ /F9 1 Tf 1.1566 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.474 12.883 Td [(U)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.205 9.1533 Td [(dm)]TJ /F11 1 Tf 1.7771 0 Td [(6)]TJ /F9 1 Tf 1.1566 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F2 1 Tf 2.0783 0 Td [(.)]TJ -38.886 -2.4036 Td [(L')24(\266)469(egalit)24(\266)475(e)-265(des)-265(termes)-265(extr)30(^)469(emes)-265(en)30(tra)108(^)385(\263ne)-259(alors)]TJ /F8 1 Tf 19.06 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.164 10.245 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.729 6.7497 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.353 11.254 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.054 6.7497 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.81 9.4435 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.434 6.7497 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.129 9.4435 Td [(U)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.566 6.7497 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\).)]TJ ET 514.22 79.448 m 521.42 79.448 l 521.42 87.872 l 514.22 87.872 l f BT /F5 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 3.8882 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(56)]TJ ET endstream endobj 203 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 202 0 R /ProcSet 2 0 R >> /Contents 201 0 R >> endobj 202 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 204 0 obj << /Length 12460 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(IV.5.)-236(F)85(onctions)-266(et)-261(parties)-266(n)25(\266)537(egligeables.)-467(Notion)-261(de)-266(presque)-261(partout.)]TJ 0 -2.5729 Td [(IV.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 59.135 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83133 0.084335 Td [(\266)]TJ -0.0060234 -0.084335 Td [(efinition.)]TJ /F12 1 Tf -5.5241 -1.9819 Td [(Une)-337(fonction)]TJ /F8 1 Tf 5.9458 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(est)-337(dite)]TJ /F8 1 Tf 3.5602 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(-)]TJ /F1 1 Tf 0.33133 0 Td [(n)24(\266)548(egligeable)]TJ /F12 1 Tf 5.8675 0 Td [(lorsque)]TJ /F9 1 Tf 3.4217 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.198 83.383 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.651 57.153 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)-283(0)]TJ /F12 1 Tf 1.5602 0 Td [(.)]TJ -29.873 -2.1024 Td [(Une)-337(partie)-337(de)]TJ /F8 1 Tf 6.3072 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(est)-343(dite)]TJ /F8 1 Tf 3.5663 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(-)]TJ /F1 1 Tf 0.33132 0 Td [(n)24(\266)548(egligeable)]TJ /F12 1 Tf 5.8614 0 Td [(lorsque)-337(sa)-337(mesure)-337(ext)24(\266)475(erieure)-343(est)-337(n)30(ulle.)]TJ -18.187 -1.6386 Td [(Une)-391(propri)24(\266)475(et)24(\266)469(e)]TJ /F8 1 Tf 6.4759 0 Td [(P)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.77711 0 Td [(d)30(\266)469(ep)-30(endan)24(t)-391(d'un)-361(\266)469(el)24(\266)469(emen)30(t)]TJ /F8 1 Tf 11.024 0 Td [(x)]TJ /F12 1 Tf 0.95783 0 Td [(de)]TJ /F8 1 Tf 1.3916 0 Td [(X)]TJ /F12 1 Tf 1.2892 0 Td [(est)-391(dite)-391(vraie)]TJ /F8 1 Tf 6.1988 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(-)]TJ /F1 1 Tf 0.33735 0 Td [(presque)-439(partout)]TJ /F12 1 Tf -31.072 -1.2048 Td [(lorsqu'elle)-337(est)-337(vraie)-337(en)-337(dehors)-337(d'une)-337(partie)]TJ /F8 1 Tf 18.867 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(-n)24(\266)469(egligeable)-331(de)]TJ /F8 1 Tf 6.7831 0 Td [(X)]TJ /F12 1 Tf 0.90361 0 Td [(.)]TJ /F1 1 Tf -27.434 -1.9639 Td [(Remarques.)]TJ 0.81325 -1.6386 Td [(1.)]TJ /F2 1 Tf 1.6867 0 Td [(S'il)-403(n'y)-397(a)-397(pas)-397(de)-397(confusion)-397(p)-30(ossible,)-415(on)-403(sous-en)24(tendra)-397(le)-403(plus)-397(souv)24(en)30(t)-397(la)-403(mesure)]TJ /F8 1 Tf 35.783 0 Td [(m)]TJ /F2 1 Tf -38.283 -1.2048 Td [(dev)54(an)24(t)-337(les)-337(expressions)-337("n)30(\266)469(egligeable")-337(et)-331("presque)-337(partout".)]TJ /F1 1 Tf 0.81325 -1.6386 Td [(2.)]TJ /F2 1 Tf 1.7771 0 Td [(Compte)-445(ten)24(u)-445(de)-445(la)-439(prop)-36(osition)]TJ /F1 1 Tf 14.289 0 Td [(IV.4.4.)]TJ /F2 1 Tf 3.8614 0 Td [(on)-439(v)24(oit)-451(qu'une)-445(partie)-445(n)24(\266)469(egligeable)-439(est)-451(une)]TJ -20.741 -1.2048 Td [(partie)-379(con)30(ten)24(ue)-379(dans)-373(une)-373(partie)-379(mesurable)-379(de)-373(mesure)-379(n)30(ulle.)-572(En)-379(particulier,)-385(si)-379(un)-349(\266)475(enonc)24(\266)475(e)]TJ /F8 1 Tf 0 -1.1988 Td [(P)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-301(est)-307(vrai)-307(presque)-301(partout)-307(on)-301(v)24(oit)-307(qu'il)-301(existe)-307(une)-301(partie)]TJ /F8 1 Tf 24.458 0 Td [(A)]TJ /F2 1 Tf 1.0482 0 Td [(mesurable)-301(et)-307(n)30(\266)469(egligeable)-301(en)]TJ -27.247 -1.1988 Td [(dehors)-337(de)-331(laquelle)-343(l')30(\266)469(enonc)24(\266)475(e)-337(est)-343(vrai.)]TJ /F1 1 Tf 0.81325 -1.6446 Td [(3.)]TJ /F2 1 Tf 1.512 0 Td [(La)-313(croissance)-313(de)-313(la)-307(mesure)-313(ext)24(\266)475(erieure)-313(assure)-313(qu'une)-313(partie)-313(d'un)-313(ensem)24(ble)-313(n)30(\266)469(egligeable)]TJ -2.3253 -1.1988 Td [(est)-337(elle-m)24(^)469(eme)-331(n)24(\266)475(egligeable.)]TJ 0 -2.2892 Td [(Le)-337(lien)-331(en)24(tre)-337(les)-337(notions)-337(pr)30(\266)469(ec)24(\266)469(eden)30(tes)-337(est)-343(donn)24(\266)469(e)-337(par)-337(la)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 28.314 Td [(IV.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 33.942 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf -4.6988 -1.6446 Td [(Une)-337(partie)-337(est)-337(n)24(\266)469(egligeable)-331(si)-337(et)-337(seulemen)30(t)-337(si)-337(sa)-337(fonction)-337(caract)24(\266)469(eristique)-337(est)-337(n)24(\266)469(egligeable.)]TJ 0 -1.6446 Td [(Une)-337(fonction)-337(est)-337(n)24(\266)469(egligeable)-331(si)-337(et)-337(seulemen)30(t)-337(si)-337(elle)-343(est)-337(n)30(ulle)-337(presque)-337(partout.)]TJ /F1 1 Tf 0 -1.9639 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6386 Td [(L')24(\266)469(equiv)54(alence)-343(de)]TJ /F8 1 Tf 7.5181 0 Td [(A)]TJ /F2 1 Tf 1.0904 0 Td [(n)30(\266)469(egligeable)-337(et)-337(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.741 38.495 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.771 27.051 Td [(n)24(\266)475(egligeable)-337(d)24(\266)469(ecoule)-337(de)-337(la)-337(d)24(\266)475(e\257nition)-337(pr)24(\266)469(ec)24(\266)475(eden)24(te)-337(et)-337(de)]TJ -16.44 -1.2048 Td [(celle)-337(de)-331(la)-337(mesure)-337(ext)30(\266)469(erieure)-337(de)]TJ /F8 1 Tf 14.048 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)]TJ -14.795 -1.9819 Td [(Si)]TJ /F8 1 Tf 1.247 0 Td [(f)]TJ /F2 1 Tf 1.012 0 Td [(est)-421(n)24(\266)469(egligeable)-415(on)-415(a)]TJ /F9 1 Tf 9.2349 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.509 35.745 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.283 23.864 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.8133 0 Td [(=)-415(0.)-698(In)30(tro)-36(duisons)-421(alors)-415(la)-421(fonction)]TJ /F8 1 Tf 16.139 0 Td [(g)]TJ /F2 1 Tf 0.92772 0 Td [(=)-415(\(+)]TJ /F6 1 Tf 2.3614 0 Td [(1)-277(\242)-277(j)]TJ /F8 1 Tf 2.1084 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(\).)]TJ -38.494 -1.6566 Td [(C'est)-343(la)-343(fonction)-343(qui)-337(v)48(aut)-343(0)-337(quand)]TJ /F8 1 Tf 15.386 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(s'ann)24(ule)-337(et)-343(+)]TJ /F6 1 Tf 5.8253 0 Td [(1)]TJ /F2 1 Tf 1.3434 0 Td [(sur)-343(l'ensem)30(ble)]TJ /F8 1 Tf 6.5 0 Td [(A)]TJ /F2 1 Tf 1.0904 0 Td [(des)-343(p)-30(oin)30(ts)-343(o)-30(\265)530(u)]TJ /F8 1 Tf 6.1446 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(ne)]TJ -38.163 -1.1988 Td [(s'ann)30(ule)-337(pas.)-451(On)-331(a)-307(\266)469(evidemmen)24(t)-337(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.69 29.849 Td [(A)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 20.976 21.009 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(,)-331(d'o)-30(\265)524(u)]TJ /F8 1 Tf -9.5783 -1.7349 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.819 28.176 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.307 19.274 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26 29.176 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.633 19.274 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.466 27.366 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.012 19.274 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.983 29.176 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.205 19.274 Td [(g)-204(dm)]TJ /F2 1 Tf 2.3554 0 Td [(=)-283(\(+)]TJ /F6 1 Tf 2.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)]TJ /F9 1 Tf 0.5 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.647 29.176 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.355 19.274 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)-277(0.)]TJ -30.024 -2.1566 Td [(P)30(ar)-228(suite)]TJ /F8 1 Tf 4.0663 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.707 25.09 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.735 17.117 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(=)-277(0)-228(et)]TJ /F8 1 Tf 3.5181 0 Td [(A)]TJ /F2 1 Tf 0.98193 0 Td [(est)-234(n)30(\266)469(egligeable.)-409(Ainsi)]TJ /F8 1 Tf 9.4096 0 Td [(f)]TJ /F2 1 Tf 0.83133 0 Td [(est)-234(n)30(ulle)-234(en)-228(dehors)-234(de)-228(la)-234(partie)-234(n)30(\266)469(egligeable)]TJ /F8 1 Tf -21.283 -1.1988 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-337(donc)-337(n)30(ulle)-337(presque)-337(partout.)]TJ -0.74699 -1.6446 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-349(supp)-30(osons)]TJ /F8 1 Tf 12.494 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(n)24(ulle)-337(presque)-343(partout)-343(donc)-343(en)-343(dehors)-343(d'une)-343(partie)]TJ /F8 1 Tf 22.247 0 Td [(A)]TJ /F2 1 Tf 1.0904 0 Td [(n)24(\266)469(egli-)]TJ -36.771 -1.7349 Td [(geable,)-439(d'o)-30(\265)530(u)]TJ /F9 1 Tf 5.741 1.3675 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.276 19.538 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.536 12.539 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.741 17.737 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.916 12.539 Td [(dm)]TJ /F2 1 Tf 1.8133 0 Td [(=)-415(0.)-698(La)-421(fonction)]TJ /F8 1 Tf 8.1566 0 Td [(g)]TJ /F2 1 Tf 0.92771 0 Td [(=)-415(\(+)]TJ /F6 1 Tf 2.3614 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.66265 0 Td [(\242)]TJ /F2 1 Tf 0.56024 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.871 17.737 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.024 12.539 Td [(est)-421(alors)-421(n)24(ulle)-415(en)-415(dehors)-421(de)]TJ /F8 1 Tf 12.723 0 Td [(A)]TJ /F2 1 Tf -38.416 -1.6506 Td [(et)-361(v)54(aut)-367(+)]TJ /F6 1 Tf 4.247 0 Td [(1)]TJ /F2 1 Tf 1.3614 0 Td [(ailleurs.)-524(Ainsi,)-367(lorsque)]TJ /F8 1 Tf 10.271 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(n'est)-361(pas)-367(n)30(ulle,)-373(c'est-)0(\265)500(a-dire)-361(sur)-367(A,)-361(la)-361(fonction)]TJ /F8 1 Tf 20.223 0 Td [(g)]TJ /F2 1 Tf 0.87349 0 Td [(est)]TJ -37.934 -1.7349 Td [(in\257nie.)-656(On)-403(a)-409(donc)]TJ /F6 1 Tf 8.7048 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.6747 0 Td [(6)]TJ /F8 1 Tf 1.1747 0 Td [(g)]TJ /F2 1 Tf 0.91566 0 Td [(et)]TJ /F9 1 Tf 1.241 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.5 14.693 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.373 9.1533 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F11 1 Tf 1.7952 0 Td [(6)]TJ /F9 1 Tf 1.1687 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.733 14.693 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.127 9.1533 Td [(g)-204(dm)]TJ /F2 1 Tf 2.4759 0 Td [(=)-397(\(+)]TJ /F6 1 Tf 2.3373 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.66265 0 Td [(\242)]TJ /F9 1 Tf 0.54819 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.871 14.693 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.608 9.1533 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.328 12.883 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.988 9.1533 Td [(dm)]TJ /F2 1 Tf 1.7952 0 Td [(=)-397(0)-403(;)-439(par)-409(suite)]TJ /F9 1 Tf -32.452 -1.0422 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 11.254 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 6.7497 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)-277(0)-331(et)]TJ /F8 1 Tf 3.0542 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(n)24(\266)475(egligeable.)]TJ ET 512.64 79.448 m 519.84 79.448 l 519.84 87.872 l 512.64 87.872 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.247 3.8882 Td [(57)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 206 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 205 0 R /ProcSet 2 0 R >> /Contents 204 0 R >> endobj 205 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 207 0 obj << /Length 14657 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(V)84(oici)-337(quelques)-331(r)24(\266)475(esultats)-337(imp)-36(ortan)24(ts)-337(concernan)30(t)-337(cette)-337(notion.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 49.344 Td [(IV.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 59.153 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Soien)24(t)-337(deux)-331(fonctions)-337(p)-30(ositiv)30(es)]TJ /F8 1 Tf 13.801 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(g)]TJ /F12 1 Tf 0.51205 0 Td [(.)]TJ -27.976 -2.1687 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(presque)-337(partout)-337(alors)]TJ /F9 1 Tf 9.6627 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.397 83.142 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.096 56.985 Td [(f)-277(dm)]TJ /F11 1 Tf 2.4458 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.491 83.142 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.054 56.985 Td [(g)-204(dm)]TJ /F12 1 Tf 2.0783 0 Td [(.)]TJ -24.801 -2.8313 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(g)]TJ /F12 1 Tf 0.84337 0 Td [(son)18(t)-307(\266)469(egales)-331(presque)-337(partout)-337(alors)]TJ /F9 1 Tf 14.705 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.983 79.09 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.608 54.153 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.078 79.09 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.566 54.153 Td [(g)-204(dm)]TJ /F12 1 Tf 2.0783 0 Td [(.)]TJ /F1 1 Tf -27.313 -2.4578 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Soit)]TJ /F8 1 Tf 2.1988 0 Td [(h)]TJ /F2 1 Tf 1.0482 0 Td [(la)-475(fonction)-481(qui)-475(v)48(aut)-481(+)]TJ /F6 1 Tf 10.271 0 Td [(1)]TJ /F2 1 Tf 1.4759 0 Td [(si)]TJ /F8 1 Tf 1.1446 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.90361 0 Td [(>)-518(g)]TJ /F2 1 Tf 1.8072 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-475(et)-475(0)-475(sinon.)-873(Cette)]TJ -30.193 -1.2048 Td [(fonction)-409(est)-409(n)24(ulle)-403(presque)-409(partout)-409(donc)-409(n)30(\266)469(egligeable)-403(et)-409(l'on)-409(a)]TJ /F8 1 Tf 27.271 0 Td [(f)]TJ /F11 1 Tf 0.99398 0 Td [(6)]TJ /F8 1 Tf 1.1747 0 Td [(g)]TJ /F2 1 Tf 0.78313 0 Td [(+)]TJ /F8 1 Tf 1.0482 0 Td [(h)]TJ /F2 1 Tf 0.57831 0 Td [(.)-656(P)24(ar)-403(suite)-409(on)-403(a)]TJ /F9 1 Tf -31.849 -0.36747 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 71.366 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 48.756 Td [(f)-277(dm)]TJ /F11 1 Tf 2.4398 0 Td [(6)]TJ /F9 1 Tf 1.0602 1.3675 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.155 71.366 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.584 48.756 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.73494 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(h)]TJ /F2 1 Tf 0.57831 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3675 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.578 71.366 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.03 48.756 Td [(g)-204(dm)]TJ /F2 1 Tf 2.3012 0 Td [(+)]TJ /F9 1 Tf 1 1.3675 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.397 71.366 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.795 48.756 Td [(h)-168(dm)]TJ /F2 1 Tf 2.4217 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3675 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.457 71.366 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.729 48.756 Td [(g)-204(dm)]TJ /F2 1 Tf 2.0783 0 Td [(.)]TJ -25.476 -2.0843 Td [(Lorsque)]TJ /F8 1 Tf 3.9337 0 Td [(f)]TJ /F2 1 Tf 1.0964 0 Td [(est)-469(\266)469(egale)-499(\265)500(a)]TJ /F8 1 Tf 5.3855 0 Td [(g)]TJ /F2 1 Tf 1.006 0 Td [(presque)-500(partout)-499(le)-500(r)24(\266)469(esultat)-500(pr)24(\266)475(ec)24(\266)469(eden)30(t)-500(s'applique)-500(p)-30(our)-493(donner)]TJ /F9 1 Tf -11.422 -0.37349 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 65.9 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7892 44.936 Td [(f)-277(dm)]TJ /F11 1 Tf 2.5181 0 Td [(6)]TJ /F9 1 Tf 1.1325 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.379 65.9 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.904 44.936 Td [(g)-204(dm)]TJ /F2 1 Tf 2.4578 0 Td [(ainsi)-379(que,)-397(en)-355(\266)475(ec)24(hangean)24(t)-385(les)-385(r)0(^)500(oles)-379(de)]TJ /F8 1 Tf 16.452 0 Td [(f)]TJ /F2 1 Tf 0.9759 0 Td [(et)-379(de)]TJ /F8 1 Tf 2.5904 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(,)]TJ /F9 1 Tf 0.66867 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.328 65.9 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.024 44.936 Td [(g)-204(dm)]TJ /F11 1 Tf 2.4337 0 Td [(6)]TJ /F9 1 Tf 1.1325 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.526 65.9 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.054 44.936 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)]TJ -38.886 -1.6566 Td [(L')24(\266)469(egalit)24(\266)475(e)-343(en)-331(d)24(\266)475(ecoule.)]TJ ET 512.64 516.06 m 519.84 516.06 l 519.84 524.48 l 512.64 524.48 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 33.54 Td [(IV.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 40.207 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F1 1 Tf -3.8855 -1.6265 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(La)-337(r)24(\266)475(eunion)-337(d'une)-343(famille)-331(d)24(\266)469(enom)30(brable)]TJ ET 0.504 w 224.14 459.43 m 290.95 459.43 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.675 38.581 Td [(de)-337(parties)-337(n)30(\266)469(egligeables)-337(est)-337(n)24(\266)469(egligeable.)]TJ /F1 1 Tf -18.53 -1.6325 Td [(2.)]TJ /F12 1 Tf 1.5843 0 Td [(Si)-343(elle)-349(est)-355(d)30(\266)469(e\257nie,)-349(la)-343(somme)-349(d'une)-349(famille)-343(d)24(\266)475(enom)24(brable)]TJ ET 313.13 439.92 m 379.94 439.92 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.133 36.948 Td [(de)-343(fonctions)-349(n)24(\266)475(egligeables)-349(est)]TJ -26.801 -1.1988 Td [(n)30(\266)469(egligeable.)]TJ /F1 1 Tf 0 -1.9518 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.6265 Td [(1.)]TJ /F2 1 Tf 1.5602 0 Td [(P)30(our)-337(une)-337(famille)-337(d)30(\266)469(enom)24(brable)-337(de)-331(parties)]TJ /F8 1 Tf 18.12 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.026 45.823 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.241 32.171 Td [(n)24(\266)475(egligeables)-343(on)-331(a)]TJ /F8 1 Tf -17.639 -1.7771 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.991 44.081 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.94 30.394 Td [(\()]TJ /F9 1 Tf 0.38554 0.95181 Td [([)]TJ /F8 1 Tf 1.2831 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.112 43.28 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.687 30.394 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.621 44.081 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.361 30.394 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.905 43.28 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.837 30.394 Td [(\))-277(=)-283(0)-331(donc)]TJ /F9 1 Tf 4.9458 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.371 43.28 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.476 30.394 Td [(est)-337(n)24(\266)469(egligeable.)]TJ /F1 1 Tf -21.331 -1.7771 Td [(2.)]TJ /F2 1 Tf 1.5602 0 Td [(P)30(our)-337(une)-337(famille)-337(d)30(\266)469(enom)24(brable)-337(de)-331(fonctions)]TJ /F8 1 Tf 19.09 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.043 40.737 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.952 28.617 Td [(n)24(\266)475(egligeables)-337(on)-337(a)]TJ /F9 1 Tf -21.62 -0.80121 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.491 39.452 Td [(\244)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7892 27.605 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ 0.5 1 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94579 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.871 37.642 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 10.723 27.605 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ /F8 1 Tf 0.50602 0.054214 Td [(dm)]TJ /F11 1 Tf 1.8735 0 Td [(6)]TJ /F9 1 Tf 1.253 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.974 39.452 Td [(\244)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 15.819 27.4 Td [(X)]TJ /F6 1 Tf 1.6084 -0.94579 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.043 37.642 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.53 26.454 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F11 1 Tf 1.8735 0 Td [(6)]TJ /F9 1 Tf 1.259 0.94579 Td [(X)]TJ 1.6084 0.41566 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.371 39.452 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.181 26.454 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.129 37.642 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.283 26.454 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F2 1 Tf 1.8795 0 Td [(=)-475(0)-1457(donc)]TJ /F9 1 Tf 6.2651 0.94579 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94579 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.905 37.642 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.753 26.454 Td [(est)-463(n)30(\266)469(egligeable.)]TJ ET 512.64 296.31 m 519.84 296.31 l 519.84 304.74 l 512.64 304.74 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 17.836 Td [(IV.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 21.382 Td [(5.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.1807 0 Td [(Si)-433(une)-439(fonction)]TJ /F8 1 Tf 7.2169 0 Td [(f)]TJ /F12 1 Tf 1.0301 0 Td [(v)48(\266)475(eri\257e)]TJ /F9 1 Tf 3.0482 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.224 32.193 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.97 21.382 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)-445(<)]TJ /F2 1 Tf 3.0663 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F12 1 Tf 1.4337 0 Td [(alors)]TJ /F8 1 Tf 2.5 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.8253 0 Td [(est)-439(\257ni)]TJ -36.114 -1.5843 Td [(presque)-331(partout.)]TJ /F1 1 Tf 0 -1.9458 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Soit)]TJ /F8 1 Tf 2.1205 0 Td [(A)]TJ /F2 1 Tf 1.1446 0 Td [(l'ensem)24(ble)-397(des)-397(p)-30(oin)24(ts)-397(o)-24(\265)524(u)]TJ /F8 1 Tf 11.193 0 Td [(f)]TJ /F2 1 Tf 0.99398 0 Td [(est)-403(in\257ni,)-415(il)-397(s'agit)-397(de)-397(prouv)24(er)-397(que)]TJ /F8 1 Tf -24.271 -1.7349 Td [(A)]TJ /F2 1 Tf 1.1205 0 Td [(est)-373(n)24(\266)469(egligeable.)-554(On)-373(a)-373(alors)]TJ /F6 1 Tf 12.223 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.62048 0 Td [(>)]TJ /F2 1 Tf 1.1205 0 Td [(\(+)]TJ /F6 1 Tf 1.1627 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F6 1 Tf 0.63856 0 Td [(\242)]TJ /F2 1 Tf 0.52409 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.026 22.849 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.831 16.117 Td [(.)-554(P)30(ar)-373(suite)]TJ /F9 1 Tf 5.1807 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.81 24.659 Td [(\244)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.47 16.117 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F11 1 Tf 1.741 0 Td [(>)]TJ /F2 1 Tf 1.1205 0 Td [(\(+)]TJ /F6 1 Tf 1.1626 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(\))]TJ /F6 1 Tf 0.63253 0 Td [(\242)]TJ /F9 1 Tf 0.52409 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.638 24.659 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.44 16.117 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.103 22.849 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.819 16.117 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)]TJ -38.886 -2.4036 Td [(Comme)-325(le)-331(premier)-325(mem)24(bre)-325(est)-325(\257ni,)]TJ /F9 1 Tf 15.711 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.543 21.219 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.5 13.714 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.009 19.409 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.886 13.714 Td [(dm)]TJ /F2 1 Tf 1.7169 0 Td [(est)-331(n)24(\266)475(ecessairemen)18(t)-325(n)24(ul)-325(et)]TJ /F8 1 Tf 11.211 0 Td [(A)]TJ /F2 1 Tf 1.0723 0 Td [(est)-331(n)30(\266)469(egligeable.)]TJ ET 512.64 144.03 m 519.84 144.03 l 519.84 152.46 l 512.64 152.46 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 9.8882 Td [(En\257n)-361(il)-355(est)-361(ais)24(\266)469(e)-361(de)-355(v)24(oir)-355(que,)-367(la)-355(r)24(\266)475(eunion)-361(de)-361(deux)-355(ensem)24(bles)-361(n)30(\266)469(egligeables)-331(\266)469(etan)24(t)-361(n)30(\266)469(egligeable,)]TJ 0 -1.1988 Td [(l')30(\266)469(egalit)24(\266)469(e)-367(presque)-367(partout)-367(est)-367(une)-367(relation)-361(d')24(\266)475(equiv)48(alence.)-530(En)-367(outre,)-373(si)-367(dans)-367(une)-361(somme)-367(ou)]TJ 0 -1.1988 Td [(un)-307(pro)-30(duit)-307(on)-307(remplace)-307(les)-307(termes)-307(par)-313(des)-307(termes)-277(\266)469(egaux)-307(presque)-313(partout,)-313(alors)-313(la)-307(nouv)30(elle)]TJ 0 -1.2048 Td [(somme)-445(\(si)-451(elle)-451(est)-451(encore)-451(d)30(\266)469(e\257nie\))-451(ou)-451(le)-451(nouv)30(eau)-451(pro)-30(duit)-445(co)108(\304)385(\263ncide)-445(presque)-451(partout)-451(a)24(v)24(ec)]TJ /F5 1 Tf 0 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(58)]TJ ET endstream endobj 209 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 208 0 R /ProcSet 2 0 R >> /Contents 207 0 R >> endobj 208 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 210 0 obj << /Length 13296 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(la)-367(somme)-367(ou)-373(le)-373(pro)-30(duit)-367(de)-367(d)24(\266)469(epart.)-554(Autremen)24(t)-373(dit)-373(l'addition)-367(et)-367(la)-373(m)30(ultiplication)-373(p)-30(euv)24(en)24(t)]TJ 0 -1.1988 Td [(s')24(\266)475(etendre)-337(aux)-343(classes)-331(de)-337(fonctions)-337(mo)-30(dulo)-337(l')30(\266)469(egalit)24(\266)469(e)-337(presque)-337(partout.)]TJ 0 -1.5783 Td [(Ces)-253(remarques)-259(nous)-253(p)-30(ermettron)24(t)-259(souv)24(en)30(t)-259(de)-259(mo)-30(di\257er)-253(les)-259(fonctions)-259(donn)24(\266)469(ees)-253(sur)-259(un)-259(ensem)24(ble)]TJ 0 -1.1988 Td [(n)30(\266)469(egligeable)-349(a\257n)-343(de)-349(remplacer)-343(une)-349(propri)24(\266)469(et)24(\266)475(e)-349(vraie)-349(presque)-349(partout)-349(par)-349(une)-343(propri)24(\266)469(et)24(\266)475(e)-349(vraie)]TJ 0 -1.1988 Td [(partout.)-451(On)-331(p)-30(ourra)-337(m)24(^)475(eme)-337(faire)-337(une)-337(quan)30(tit)24(\266)469(e)-337(d)30(\266)469(enom)24(brable)-337(de)-331(telles)-343(mo)-30(di\257cations.)]TJ 0 -1.5783 Td [(En)-457(particulier,)-487(p)-30(our)-463(mo)-30(di\257er)-457(une)-457(fonction)]TJ /F8 1 Tf 19.5 0 Td [(f)]TJ /F2 1 Tf 1.0542 0 Td [(sur)-463(une)-457(partie)-463(n)24(\266)475(egligeable)]TJ /F8 1 Tf 12.072 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-487(on)-463(utilisera)]TJ -33.373 -1.1988 Td 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Tf -11.072 -1.5783 Td [(\(i\))]TJ /F12 1 Tf 1.8795 0 Td [(toutes)-337(les)-337(parties)-337(n)30(\266)469(egligeables)-337(son)24(t)-337(mesurables)-337(;)]TJ /F1 1 Tf -2.1988 -1.5783 Td [(\(ii\))]TJ /F12 1 Tf 2.1988 0 Td [(toutes)-337(les)-337(fonctions)-337(n)30(\266)469(egligeables)-337(son)24(t)-337(mesurables.)]TJ -2.3735 -1.5723 Td [(Lorsque)-331(ces)-307(\266)469(enonc)30(\266)469(es)-337(son)24(t)-337(r)30(\266)469(ealis)24(\266)475(es)-337(on)-337(dit)-337(que)-337(l'espace)-337(mesur)24(\266)475(e)-337(est)]TJ /F1 1 Tf 28.114 0 Td [(complet)]TJ /F12 1 Tf 3.9759 0 Td [(.)]TJ /F1 1 Tf -32.09 -1.8554 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Supp)-24(osons)]TJ /F1 1 Tf 5.0602 0 Td [(\(ii\))]TJ /F2 1 Tf 1.5301 0 Td [(.)-837(Si)]TJ /F8 1 Tf 2.4096 0 Td [(A)]TJ /F2 1 Tf 1.2169 0 Td [(est)-463(n)24(\266)469(egligeable)-463(alors)-463(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.216 52.314 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.506 36.707 Td 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11.952 0 0 11.952 0 0 Tm 22.855 32.732 Td [(est)-355(n)30(\266)469(egligeable)-349(ainsi)-343(donc)-349(que)-349(toute)-343(partie)-349(de)]TJ /F8 1 Tf 20.205 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.69 47.426 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 32.732 Td [(.)]TJ -38.886 -1.1988 Td [(Ces)-337(parties)-337(son)18(t)-337(donc)-343(mesurables.)-463(P)30(ar)-337(passage)-337(au)-343(compl)24(\266)475(emen)24(taire)]TJ /F8 1 Tf 29.373 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(est)-343(aussi)-343(mesurable.)]TJ -30.458 -1.2048 Td [(Main)24(tenan)24(t,)-439(p)-30(our)-415(tout)-415(r)24(\266)475(eel)]TJ /F8 1 Tf 12.41 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)-433(l'ensem)30(ble)]TJ /F6 1 Tf 5.6024 0 Td [(f)]TJ /F8 1 Tf 0.50603 0 Td [(x)]TJ /F6 1 Tf 0.98193 0 Td [(2)]TJ /F8 1 Tf 1.0783 0 Td [(X)]TJ /F6 1 Tf 1.3193 0 Td [(j)]TJ /F8 1 Tf 0.69277 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.8012 0 Td [(>)-409(a)]TJ /F6 1 Tf 1.7169 0 Td [(g)]TJ /F2 1 Tf 0.91566 0 Td [(con)24(tien)24(t)]TJ /F8 1 Tf 3.9157 0 Td [(A)]TJ /F2 1 Tf 1.1627 0 Td [(\(si)]TJ /F8 1 Tf 1.4759 0 Td [(a)-421(<)]TJ /F2 1 Tf 2.1325 0 Td [(0\))-421(ou)]TJ -36.801 -1.1988 Td [(ne)-457(le)-463(rencon)24(tre)-457(pas)-463(\(si)]TJ /F8 1 Tf 10.614 0 Td [(a)]TJ /F11 1 Tf 1.0181 0 Td [(>)]TJ /F2 1 Tf 1.2651 0 Td [(0\))-457(;)-524(de)-457(toute)-457(fa)-6(\273)439(con)-457(c'est)-463(la)-463(r)30(\266)469(eunion)-463(d'une)-457(partie)-463(de)]TJ /F8 1 Tf 23.813 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.241 42.271 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 29.129 Td [(et)]TJ -38.361 -1.1988 Td [(\266)469(ev)24(en)30(tuellemen)24(t)-313(de)]TJ /F8 1 Tf 8.1566 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(.)-439(Dans)-313(tous)-319(les)-313(cas)-313(il)-313(s'agit)-319(d'une)-319(partie)-313(mesurable)-319(et)]TJ /F8 1 Tf 23.434 0 Td [(f)]TJ /F2 1 Tf 0.90963 0 Td [(est)-319(mesurable)]TJ -33.223 -1.2048 Td [(ce)-337(qui)-331(prouv)24(e)]TJ /F1 1 Tf 6.1988 0 Td [(\(ii\))]TJ /F2 1 Tf 1.5361 0 Td [(.)]TJ ET 512.64 318.27 m 519.84 318.27 l 519.84 326.7 l 512.64 326.7 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 23.774 Td [(Remarque.)]TJ /F2 1 Tf 6.7651 0 Td [(Sur)-469(un)-463(espace)-463(mesur)30(\266)469(e,)-500(soit)]TJ /F8 1 Tf 12.651 0 Td [(f)]TJ /F2 1 Tf 1.0602 0 Td [(une)-463(fonction)-463(mesurable)-463(et)]TJ /F8 1 Tf 12.163 0 Td [(g)]TJ /F2 1 Tf 0.97591 0 Td [(une)-463(fonction)]TJ -33.645 -1.2048 Td [(\266)469(egale)-403(presque)-397(partout)-403(\265)500(a)]TJ /F8 1 Tf 10.886 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)-415(notons)]TJ /F8 1 Tf 3.9819 0 Td [(A)]TJ /F2 1 Tf 1.1506 0 Td [(l'ensem)30(ble)-397(des)-403(p)-30(oin)24(ts)-397(o)-24(\265)524(u)]TJ /F8 1 Tf 11.193 0 Td [(f)]TJ /F2 1 Tf 0.99398 0 Td [(et)]TJ /F8 1 Tf 1.2349 0 Td [(g)]TJ /F2 1 Tf 0.90964 0 Td [(di\256)30(\265)469(eren)24(t)-403(et)]TJ /F8 1 Tf 5.241 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.448 32.892 Td 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[(est)-349(mesurable)-349(ainsi)-343(que)]TJ /F8 1 Tf 10.542 0 Td [(g)]TJ /F6 1 Tf 0.74096 0 Td [(\242)]TJ /F2 1 Tf 0.50602 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.983 26.935 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 43.096 38.042 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.319 18.973 Td [(qui)-343(lui)-349(est)-325(\266)475(egale.)-487(P)30(ar)-343(ailleurs)]TJ /F8 1 Tf 13.012 0 Td [(g)]TJ /F6 1 Tf 0.74699 0 Td [(\242)]TJ /F2 1 Tf 0.50602 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.155 26.935 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.145 18.973 Td [(est)-349(n)30(\266)469(egligeable)]TJ -32.813 -1.2048 Td [(donc)-331(mesurable.)-451(Il)-331(s'ensuit)-337(que)]TJ /F8 1 Tf 13.97 0 Td [(g)]TJ /F2 1 Tf 0.84337 0 Td [(est)-343(mesurable.)]TJ -14.813 -2.1265 Td [(Ainsi,)-475(sur)-451(un)-451(espace)-445(mesur)30(\266)469(e)-451(complet,)-475(la)-445(mesurabilit)24(\266)475(e)-451(d'une)-451(fonction)-451(n'est)-445(pas)-451(mo)-30(di\257)24(\266)469(ee)]TJ 0 -1.1988 Td [(apr)30(\265)469(es)-391(un)-391(c)24(hangemen)24(t)-391(de)-391(la)-385(fonction)-391(sur)-397(un)-391(ensem)24(ble)-385(n)24(\266)469(egligeable.)-608(Cela)-391(explique)-391(l'in)24(t)24(\266)469(er)24(^)469(et)]TJ 0 -1.2048 Td [(que)-337(l'on)-337(p)-30(eut)-337(a)30(v)24(oir)-337(\265)500(a)-331(se)-337(ramener)-337(\265)500(a)-331(des)-337(espaces)-337(mesur)24(\266)475(es)-337(complet.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 8.565 Td [(IV.6.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 10.268 Td [(2.)-289(Compl)18(\266)548(etion)-385(d'un)-385(espace)-385(mesur)18(\266)548(e.)]TJ /F2 1 Tf -2.9458 -1.5783 Td [(T)78(out)-325(espace)-319(mesur)24(\266)475(e)-325(p)-30(eut)-295(^)469(etre)-319(compl)24(\266)469(et)30(\266)469(e)-325(en)-319(prolongean)24(t)-325(la)-319(mesure)-325(\265)499(a)-319(la)-319(tribu)-325(engendr)24(\266)475(ee)-325(par)]TJ 0 -1.1988 Td [(l'ensem)30(ble)-385(des)-379(parties)-385(mesurables)-385(et)-379(des)-385(parties)-379(n)24(\266)469(egligeables.)-590(On)-379(a)-379(\265)500(a)-379(ce)-385(sujet)-385(le)-385(r)24(\266)475(esultat)]TJ 0 -1.2048 Td [(suiv)54(an)24(t.)]TJ 18.916 -2.3976 Td [(59)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 212 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 211 0 R /ProcSet 2 0 R >> /Contents 210 0 R >> endobj 211 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 213 0 obj << /Length 21899 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F15 1 Tf 0 -2.4036 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.006 0 Td [(Soit)]TJ /F2 1 Tf 2.1024 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))]TJ /F12 1 Tf 0.77108 0 Td [(un)-379(espace)-379(mesur)30(\266)469(e.)-584(Notons)]TJ ET 0.504 w 371.23 753.55 m 380.81 753.55 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.06 62.22 Td [(T)]TJ /F12 1 Tf 1.1747 0 Td [(la)-379(tribu)-379(engendr)24(\266)469(ee)-379(par)]TJ /F6 1 Tf 10.247 0 Td [(T)]TJ /F12 1 Tf 1.1807 0 Td [(et)]TJ -38.331 -1.1988 Td [(l'ensem)30(ble)-337(des)-337(parties)-331(n)24(\266)469(egligeables.)-445(On)-337(a)-337(alors)-337(:)]TJ /F1 1 Tf 0.81325 -1.6386 Td [(1.)]TJ /F12 1 Tf 1.4277 0 Td [(Les)-246(\266)475(el)18(\266)475(emen)24(ts)-271(de)]TJ ET 174.53 719.56 m 184.1 719.56 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.602 59.382 Td [(T)]TJ /F12 1 Tf 1.0663 0 Td [(son)18(t)-271(les)-271(r)30(\266)469(eunions)]TJ /F8 1 Tf 7.4157 0 Td [(A)]TJ /F6 1 Tf 0.84337 0 Td [([)]TJ /F8 1 Tf 0.75904 0 Td [(N)]TJ /F12 1 Tf 1.1807 0 Td [(d'une)-271(partie)]TJ /F8 1 Tf 5.488 0 Td [(A)]TJ /F6 1 Tf 1.0181 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurable)-271(et)-271(d'une)-271(partie)]TJ /F8 1 Tf -27.843 -1.1988 Td [(N)]TJ /F12 1 Tf 1.247 0 Td [(n)30(\266)469(egligeable.)]TJ /F1 1 Tf -0.43373 -1.6386 Td [(2.)]TJ /F12 1 Tf 1.4819 0 Td [(Les)-295(fonctions)]TJ ET 162.58 685.65 m 172.15 685.65 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.602 56.545 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurables)-295(son)24(t)-301(celles)-295(qui)-295(son)24(t)-271(\266)469(egales)-295(presque)-295(partout)-301(\265)499(a)-295(une)-295(fonction)]TJ /F6 1 Tf -9.0723 -1.1988 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(-mesurable,)-337(ce)-337(son)24(t)-331(ausi)-337(celles)-331(qui)-331(s')24(\266)469(ecriv)30(en)24(t)]TJ /F8 1 Tf 19.494 0 Td [(f)]TJ /F2 1 Tf 0.81325 0 Td [(+)]TJ /F8 1 Tf 0.99398 0 Td [(n)]TJ /F12 1 Tf 0.93373 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.1928 0 Td [(f)]TJ /F6 1 Tf 0.92771 0 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(-mesurable)-337(et)]TJ /F8 1 Tf 6.2289 0 Td [(n)]TJ /F12 1 Tf 0.93373 0 Td [(n)30(\266)469(egligeable.)]TJ /F1 1 Tf -33.295 -1.6386 Td [(3.)]TJ /F12 1 Tf 1.3855 0 Td [(La)-253(mesure)-253(ext)24(\266)475(erieure)]TJ /F8 1 Tf 9.247 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.267 77.443 Td [(\244)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 18.361 53.707 Td [(d)24(\266)469(e\257nit)-246(par)-246(restriction)-253(une)-253(mesure)]TJ ET 393.34 648.72 m 403.85 648.72 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.91 53.707 Td [(m)]TJ /F12 1 Tf 1.1265 0 Td [(sur)]TJ ET 425.74 651.74 m 435.31 651.74 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 35.62 53.707 Td [(T)]TJ /F12 1 Tf 0.80121 0 Td [(.)-415(Les)-246(mesures)]TJ /F8 1 Tf 6.1145 0 Td [(m)]TJ /F12 1 Tf 1.1265 0 Td [(et)]TJ ET 63.72 634.39 m 74.232 634.39 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 52.509 Td [(m)]TJ /F12 1 Tf 1.1325 0 Td [(d)30(\266)469(e\257nissen)24(t)-259(la)-253(m)24(^)469(eme)-253(mesure)-259(ext)24(\266)475(erieure)-259(et)-259(la)-253(m)24(^)469(eme)-253(in)24(t)24(\266)475(egrale)-265(sup)0(\266)475(erieure.)-421(Les)-253(parties)-259(\(resp.)]TJ -1.1325 -1.2048 Td [(fonctions\))]TJ /F8 1 Tf 4.6446 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(-n)24(\266)469(egligeables)-337(son)24(t)-337(les)-337(m)30(^)469(emes)-337(que)-337(les)-337(parties)-337(\(resp.)-445(fonctions\))]TJ ET 450.86 620.06 m 461.38 620.06 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.723 51.304 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(-n)24(\266)469(egligeables.)]TJ /F1 1 Tf -32.458 -1.6386 Td [(4.)]TJ /F12 1 Tf 1.5602 0 Td [(L'espace)-337(mesur)30(\266)469(e)]TJ /F2 1 Tf 7.4217 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ ET 200.95 603.5 m 210.53 603.5 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.813 49.665 Td [(T)]TJ /F8 1 Tf 0.79518 0 Td [(;)]TJ ET 215.78 600.48 m 226.3 600.48 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.054 49.665 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-337(complet,)-337(on)-331(dit)-343(que)-337(c'est)-337(le)]TJ /F1 1 Tf 13.765 0 Td [(compl)18(\266)548(et)18(\266)548(e)]TJ /F12 1 Tf 4.8434 0 Td [(de)]TJ /F2 1 Tf 1.3313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.80121 0 Td [(;)-162(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -38.464 -1.9518 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.6386 Td [(1.)]TJ /F2 1 Tf 1.512 0 Td [(La)-313(tribu)-307(engendr)24(\266)469(ee)-313(par)-307(les)]TJ /F8 1 Tf 11.741 0 Td [(A)]TJ /F2 1 Tf 1.0602 0 Td [(appartenan)24(t)-313(\265)500(a)]TJ /F6 1 Tf 6.4277 0 Td [(T)]TJ /F2 1 Tf 1.1084 0 Td [(et)-313(les)]TJ /F8 1 Tf 2.5663 0 Td [(N)]TJ /F2 1 Tf 1.2229 0 Td [(n)24(\266)475(egligeables)-313(con)24(tien)24(t)-283(\266)469(evidem-)]TJ -26.452 -1.1988 Td [(men)24(t)-373(l'ensem)30(ble)]TJ /F6 1 Tf 7.4157 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.379 64.806 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.199 44.876 Td [(des)-373(r)30(\266)469(eunions)]TJ /F8 1 Tf 5.8072 0 Td [(A)]TJ /F6 1 Tf 1 0 Td [([)]TJ /F8 1 Tf 0.91566 0 Td [(N)]TJ /F2 1 Tf 0.90964 0 Td [(.)-560(P)30(our)-373(mon)24(trer)-373(l')24(\266)475(egalit)24(\266)469(e)-373(il)-373(su\261t)-373(donc)-373(de)-373(mon)24(trer)]TJ -17.5 -1.2048 Td [(que)]TJ /F6 1 Tf 1.8614 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.44 63.09 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6024 43.671 Td [(est)-343(une)-331(tribu.)]TJ -3.2711 -1.6325 Td [(Il)-349(est)-325(\266)469(eviden)24(t)-355(que)]TJ /F8 1 Tf 7.9578 0 Td [(X)]TJ /F2 1 Tf 1.259 0 Td [(est)-355(dans)]TJ /F6 1 Tf 3.9337 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.595 60.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.91 42.039 Td [(et)-355(que)]TJ /F6 1 Tf 3.0602 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.017 60.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.404 42.039 Td [(est)-355(stable)-349(par)-349(r)24(\266)475(eunion)-355(d)24(\266)475(enom)24(brable,)-361(prouv)30(ons)]TJ -19.072 -1.2048 Td [(sa)-337(stabilit)30(\266)469(e)-337(par)-337(passage)-337(au)-331(compl)24(\266)469(emen)24(taire.)]TJ 0 -1.6325 Td [(Soit)]TJ /F8 1 Tf 1.9518 0 Td [(A)]TJ /F6 1 Tf 0.77109 0 Td [([)]TJ /F8 1 Tf 0.68675 0 Td [(N)]TJ /F2 1 Tf 1.1506 0 Td [(un)-204(\266)469(el)24(\266)469(emen)30(t)-234(de)]TJ /F6 1 Tf 6.1687 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.129 56.685 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.373 39.201 Td [(\(a)24(v)24(ec)]TJ /F8 1 Tf 2.4819 0 Td [(A)]TJ /F2 1 Tf 0.98193 0 Td [(mesurable)-234(et)]TJ /F8 1 Tf 5.6988 0 Td [(N)]TJ /F2 1 Tf 1.1446 0 Td [(n)24(\266)469(egligeable\).)-415(Comme)]TJ /F8 1 Tf 9.4277 0 Td [(N)]TJ /F2 1 Tf 1.1446 0 Td [(est)-240(n)30(\266)469(egligeable)]TJ -32.922 -1.2048 Td [(il)-277(est)-283(con)30(ten)24(u)-283(dans)-277(une)-277(partie)]TJ /F8 1 Tf 12.958 0 Td [(B)]TJ /F2 1 Tf 1.0843 0 Td [(mesurable)-283(de)-277(mesure)-283(n)30(ulle.)-433(Alors)-277(les)-253(\266)475(el)18(\266)475(emen)24(ts)-277(de)-277(\()]TJ /F8 1 Tf 21.777 0 Td [(A)]TJ /F6 1 Tf 0.85542 0 Td [([)]TJ /F8 1 Tf 0.77711 0 Td [(N)]TJ /F2 1 Tf 0.91567 0 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.095 54.969 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 36.798 Td [(qui)-307(ne)-313(son)24(t)-313(pas)-319(dans)]TJ /F8 1 Tf 9.1807 0 Td [(B)]TJ /F2 1 Tf 1.1205 0 Td [(formen)24(t)-313(l'ensem)24(ble)-307(mesurable)-313(\()]TJ /F8 1 Tf 13.596 0 Td [(A)]TJ /F6 1 Tf 0.92771 0 Td [([)]TJ /F8 1 Tf 0.84337 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.078 53.245 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.602 36.798 Td [(,)-319(les)-283(\266)469(el)24(\266)469(emen)30(ts)-313(qui)-313(son)24(t)-313(dans)]TJ /F8 1 Tf -27.271 -1.1988 Td [(B)]TJ /F2 1 Tf 1.1928 0 Td [(formen)24(t)-385(un)-391(ensem)24(ble)-385(con)30(ten)24(u)-391(dans)]TJ /F8 1 Tf 15.753 0 Td [(B)]TJ /F2 1 Tf 1.1928 0 Td [(donc)-385(n)24(\266)475(egligeable.)-602(P)30(ar)-385(suite)-391(\()]TJ /F8 1 Tf 12.861 0 Td [(A)]TJ /F6 1 Tf 1.006 0 Td [([)]TJ /F8 1 Tf 0.92771 0 Td [(N)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.621 51.53 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.355 35.599 Td [(qui)-385(est)-391(la)]TJ -35.024 -1.2048 Td [(r)30(\266)469(eunion)-337(de)-331(ces)-337(ensem)24(bles)-337(appartien)24(t)-337(\265)500(a)]TJ /F6 1 Tf 16.916 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.983 49.814 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.331 34.394 Td [(.)]TJ /F1 1 Tf -17.187 -1.6325 Td [(2.)]TJ /F2 1 Tf 1.3373 0 Td [(Notons)]TJ /F6 1 Tf 3.3133 0 Td [(M)]TJ /F2 1 Tf 1.4217 0 Td [(\(resp.)]TJ ET 180.22 401.4 m 194.62 401.4 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.078 32.762 Td [(M)]TJ /F2 1 Tf 1.2048 0 Td [(,)]TJ /F6 1 Tf 0.5241 0 Td [(M)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.767 47.469 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.289 32.762 Td [(\))-222(l'ensem)24(ble)-222(des)-222(fonctions)]TJ /F6 1 Tf 11.072 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurables)-222(\(resp.)]TJ ET 458.64 401.4 m 468.22 401.4 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.373 32.762 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurables,)]TJ -33.873 -1.2048 Td [(\266)469(egales)-373(presque)-379(partout)-379(\265)500(a)-379(une)-373(fonction)]TJ /F6 1 Tf 17.03 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable\))-379(et)]TJ /F6 1 Tf 6.7108 0 Td [(N)]TJ /F2 1 Tf 1.3434 0 Td [(l'ensem)30(ble)-379(des)-379(fonctions)-373(n)24(\266)469(egli-)]TJ -25.849 -1.1988 Td [(geables.)-445(Il)-337(s'agit)-337(de)-331(prouv)24(er)-337(l')24(\266)475(egalit)24(\266)469(e)-337(des)-337(ensem)24(bles)]TJ ET 333.79 372.67 m 348.19 372.67 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.928 30.358 Td [(M)]TJ /F2 1 Tf 1.2048 0 Td [(,)]TJ /F6 1 Tf 0.60843 0 Td [(M)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.284 44.03 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.56 30.358 Td [(et)]TJ /F6 1 Tf 1.1627 0 Td [(M)]TJ /F2 1 Tf 1.4277 0 Td [(+)]TJ /F6 1 Tf 1 0 Td [(N)]TJ /F2 1 Tf 0.96988 0 Td [(.)]TJ -30.789 -1.6386 Td [(Une)-331(partie)]TJ ET 123.05 353.16 m 132.62 353.16 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.295 28.72 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable)]TJ /F8 1 Tf 5.0602 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(p)-30(eut)-331(donc)-331(s')24(\266)469(ecrire)]TJ /F8 1 Tf 8.0843 0 Td [(A)]TJ /F6 1 Tf 0.96385 0 Td [([)]TJ /F8 1 Tf 0.87952 0 Td [(N)]TJ /F2 1 Tf 1.241 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.1928 0 Td [(A)]TJ /F6 1 Tf 1.0241 0 Td [(2)-283(T)]TJ /F2 1 Tf 2.0723 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(N)]TJ /F2 1 Tf 1.241 0 Td [(n)24(\266)469(egligeable.)-445(Quitte)]TJ -30.825 -1.1988 Td [(\265)499(a)-301(remplacer)]TJ /F8 1 Tf 5.3916 0 Td [(N)]TJ /F2 1 Tf 1.2169 0 Td [(par)]TJ /F8 1 Tf 1.747 0 Td [(N)]TJ /F6 1 Tf 1.0723 0 Td [(\\)]TJ /F8 1 Tf 0.83133 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.379 39.969 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.048 27.521 Td [(on)-307(p)-30(eut)-307(m)24(^)475(eme)-301(supp)-30(oser)]TJ /F8 1 Tf 10.62 0 Td [(N)]TJ /F2 1 Tf 1.2169 0 Td [(disjoin)30(t)-307(de)]TJ /F8 1 Tf 4.8374 0 Td [(A)]TJ /F2 1 Tf 1.0482 0 Td [(et)-307(alors)-307(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.578 39.168 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.807 27.521 Td [(=)-277(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.276 39.168 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.235 27.521 Td [(+)-162(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.586 39.168 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 26.322 Td [(est)-379(la)-379(somme)-379(d'une)-379(fonction)]TJ /F6 1 Tf 12.819 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable)-379(p)-30(ositiv)30(e)-385(et)-379(d'une)-379(fonction)-385(n)30(\266)469(egligeable)-379(p)-30(ositiv)30(e.)]TJ -13.62 -1.2048 Td [(Il)-349(en)-355(v)54(a)-349(donc)-355(de)-349(m)24(^)469(eme)-349(p)-30(our)-355(toute)-349(fonction)]TJ ET 293.54 310.1 m 303.12 310.1 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.56 25.117 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable)-355(p)-30(ositiv)30(e)-361(puisqu'elle)-349(p)-30(eut)-355(s')24(\266)469(ecrire)]TJ /F9 1 Tf -20.024 -0.39157 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7759 32.306 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 23.774 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.853 33.806 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.9157 23.774 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.129 33.806 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 18.157 47.042 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7831 23.774 Td [(ab)-30(ec)]TJ /F8 1 Tf 2.3072 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.388 33.806 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.458 23.774 Td [(2)]TJ ET 172.15 293.97 m 181.73 293.97 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.404 23.774 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)]TJ -9.8735 -2.7711 Td [(Plus)-319(g)24(\266)475(en)24(\266)475(eralemen)24(t,)-325(p)-30(our)]TJ /F8 1 Tf 10.928 0 Td [(g)]TJ ET 204.19 260.92 m 213.77 260.92 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.084 21.003 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable)-319(on)-319(p)-30(eut)-289(\266)469(ecrire)]TJ /F8 1 Tf 11.422 0 Td [(g)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.267 29.84 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.488 21.003 Td [(\241)]TJ /F8 1 Tf 0.96386 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.552 29.84 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.922 21.003 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.302 29.84 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.488 21.003 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 55.862 29.84 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.675 21.003 Td [(+)]TJ /F8 1 Tf 0.96386 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 59.017 29.84 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.687 21.003 Td [(,)]TJ /F8 1 Tf 0.60241 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.198 29.84 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.717 21.003 Td [(=)]TJ /F8 1 Tf -38.386 -1.1988 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3276 28.125 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4518 19.804 Td [(+)]TJ /F8 1 Tf 0.96386 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 11.474 28.125 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.7771 19.804 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1807 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.379 28.125 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.211 19.804 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.81 28.125 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.608 19.804 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurables)-313(p)-30(ositiv)30(es,)]TJ /F8 1 Tf 9.8012 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.931 28.125 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.572 19.804 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.526 28.125 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.084 19.804 Td [(n)24(\266)475(egligeables)-319(p)-30(ositiv)24(es.)-439(Remarquons)]TJ -23.753 -1.1988 Td [(que)]TJ /F8 1 Tf 1.9157 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 11.078 26.409 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5783 18.605 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.888 26.409 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.211 18.605 Td [(\266)469(etan)24(t)-391(p)-30(ositiv)24(es)-391(s'ann)30(ulen)24(t)-397(quand)]TJ /F8 1 Tf 14.753 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.836 26.409 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.494 18.605 Td [(s'ann)24(ule,)-403(de)-391(m)24(^)469(eme)]TJ /F8 1 Tf 8.5542 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.293 26.409 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.38 18.605 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.103 26.409 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.042 18.605 Td [(s'ann)24(ulen)24(t)]TJ -34.711 -1.2048 Td [(quan)30(t)]TJ /F8 1 Tf 2.8494 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.388 24.685 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6807 17.4 Td [(s'ann)24(ule.)-487(Il)-349(s'ensuit)-349(qu'en)-349(tout)-355(p)-24(oin)24(t)-355(l'une)-349(au)-349(moins)-349(de)]TJ /F8 1 Tf 24.572 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.724 24.685 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.542 17.4 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 53.259 24.685 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.012 17.4 Td [(s'ann)24(ule)-349(ce)-349(qui)]TJ -32.681 -1.1988 Td [(p)-30(ermet)-337(de)-331(d)24(\266)475(e\257nir)-337(\(sans)-343(forme)-331(ind)24(\266)469(etermin)30(\266)469(ee\))-331(la)-337(fonction)-337(mesurable)]TJ /F8 1 Tf 29.295 0 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 53.017 22.969 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.717 16.201 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.112 22.969 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.994 16.201 Td [(;)-331(de)-337(m)24(^)475(eme)]TJ -34.663 -1.1988 Td [(on)-337(p)-24(eut)-343(d)30(\266)469(e\257nir)-337(la)-337(fonction)-337(n)24(\266)475(egligeable)]TJ /F8 1 Tf 16.904 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.457 21.254 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.446 15.003 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(n)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.707 21.254 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.494 15.003 Td [(.)-451(On)-337(a)-331(alors)]TJ /F8 1 Tf 5.6325 0 Td [(g)]TJ /F2 1 Tf 0.78915 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F2 1 Tf 0.81927 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(n)]TJ /F2 1 Tf 0.93976 0 Td [(\(on)-337(remarquera)]TJ -32.404 -1.2048 Td [(que)]TJ /F8 1 Tf 1.8434 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(n)]TJ /F2 1 Tf 0.88554 0 Td [(\266)469(etan)24(t)-319(toujours)-319(du)-319(m)24(^)475(eme)-319(signe)-319(que)]TJ /F8 1 Tf 15.139 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(,)-319(il)-319(n'y)-313(a)-319(pas)-319(de)-319(forme)-313(ind)24(\266)469(etermin)30(\266)469(ee\).)-439(On)-319(a)]TJ -20.452 -1.1988 Td [(donc)]TJ ET 92.232 160.48 m 105.19 160.48 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7169 12.599 Td [(M)]TJ /F6 1 Tf 1.3614 0 Td [(\275)-277(M)]TJ /F2 1 Tf 2.4819 0 Td [(+)]TJ /F6 1 Tf 1 0 Td [(N)]TJ /F2 1 Tf 0.96386 0 Td [(.)]TJ -8.1928 -1.9578 Td [(Si)]TJ /F8 1 Tf 1.1928 0 Td [(g)]TJ /F2 1 Tf 0.87952 0 Td [(s')24(\266)469(ecrit)]TJ /F8 1 Tf 2.9819 0 Td [(f)]TJ /F2 1 Tf 0.84337 0 Td [(+)]TJ /F8 1 Tf 1.0181 0 Td [(n)]TJ /F2 1 Tf 0.96385 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.2229 0 Td [(f)]TJ /F6 1 Tf 0.96386 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable)-367(et)]TJ /F8 1 Tf 6.2952 0 Td [(n)]TJ /F2 1 Tf 0.96385 0 Td [(n)24(\266)475(egligeable)-367(alors)]TJ /F8 1 Tf 7.5723 0 Td [(g)]TJ /F2 1 Tf 0.87349 0 Td [(est)-343(\266)469(egale)-367(\265)500(a)]TJ /F8 1 Tf 4.988 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(lorsque)]TJ /F8 1 Tf 3.4518 0 Td [(n)]TJ /F2 1 Tf 0.96386 0 Td [(est)]TJ -37.934 -1.1988 Td [(n)30(ulle,)-337(c'est-)0(\265)500(a-dire)-337(presque)-337(partout.)-451(On)-337(a)-331(donc)]TJ /F6 1 Tf 20.361 0 Td [(M)]TJ /F2 1 Tf 1.4217 0 Td [(+)]TJ /F6 1 Tf 1 0 Td [(N)-427(\275)-283(M)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.25 14.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.904 9.4424 Td [(.)]TJ -26.572 -1.9518 Td [(Si)]TJ /F8 1 Tf 1.1205 0 Td [(g)]TJ /F2 1 Tf 0.80723 0 Td [(est)-265(\266)469(egale)-295(presque)-289(partout)-295(\265)500(a)]TJ /F8 1 Tf 11.94 0 Td [(f)]TJ /F6 1 Tf 0.88554 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable,)-301(il)-295(existe)-289(une)-295(partie)]TJ /F8 1 Tf 13.62 0 Td [(N)]TJ /F2 1 Tf 1.2048 0 Td [(n)30(\266)469(egligeable,)-301(que)-295(l'on)]TJ -30.38 -1.2048 Td [(p)-30(eut)-337(toujours)-331(supp)-30(oser)]TJ /F6 1 Tf 10.386 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable,)-337(en)-331(dehors)-337(de)-337(laquelle)]TJ /F8 1 Tf 14.825 0 Td [(g)]TJ /F2 1 Tf 0.84338 0 Td [(co)102(\304)391(\263ncide)-337(a)30(v)24(ec)]TJ /F8 1 Tf 6.0301 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)-445(Alors,)-337(p)-30(our)]TJ /F5 1 Tf -33.482 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(60)]TJ ET endstream endobj 215 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 214 0 R /ProcSet 2 0 R >> /Contents 213 0 R >> endobj 214 0 obj << /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 216 0 obj << /Length 18760 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(toute)-277(partie)-277(b)-30(or)24(\266)475(elienne)]TJ /F8 1 Tf 10.139 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(de)]TJ ET 0.504 w 212.4 753.55 m 221.04 753.55 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 17.771 62.22 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-289(l'image)-283(r)30(\266)469(ecipro)-24(que)]TJ /F8 1 Tf 8.7952 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.784 89.633 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.873 62.22 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(est)-277(la)-277(r)24(\266)475(eunion)-283(de)]TJ /F8 1 Tf 8.0602 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.216 89.633 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.657 62.22 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F6 1 Tf 0.49398 0 Td [(\\)]TJ /F8 1 Tf 0.77711 0 Td [(N)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.5 89.633 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 62.22 Td [(et)]TJ -38.331 -1.1988 Td [(d'une)-259(partie)-265(de)]TJ /F8 1 Tf 6.7229 0 Td [(N)]TJ /F2 1 Tf 0.90964 0 Td [(.)-421(Cette)-265(derni)24(\265)475(ere)-265(partie)-259(est)-234(\266)469(evidemmen)24(t)-259(n)24(\266)469(egligeable)-259(tandis)-259(que)-265(la)-259(premi)24(\265)469(ere)]TJ -7.6325 -1.1988 Td [(est)-271(\266)475(egale)-301(\265)500(a)]TJ /F8 1 Tf 4.7711 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.319 86.193 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.777 59.822 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F6 1 Tf 0.53012 0 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(N)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.707 86.193 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.867 59.822 Td [(donc)]TJ /F6 1 Tf 2.3494 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable.)-433(Il)-295(s'ensuit)]TJ /F8 1 Tf 9.9518 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.198 86.193 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.56 59.822 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(2)]TJ ET 398.16 724.82 m 407.74 724.82 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.313 59.822 Td [(T)]TJ /F2 1 Tf 1.0904 0 Td [(et)-295(donc)]TJ /F8 1 Tf 3.4759 0 Td [(g)]TJ ET 462.38 724.82 m 471.96 724.82 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.687 59.822 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable.)]TJ -34.151 -1.2048 Td [(On)-337(a)-331(donc)]TJ /F6 1 Tf 4.8916 0 Td [(M)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.345 84.478 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.982 58.617 Td [(\275)]TJ ET 155.81 710.49 m 170.21 710.49 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.036 58.617 Td [(M)]TJ /F2 1 Tf 1.5361 0 Td [(et)-337(l')30(\266)469(egalit)24(\266)469(e)-337(des)-337(trois)-337(ensem)24(bles)]TJ ET 336.24 710.49 m 350.64 710.49 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.133 58.617 Td [(M)]TJ /F2 1 Tf 1.1988 0 Td [(,)]TJ /F6 1 Tf 0.61446 0 Td [(M)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.569 84.478 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.759 58.617 Td [(et)]TJ /F6 1 Tf 1.1687 0 Td [(M)]TJ /F2 1 Tf 1.4217 0 Td [(+)]TJ /F6 1 Tf 1 0 Td [(N)]TJ /F2 1 Tf 1.3012 0 Td [(en)-337(r)30(\266)469(esulte.)]TJ /F1 1 Tf -30.506 -1.8373 Td [(3.)]TJ /F2 1 Tf 1.4337 0 Td [(Soit)-265(\()]TJ /F8 1 Tf 2.3795 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.931 81.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.765 56.78 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.966 81.038 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.705 56.78 Td [(une)-271(famille)-271(d)24(\266)469(enom)24(brable)-271(de)-271(fonctions)]TJ ET 347.11 688.53 m 356.69 688.53 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.042 56.78 Td [(T)]TJ /F2 1 Tf 1.0663 0 Td [(mesurable)-271(p)-30(ositiv)30(es.)-427(On)-271(a)-271(vu)-271(que)]TJ -24.777 -1.1988 Td [(l'on)-301(p)-24(ouv)48(ait)-277(\266)475(ecrire)-301(ces)-301(fonctions)]TJ /F8 1 Tf 13.976 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.31 79.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.398 55.581 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.397 79.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.422 55.581 Td [(+)]TJ /F8 1 Tf 0.92771 0 Td [(n)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.276 79.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.584 55.581 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.1566 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.966 79.323 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.861 55.581 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable)-301(p)-30(ositiv)24(e)-301(et)]TJ /F8 1 Tf 9.8253 0 Td [(n)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.931 79.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.717 55.581 Td [(n)30(\266)469(egligeable)]TJ -34.386 -1.3434 Td [(p)-30(ositiv)30(e)-403(;)-433(on)-403(a)-397(alors)]TJ /F9 1 Tf 9.3012 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.922 77.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.44 54.238 Td [(=)]TJ /F9 1 Tf 1.1687 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.638 77.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.307 54.238 Td [(+)]TJ /F9 1 Tf 1.0482 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(n)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.155 77.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.301 54.238 Td [(a)30(v)24(ec)]TJ /F9 1 Tf 2.259 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(n)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.603 77.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.506 54.238 Td [(n)24(\266)475(egligeable.)-650(Comme)-403(les)]TJ /F8 1 Tf 10.952 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.034 77.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.681 54.238 Td [(son)18(t)]TJ /F6 1 Tf -37.349 -2.0904 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurables)-337(on)-337(conclut)]TJ /F9 1 Tf 10.355 1.3614 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.017 76.219 Td [(\244)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 17.94 53.093 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.664 74.409 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.53 52.147 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ 1 -0.41566 Td [(X)]TJ /F8 1 Tf 1.6084 -0.94578 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.733 74.409 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.867 52.147 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ 1.6145 0.41566 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.793 74.409 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.199 52.147 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ 1.6145 0.41566 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.164 76.219 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.006 52.147 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.5 74.409 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.982 52.147 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(.)]TJ -36.054 -2.2952 Td [(Autremen)24(t)-259(dit)-259(l'in)24(t)24(\266)469(egrale)-259(sup)0(\266)469(erieure)-259(est)-259(d)24(\266)475(enom)24(brablemen)24(t)-253(additiv)24(e)-259(sur)-259(l'ensem)24(ble)-253(des)-259(fonc-)]TJ 0 -1.1988 Td [(tions)]TJ ET 91.944 591.33 m 101.52 591.33 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6928 48.653 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurables.)-421(Comme)-246(il)-240(s'agit)-222(\266)475(evidemmen)24(t)-246(d'une)-253(fonctionnelle)-240(croissan)24(te)-246(et)-246(p)-30(ositiv)30(e-)]TJ -3.1566 -1.1988 Td [(men)24(t)-265(homog)24(\265)469(ene)-271(sur)]TJ /F6 1 Tf 8.6627 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 21.06 67.693 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.38 47.454 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-271(et)-265(que,)-283(par)-265(d)24(\266)469(e\257nition,)-277(p)-30(our)]TJ /F8 1 Tf 12.518 0 Td [(f)]TJ /F6 1 Tf 0.8735 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.405 67.693 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.392 47.454 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-265(il)-265(existe)]TJ /F8 1 Tf 4.2169 0 Td [(g)]TJ /F6 1 Tf 0.77711 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable,)]TJ -34.151 -1.7952 Td [(donc)]TJ ET 92.808 555.62 m 102.38 555.62 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7651 45.659 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable,)-391(ma)-60(joran)30(t)]TJ /F8 1 Tf 9.7831 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(et)-379(telle)-379(que)]TJ /F9 1 Tf 5.3253 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.69 66.935 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.102 45.659 Td [(f)-271(dm)]TJ /F2 1 Tf 2.512 0 Td [(=)]TJ /F9 1 Tf 1.1265 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(g)-204(dm)]TJ /F2 1 Tf 2.4277 0 Td [(=)]TJ /F9 1 Tf 1.1265 1.3615 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.509 66.935 Td [(\244)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.759 45.659 Td [(g)-198(dm)]TJ /F2 1 Tf 2.4518 0 Td [(on)-379(p)-30(eut)-379(appli-)]TJ -32.88 -1.8434 Td [(quer)-307(la)-307(prop)-30(osition)]TJ /F1 1 Tf 8.5361 0 Td [(IV.3.3.)]TJ /F2 1 Tf 3.4157 0 Td [(.)-433(On)-307(conclut)-313(qu'il)-301(existe)-307(une)-307(mesure)]TJ ET 398.45 530.49 m 408.96 530.49 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.337 43.816 Td [(m)]TJ /F2 1 Tf 1.1867 0 Td [(sur)-307(la)-307(tribu)]TJ ET 474.77 533.52 m 484.34 533.52 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.723 43.816 Td [(T)]TJ /F2 1 Tf 1.1084 0 Td [(telle)-301(que)]TJ -35.5 -1.1988 Td [(l'in)24(t)24(\266)475(egrale)-421(sup)0(\266)469(erieure)-415(d)24(\266)469(e\257nie)-415(par)]TJ ET 239.83 516.16 m 250.34 516.16 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.066 42.617 Td [(m)]TJ /F2 1 Tf 1.2892 0 Td [(co)102(\304)391(\263ncide)-415(a)24(v)24(ec)-415(celle)-415(d)24(\266)469(e\257nie)-415(par)]TJ /F8 1 Tf 13.608 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-686(En)-415(particulier)-415(les)]TJ -30.512 -1.2048 Td [(fonctions)-253(n)24(\266)475(egligeables)-259(d)30(\266)469(e\257nies)-253(par)-253(ces)-253(mesures)-253(son)18(t)-253(les)-253(m)24(^)475(emes.)-421(Il)-253(y)-259(a)-246(alors)-259(aussi)-253(co)108(\304)385(\263ncidence)]TJ 0 -1.1988 Td [(des)-373(mesures)-379(ext)30(\266)469(erieures)-373(d)24(\266)475(e\257nies)-379(par)]TJ ET 256.25 487.51 m 266.76 487.51 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.44 40.214 Td [(m)]TJ /F2 1 Tf 1.247 0 Td [(et)]TJ /F8 1 Tf 1.2108 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(,)-379(par)-379(suite)]TJ ET 354.89 487.51 m 365.4 487.51 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.693 40.214 Td [(m)]TJ /F2 1 Tf 1.253 0 Td [(est)-379(la)-373(restriction)-379(\265)500(a)]TJ ET 470.95 490.53 m 480.53 490.53 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.404 40.214 Td [(T)]TJ /F2 1 Tf 1.1747 0 Td [(de)]TJ /F8 1 Tf 1.3735 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.284 58.142 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 40.214 Td [(et)]TJ -38.331 -1.1988 Td [(les)-337(parties)-331(n)24(\266)469(egligeables)-337(d)30(\266)469(e\257nies)-337(par)-337(ces)-331(mesures)-337(son)24(t)-337(les)-337(m)24(^)475(emes.)]TJ /F1 1 Tf 0.81325 -1.8373 Td [(4.)]TJ /F2 1 Tf 1.3614 0 Td [(Les)-234(parties)]TJ ET 148.1 451.15 m 158.62 451.15 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.392 37.177 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(-n)24(\266)475(egligeables)-240(son)24(t)-234(alors)]TJ /F8 1 Tf 10.084 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(-n)24(\266)469(egligeables)-240(donc)-234(appartiennen)24(t)-234(\265)499(a)]TJ ET 469.94 454.17 m 479.52 454.17 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.319 37.177 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)-409(L'espace)]TJ -34.789 -1.1988 Td [(mesur)30(\266)469(e)-337(\()]TJ /F8 1 Tf 3.7892 0 Td [(X)-24(;)]TJ ET 124.49 439.84 m 134.06 439.84 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.416 35.979 Td [(T)]TJ /F8 1 Tf 0.79518 0 Td [(;)]TJ ET 139.32 436.82 m 149.83 436.82 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.657 35.979 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\))-337(est)-337(donc)-337(complet.)]TJ /F1 1 Tf -7.2048 -2.6747 Td [(Exemple.)]TJ /F2 1 Tf 5.6566 0 Td [(Nous)-415(v)24(errons)-421(\(cf.)-698(prop)-30(osition)]TJ /F1 1 Tf 13.669 0 Td [(VI)-36(I.2.2.)]TJ /F2 1 Tf 3.8795 0 Td [(\))-698(que)-421(la)-421(mesure)-421(de)-415(Leb)-30(esgue)-421(est)-421(la)]TJ -23.205 -1.1988 Td [(compl)24(\266)475(et)24(\266)469(ee)-337(de)-331(celle)-337(de)-337(Borel.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 22.776 Td [(IV.7.)-236(Les)-381(fonctions)-376(in)30(t)20(\266)537(egrables.)-502(L'espace)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.494 27.304 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.198 39.659 Td [(1)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 25.553 22.776 Td [(.)]TJ -21.106 -2.8191 Td [(IV.7.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 23.924 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83133 0.084337 Td [(\266)]TJ -0.0060234 -0.084337 Td [(efinition.)]TJ /F12 1 Tf 5.4337 0 Td [(Une)-391(fonction)]TJ /F1 1 Tf 6.0542 0 Td [(in)30(t)24(\266)548(egrable)]TJ /F12 1 Tf 5.3795 0 Td [(est)-391(une)-391(fonction)-391(mesurable)]TJ /F8 1 Tf 12.259 0 Td [(f)]TJ /F12 1 Tf 0.98193 0 Td [(p)-30(our)-391(la-)]TJ -35.633 -1.6566 Td [(quelle)]TJ /F9 1 Tf 2.8614 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F12 1 Tf 1.7349 0 Td [(est)-337(\257ni.)]TJ /F2 1 Tf -6.9157 -2.7651 Td [(Si)]TJ /F8 1 Tf 1.1807 0 Td [(f)]TJ /F2 1 Tf 0.95181 0 Td [(est)-355(in)24(t)24(\266)475(egrable)-355(alors)-355(sa)-355(partie)-355(p)-30(ositiv)30(e)]TJ /F8 1 Tf 16.53 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.034 27.693 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.5 19.503 Td [(est)-355(encore)-355(mesurable)-355(et)-349(de)-355(0)]TJ /F11 1 Tf 12.801 0 Td [(6)]TJ /F8 1 Tf 1.0904 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.069 27.693 Td [(+)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 40.849 19.503 Td [(6)]TJ /F6 1 Tf 1.0843 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F2 1 Tf 0.62651 0 Td [(on)]TJ -38.108 -1.7169 Td [(d)30(\266)469(eduit)]TJ /F9 1 Tf 3.1084 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.216 25.237 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.765 17.786 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F2 1 Tf 1.7349 0 Td [(et)-331(donc)]TJ /F9 1 Tf 3.5542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32.328 25.237 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.422 17.786 Td [(dm)]TJ /F2 1 Tf 1.7349 0 Td [(\257ni.)-445(De)-337(m)24(^)469(eme,)-331(p)-30(our)-331(la)-337(partie)-337(n)30(\266)469(egativ)24(e)-337(de)]TJ /F8 1 Tf 18.464 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)]TJ /F9 1 Tf -38.886 -0.95783 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.7586 21.918 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6627 15.467 Td [(dm)]TJ /F2 1 Tf 1.7349 0 Td [(est)-337(\257ni.)]TJ -4.0663 -2.9639 Td [(On)-337(p)-24(ose)-337(alors)]TJ /F9 1 Tf 6.3193 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4458 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 25.241 17.685 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.47 12.503 Td [(dm)]TJ /F6 1 Tf 1.6205 0 Td [(\241)]TJ /F9 1 Tf 1.006 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.319 17.685 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.428 12.503 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-445(C'est)-337(un)-337(r)24(\266)475(eel)-337(\(\257ni\))-343(app)-30(el)24(\266)469(e)]TJ /F1 1 Tf 12.265 0 Td [(in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 4.6867 0 Td [(de)]TJ /F8 1 Tf 1.3374 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)]TJ -38.38 -2.2892 Td [(Si)-403(l'on)-403(a)-403(c)24(hoisi)-403(une)-403(lettre,)-421(par)-403(exemple)]TJ /F8 1 Tf 17.753 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-421(p)-24(our)-409(d)30(\266)469(esigner)-403(une)-403(v)48(ariable)-403(dans)]TJ /F8 1 Tf 15.289 0 Td [(X)]TJ /F2 1 Tf 1.3072 0 Td [(on)-403(notera)]TJ -34.922 -1.6566 Td [(aussi)-421(l'in)24(t)24(\266)469(egrale)-421(de)]TJ /F8 1 Tf 8.6928 0 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(par)]TJ /F9 1 Tf 1.8735 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-427(et)-421(m)24(^)475(eme)]TJ /F9 1 Tf 5.0422 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.5181 0 Td [(dans)-415(le)-427(cas)-421(de)-421(la)-421(mesure)-427(de)]TJ -26.729 -1.6566 Td [(Leb)-30(esgue.)-451(On)-331(dit)-343(alors)-337(que)]TJ /F8 1 Tf 12.199 0 Td [(x)]TJ /F2 1 Tf 0.90361 0 Td [(est)-343(la)]TJ /F1 1 Tf 2.6747 0 Td [(v)60(ariable)-385(d'in)30(t)24(\266)548(egration)]TJ /F2 1 Tf 10.777 0 Td [(.)]TJ -7.6386 -3.012 Td [(61)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 218 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 217 0 R /ProcSet 2 0 R >> /Contents 216 0 R >> endobj 217 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 219 0 obj << /Length 14649 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ 0 -2.4036 Td [(Remarque.)]TJ /F2 1 Tf 6.4398 0 Td [(Cette)-385(d)24(\266)469(e\257nition)-379(de)-385(l'in)24(t)24(\266)475(egrale)-391(de)]TJ /F8 1 Tf 14.723 0 Td [(f)]TJ /F2 1 Tf 0.9759 0 Td [(co)102(\304)391(\263ncide)-385(a)30(v)24(ec)-385(la)-385(d)30(\266)469(e\257nition)-385(d)30(\266)469(ej)0(\265)499(a)-385(donn)24(\266)469(ee)]TJ -22.139 -1.1988 Td [(lorsque)]TJ /F8 1 Tf 3.4277 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-337(p)-30(ositiv)24(e.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 47.937 Td [(IV.7.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 57.467 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td 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0.90964 0 Td [(\))-469(si)]TJ -37.639 -1.1988 Td [(le)-325(mesure)]TJ /F8 1 Tf 4.4217 0 Td [(m)]TJ /F2 1 Tf 1.1988 0 Td [(p)-30(eut)-295(^)475(etre)-319(sous-en)18(tendue\))-319(l'ensem)30(ble)-325(des)-319(fonctions)-319(in)24(t)24(\266)469(egrables)-319(\265)500(a)-319(v)54(aleurs)-325(\257nies.)]TJ -5.6205 -1.1988 Td [(T)78(outefois,)-427(par)-409(abus,)-433(on)-379(\266)469(ecrira)-415(parfois)]TJ /F8 1 Tf 16.94 0 Td [(f)]TJ /F6 1 Tf 1.006 0 Td [(2)-403(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.828 38.107 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.488 26.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-409(p)-30(our)-409(dire)-415(que)]TJ /F8 1 Tf 7.253 0 Td [(f)]TJ /F2 1 Tf 1.006 0 Td [(est)-415(in)24(t)24(\266)469(egrable)-409(sans)-409(se)]TJ -29.711 -1.2048 Td [(soucier)-337(de)-331(sa)24(v)24(oir)-337(si)]TJ /F8 1 Tf 8.5542 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-343(bien)-331(\265)500(a)-337(v)54(aleurs)-337(\257nies.)]TJ -9.4819 -2.2952 Td [(Plus)-325(g)24(\266)475(en)24(\266)469(eralemen)30(t,)-331(si)]TJ /F8 1 Tf 9.5843 0 Td [(f)]TJ /F2 1 Tf 0.92169 0 Td [(est)-331(une)-325(fonction)-331(d)30(\266)469(e\257nie)-325(presque)-325(partout)-331(sur)]TJ /F8 1 Tf 19.355 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-325(on)-325(dira)-331(qu'elle)-325(est)]TJ -30.771 -1.2048 Td [(in)24(t)24(\266)475(egrable)-313(s'il)-313(existe)]TJ /F9 1 Tf 9.1386 0.26506 Td [(e)]TJ /F8 1 Tf -0.19277 -0.26506 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(fonction)-319(in)30(t)24(\266)469(egrable)-313(qui)-307(co)102(\304)391(\263ncide)-313(presque)-313(partout)-313(a)24(v)24(ec)]TJ /F8 1 Tf 23.337 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)-439(L'in)24(t)24(\266)475(egrale)]TJ -33.789 -1.6566 Td [(de)]TJ /F9 1 Tf 1.6084 0.26506 Td [(e)]TJ /F8 1 Tf -0.19277 -0.26506 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(ne)-421(d)24(\266)475(ep)-30(end)-421(pas)-421(alors)-427(du)-421(c)24(hoix)-421(de)-415(cette)-421(fonction,)-445(on)-421(la)-415(notera)-427(encore)]TJ /F9 1 Tf 30.837 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5843 0 Td [(et)-421(on)]TJ -36.886 -1.8253 Td [(\266)469(ecrira)]TJ /F8 1 Tf 2.8133 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.207 26.4 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.072 18.033 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 12.46 Td [(IV.7.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 14.936 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084337 Td [(\266)]TJ 0 -0.084337 Td [(eor)]TJ 1.8012 0.084337 Td [(\265)]TJ -0.0060244 -0.084337 Td [(eme.)]TJ /F12 1 Tf 2.7952 0 Td [(L'ensem)18(ble)]TJ /F6 1 Tf 5.1988 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 31.353 21.969 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.361 14.936 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.75904 0 Td [(est)-379(un)-373(sous-espace)-373(v)24(ectoriel)-373(de)-373(l'ensem)24(ble)-373(des)]TJ -19.084 -1.1988 Td [(fonctions)-337(de)]TJ /F8 1 Tf 5.5904 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(dans)]TJ /F3 1 Tf 2.3373 0 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -9.8916 -1.6506 Td [(L'in)30(t)24(\266)469(egrale)-337(d)24(\266)469(e\257nit)-331(une)-337(forme)-331(lin)24(\266)469(eaire)-337(sur)-337(cet)-337(espace.)]TJ /F1 1 Tf 0 -1.9759 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6506 Td [(P)30(our)]TJ /F8 1 Tf 2.4217 0 Td [(f)]TJ /F2 1 Tf 0.92169 0 Td [(dans)]TJ /F6 1 Tf 2.3313 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.741 12.702 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.145 8.4605 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-325(et)]TJ /F8 1 Tf 1.8675 0 Td [(\270)]TJ /F2 1 Tf 0.90964 0 Td [(dans)]TJ /F3 1 Tf 2.3313 0 Td [(R)]TJ /F2 1 Tf 1.0482 0 Td [(on)-325(a)]TJ /F8 1 Tf 2.2048 0 Td [(\270f)]TJ /F2 1 Tf 1.506 0 Td [(mesurable)-325(\(puisque)]TJ /F8 1 Tf 8.747 0 Td [(f)]TJ /F2 1 Tf 0.92169 0 Td [(l'est\))-331(et)]TJ /F6 1 Tf 3.6566 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\270f)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F2 1 Tf 0.55421 0 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\270)]TJ /F6 1 Tf 0.58434 0 Td [(jj)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(,)-325(d'o)-36(\265)530(u)]TJ /F9 1 Tf -36.669 -0.3494 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\270f)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\270)]TJ /F6 1 Tf 0.58434 0 Td [(j)]TJ /F9 1 Tf 0.43976 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.7289 0 Td [(\257ni.)-451(P)30(ar)-337(suite)]TJ /F8 1 Tf 6.3855 0 Td [(\270f)]TJ /F2 1 Tf 1.5181 0 Td [(est)-337(in)24(t)24(\266)475(egrable.)]TJ /F5 1 Tf -18.898 -2.8614 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(62)]TJ ET endstream endobj 221 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 220 0 R /ProcSet 2 0 R >> /Contents 219 0 R >> endobj 220 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 222 0 obj << /Length 19767 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(P)30(our)]TJ /F8 1 Tf 2.5181 0 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(et)]TJ /F8 1 Tf 1.253 0 Td [(g)]TJ /F2 1 Tf 0.93373 0 Td [(dans)]TJ /F6 1 Tf 2.4277 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 20.276 89.633 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.62 62.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\),)]TJ /F8 1 Tf 1.1084 0 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(et)]TJ /F8 1 Tf 1.253 0 Td [(g)]TJ /F2 1 Tf 0.93373 0 Td [(son)24(t)-421(\265)500(a)-421(v)54(aleurs)-427(\257nies)-421(donc)]TJ /F8 1 Tf 11.729 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(+)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.93373 0 Td [(est)-427(d)30(\266)469(e\257ni)-421(et)-421(\265)500(a)-421(v)54(aleurs)]TJ -29.494 -1.1988 Td [(\257nies.)-849(Cette)-475(fonction)-475(est)-469(mesurable)-475(comme)]TJ /F8 1 Tf 20.307 0 Td [(f)]TJ /F2 1 Tf 1.0663 0 Td [(et)]TJ /F8 1 Tf 1.3012 0 Td [(g)]TJ /F2 1 Tf 0.98193 0 Td [(et)-475(l'on)-469(a)]TJ /F6 1 Tf 4.3554 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(+)]TJ /F8 1 Tf 1.0904 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(j)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F6 1 Tf 1.2831 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(j)]TJ /F2 1 Tf 0.59036 0 Td [(+)]TJ /F6 1 Tf 1.0904 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(,)-500(d'o)-36(\265)530(u)]TJ /F9 1 Tf -36.494 -0.3494 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F6 1 Tf 0.51807 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F2 1 Tf 1.7349 0 Td [(est)-337(\257ni)-337(et)]TJ /F8 1 Tf 4.4518 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.8494 0 Td [(est)-337(in)24(t)24(\266)469(egrable.)]TJ -22.819 -2.1506 Td [(Comme)-228(la)-222(fonction)-228(n)24(ulle)-222(est)-198(\266)469(evidemmen)24(t)-228(in)24(t)24(\266)475(egrable,)]TJ /F6 1 Tf 22.41 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.69 82.383 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.88 57.159 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-222(est)-228(bien)-228(un)-228(sous-espace)-228(v)24(ectoriel)]TJ -24.849 -1.2048 Td [(de)-331(l'espace)-337(des)-337(fonctions)-337(de)]TJ /F8 1 Tf 12.325 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(dans)]TJ /F3 1 Tf 2.3434 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)]TJ -16.633 -1.747 Td [(Soit)]TJ /F8 1 Tf 2.0542 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.164 78.159 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.343 54.207 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\).)-445(P)24(our)]TJ /F8 1 Tf 3.5422 0 Td [(\270)]TJ /F11 1 Tf 0.86145 0 Td [(>)]TJ /F2 1 Tf 1.0602 0 Td [(0)-331(on)-337(a)-331(\()]TJ /F8 1 Tf 3.4458 0 Td [(\270f)]TJ /F2 1 Tf 1.1807 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 31.655 77.357 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.066 54.207 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\270f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.052 77.357 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.193 54.207 Td [(et)-337(\()]TJ /F8 1 Tf 1.5542 0 Td [(\270f)]TJ /F2 1 Tf 1.1807 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.957 77.357 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.271 54.207 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\270f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.371 77.357 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.078 54.207 Td [(,)]TJ -23.476 -2.0361 Td [(il)-331(s'ensuit)]TJ /F9 1 Tf 4.512 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\270f)-277(dm)]TJ /F2 1 Tf 3.0241 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.672 74.443 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.265 52.171 Td [(dm)]TJ /F6 1 Tf 1.6205 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.819 74.443 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.97 52.171 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\270)]TJ /F9 1 Tf 0.74699 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)-271(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)]TJ -29.289 -2.0422 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(\270)-283(<)]TJ /F2 1 Tf 1.9157 0 Td [(0)-337(on)-331(a)-337(\()]TJ /F8 1 Tf 3.4458 0 Td [(\270f)]TJ /F2 1 Tf 1.1807 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 21.034 71.521 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.639 50.129 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(\270f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.543 71.521 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.554 50.129 Td [(et)-337(\()]TJ /F8 1 Tf 1.5602 0 Td [(\270f)]TJ /F2 1 Tf 1.1807 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.457 71.521 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.639 50.129 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(\270f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.983 71.521 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.211 50.129 Td [(,)]TJ -17.608 -2.0361 Td [(il)-331(s'ensuit)]TJ /F9 1 Tf 4.512 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(\270f)-277(dm)]TJ /F2 1 Tf 3.0241 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.784 68.607 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.054 48.093 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.948 68.607 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.747 48.093 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\270)]TJ /F9 1 Tf 0.75301 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)]TJ -30.066 -2.1506 Td [(P)30(our)]TJ /F8 1 Tf 2.4036 0 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(et)]TJ /F8 1 Tf 1.1386 0 Td [(g)]TJ /F2 1 Tf 0.81928 0 Td [(dans)]TJ /F6 1 Tf 2.3133 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.474 66.34 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.054 45.942 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-313(on)-307(a)]TJ /F8 1 Tf 2.8675 0 Td [(f)]TJ /F2 1 Tf 0.77108 0 Td [(+)]TJ /F8 1 Tf 0.94578 0 Td [(g)]TJ /F2 1 Tf 0.78916 0 Td [(=)-277(\()]TJ /F8 1 Tf 1.4458 0 Td [(f)]TJ /F2 1 Tf 0.76506 0 Td [(+)]TJ /F8 1 Tf 0.95181 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.466 65.53 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.62 45.942 Td [(\241)]TJ /F2 1 Tf 0.94578 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.76506 0 Td [(+)]TJ /F8 1 Tf 0.94579 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.319 65.53 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.524 45.942 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.897 65.53 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.904 45.942 Td [(\241)]TJ /F8 1 Tf 0.94578 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.147 65.53 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.187 45.942 Td [(+)]TJ /F8 1 Tf 0.94578 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.388 65.53 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.446 45.942 Td [(\241)]TJ /F8 1 Tf 0.94578 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.621 65.53 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.542 45.942 Td [(.)-439(Il)-307(s'ensuit)]TJ -34.211 -1.1988 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.8012 0 Td [(+)]TJ /F8 1 Tf 0.98795 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 12.026 63.814 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.2771 44.744 Td [(+)]TJ /F8 1 Tf 0.98193 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.379 63.814 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.627 44.744 Td [(+)]TJ /F8 1 Tf 0.98193 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.724 63.814 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.042 44.744 Td [(=)-277(\()]TJ /F8 1 Tf 1.4458 0 Td [(f)]TJ /F2 1 Tf 0.80121 0 Td [(+)]TJ /F8 1 Tf 0.98193 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.009 63.814 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.054 44.744 Td [(+)]TJ /F8 1 Tf 0.98193 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 29.371 63.814 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.392 44.744 Td [(+)]TJ /F8 1 Tf 0.98795 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32.707 63.814 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.518 44.744 Td [(.)-439(S'agissan)18(t)-325(de)-325(fonctions)-331(mesurables)-325(p)-30(ositiv)24(es)-325(et)]TJ -18.187 -1.1988 Td [(d'in)24(t)24(\266)469(egrale)-355(\257nie,)-361(en)-349(appliquan)24(t)-355(les)-349(propri)18(\266)475(et)24(\266)469(es)-349(de)-355(l'in)24(t)24(\266)469(egrale)-355(p)-30(our)-355(des)-355(fonctions)-349(mesurables)]TJ 0 -1.2048 Td [(p)-30(ositiv)30(es)-337(il)-331(vien)18(t)-337(:)]TJ /F9 1 Tf 2.7108 -0.29518 Td [(Z)]TJ /F2 1 Tf 0.83133 -1.3614 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.155 58.004 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.819 40.683 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.224 58.004 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.771 40.683 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.302 58.004 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.711 40.683 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83133 -1.3614 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.06 58.004 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.235 40.683 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.147 58.004 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.175 40.683 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1.006 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.207 58.004 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.108 40.683 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(,)]TJ -34.44 -2.3193 Td [(d'o)-30(\265)524(u)]TJ /F9 1 Tf 2.2229 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83735 -1.3614 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.94 54.685 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.066 38.364 Td [(dm)]TJ /F6 1 Tf 1.6205 0 Td [(\241)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83735 -1.3614 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.8253 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.526 54.685 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.476 38.364 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.776 54.685 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.53 38.364 Td [(dm)]TJ /F6 1 Tf 1.6205 0 Td [(\241)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.853 54.685 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.488 38.364 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 50.922 54.685 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.416 38.364 Td [(dm)]TJ /F6 1 Tf 1.6205 0 Td [(\241)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.983 54.685 Td [(\241)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.355 38.364 Td [(dm)]TJ /F2 1 Tf -26.301 -2.3253 Td [(c'est-)-6(\265)500(a-dire)]TJ /F9 1 Tf 5.1265 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83133 -1.3614 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.3855 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F2 1 Tf 2.0783 0 Td [(.)]TJ -29.157 -1.9819 Td [(La)-337(lin)24(\266)475(earit)24(\266)469(e)-337(de)-337(l'in)24(t)24(\266)475(egrale)-343(est)-337(donc)-307(\266)469(etablie.)]TJ ET 512.64 405.82 m 519.84 405.82 l 519.84 414.25 l 512.64 414.25 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 32.081 Td [(En)-331(application)-337(on)-337(a)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 23.982 Td [(IV.7.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 28.75 Td [(5.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8614 0 Td [(P)24(our)]TJ /F8 1 Tf 2.4277 0 Td [(f)]TJ /F12 1 Tf 0.92169 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(g)]TJ /F12 1 Tf 0.83735 0 Td [(in)24(t)24(\266)475(egrables)-331(telles)-325(que)]TJ /F8 1 Tf 9.4337 0 Td [(f)]TJ /F11 1 Tf 0.87349 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F12 1 Tf 0.83735 0 Td [(on)-325(a)]TJ /F9 1 Tf 2.2048 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F11 1 Tf 2.4458 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F12 1 Tf 2.0783 0 Td [(.)]TJ /F1 1 Tf -38.886 -2.2048 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(P)30(ar)-313(d)30(\266)469(e\257nition,)-319(l'in)24(t)24(\266)475(egrale)-319(d'une)-313(fonction)-313(p)-30(ositiv)30(e)-319(est)-313(p)-30(ositiv)30(e.)-445(En)-307(par-)]TJ -8.8193 -1.6566 Td [(ticulier,)-457(p)-30(our)]TJ /F8 1 Tf 6.259 0 Td [(f)]TJ /F11 1 Tf 1.0422 0 Td [(6)]TJ /F8 1 Tf 1.2169 0 Td [(g)]TJ /F2 1 Tf 0.95181 0 Td [(on)-433(a)]TJ /F9 1 Tf 2.4217 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83132 -1.3614 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F6 1 Tf 0.8012 0 Td [(\241)]TJ /F8 1 Tf 1.0663 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F11 1 Tf 1.8434 0 Td [(>)]TJ /F2 1 Tf 1.2229 0 Td [(0,)-457(donc)]TJ /F9 1 Tf 3.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F6 1 Tf 2.3675 0 Td [(\241)]TJ /F9 1 Tf 1.0663 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)-271(dm)]TJ /F11 1 Tf 2.6024 0 Td [(>)]TJ /F2 1 Tf 1.2229 0 Td [(0)-433(et)-433(le)-439(r)30(\266)469(esultat.)]TJ ET 512.64 277.59 m 519.84 277.59 l 519.84 286.02 l 512.64 286.02 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 21.352 Td [(L'ensem)24(ble)]TJ /F6 1 Tf 5.2108 0 Td [(N)]TJ /F2 1 Tf 1.3494 0 Td [(des)-391(fonctions)-385(mesurables)-385(et)-391(n)30(\266)469(egligeables)-385(\265)500(a)-385(v)54(aleurs)-391(\257nies)-391(est)-385(clairemen)24(t)-391(un)]TJ -6.5602 -1.1988 Td [(sous-espace)-403(v)24(ectoriel)-403(de)]TJ /F6 1 Tf 10.831 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 24.121 29.426 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.301 20.153 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-403(sur)-403(lequel)-403(l'in)24(t)24(\266)469(egrale)-403(est)-403(une)-403(forme)-397(n)24(ulle.)-644(Il)-403(s'ensuit)-397(que)]TJ -13.271 -1.2048 Td [(celle-ci)-361("passe)-367(au)-367(quotien)24(t")-367(p)-30(our)-367(d)24(\266)475(e\257nir)-373(une)-361(forme)-367(lin)24(\266)469(eaire)-367(sur)-367(le)-373(quotien)24(t)]TJ /F6 1 Tf 32.976 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 55.81 27.711 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.446 18.948 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(=)]TJ /F6 1 Tf 0.5 0 Td [(N)]TJ /F2 1 Tf 1.3313 0 Td [(que)]TJ -37.633 -1.1988 Td [(l'on)-337(note)-337(aussi)]TJ /F8 1 Tf 6.6265 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.086 25.987 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.084 17.75 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\).)]TJ -9.0482 -1.5241 Td [(En)-373(termes)-373(plus)-343(\266)469(el)24(\266)469(emen)30(taires,)-385(deux)-373(fonctions)-373(de)]TJ /F6 1 Tf 21.193 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.94 23.806 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.663 16.226 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-373(qui)-373(di\256)24(\265)469(eren)24(t)-373(d'une)-379(fonction)-373(n)24(\266)469(egli-)]TJ -23.627 -1.2048 Td [(geable)-439(son)24(t)-415(\266)469(egales)-445(presque)-439(partout,)-475(elles)-439(on)24(t)-445(donc)-439(m)24(^)469(eme)-439(in)24(t)24(\266)475(egrale.)-771(Si)-439(l'on)-445(iden)24(ti\257e)-445(ces)]TJ 0 -1.1988 Td [(fonctions)-337(en)-337(les)-343(mettan)24(t)-343(dans)-337(la)-337(m)24(^)469(eme)-337(classe,)-343(l'in)24(t)24(\266)469(egrale)-343(ne)-337(d)30(\266)469(ep)-30(end)-337(que)-343(de)-337(la)-337(classe)-343(et)-343(les)]TJ 0 -1.1988 Td [(classes)-313(de)-307(fonctions)-313(formen)24(t)-313(l'espace)-313(v)24(ectoriel)-313(quotien)24(t)]TJ /F8 1 Tf 23.964 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 42.897 18.65 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.422 12.623 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\).)-439(On)-313(conserv)30(era)-319(usuellemen)24(t)]TJ -26.392 -1.7108 Td [(la)-301(m)24(^)475(eme)-307(notation)-307(p)-30(our)-301(une)-307(fonction)-307(et)-301(p)-30(our)-307(sa)-301(classe)-307(et)-307(la)-301(notation)]TJ /F9 1 Tf 29.542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4639 0 Td [(sera)-307(conserv)54(\266)469(ee)]TJ -33.006 -1.6566 Td [(p)-30(our)-331(d)24(\266)469(esigner)-337(l'in)24(t)24(\266)475(egrale)-337(d'une)-343(classe.)]TJ 18.916 -5.3675 Td [(63)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 224 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 223 0 R /ProcSet 2 0 R >> /Contents 222 0 R >> endobj 223 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 225 0 obj << /Length 14414 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(IV.8.)-236(In)30(t)20(\266)537(egration)-371(des)-376(fonctions)-381(v)30(ectorielles.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 60.599 Td [(Dans)-379(cette)-379(section)-373(nous)-379(nous)-373(limiterons)-379(au)-379(cas)-373(des)-379(fonctions)-379(\265)500(a)-379(v)54(aleurs)-379(dans)-379(un)]TJ /F3 1 Tf 35.325 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(-espace)]TJ -36.048 -1.1988 Td [(v)24(ectoriel)-240(de)-240(dimension)-246(\257nie.)-421(Ce)-246(cas)-240(con)30(tien)18(t)-240(donc)-246(en)-240(particulier)-240(celui)-246(des)-240(fonctions)-246(\265)499(a)-240(v)54(aleurs)]TJ 0 -1.2048 Td [(complexes.)]TJ 0 -1.6205 Td [(Outre)-259(l'espace)-259(mesur)30(\266)469(e)-259(\()]TJ /F8 1 Tf 10.114 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.80121 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3253 0 Td [(\))-253(nous)-259(supp)-30(osons)-253(donc)-259(donn)24(\266)469(e)-259(un)]TJ /F3 1 Tf 14.133 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(-espace)-259(v)24(ectoriel)]TJ /F8 1 Tf 7.3012 0 Td [(E)]TJ /F2 1 Tf 1.0482 0 Td [(de)-253(di-)]TJ -36.741 -1.1988 Td [(mension)-265(\257nie.)-427(Nous)-265(utiliserons)-234(\266)469(ev)24(en)30(tuellemen)24(t)-265(une)-271(base)-265(\()]TJ /F8 1 Tf 24.687 0 Td [(e)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.621 79.03 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.928 55.376 Td [(;)-168(e)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.569 79.03 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.289 55.376 Td [(;)-168(:)-168(:)-162(:)-168(;)-168(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.052 79.03 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.524 55.376 Td [(\))-265(de)]TJ /F8 1 Tf 1.9157 0 Td [(E)]TJ /F2 1 Tf 1.0602 0 Td [(qui)-265(p)-30(ermettra)]TJ -33.169 -1.3976 Td [(d'asso)-30(cier)-367(\265)500(a)-361(un)-367(v)24(ecteur)]TJ /F8 1 Tf 10.446 0 Td [(u)]TJ /F2 1 Tf 0.93374 0 Td [(ses)-367(co)-30(ordonn)18(\266)475(ees)-367(\()]TJ /F8 1 Tf 7.6627 0 Td [(u)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.698 77.03 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.398 53.979 Td [(;)-168(u)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.802 77.03 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.861 53.979 Td [(;)-168(:)-168(:)-168(:)-168(;)-162(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.44 77.03 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.205 53.979 Td [(\))-367(de)-361(fa)-6(\273)439(con)-361(que)]TJ /F8 1 Tf 6.6747 0 Td [(u)]TJ /F2 1 Tf 0.8494 0 Td [(=)]TJ /F9 1 Tf 1.0602 0.94578 Td [(X)]TJ /F8 1 Tf 1.6084 -0.94578 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.629 77.03 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.307 53.979 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.776 77.03 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.102 53.979 Td [(.)-367(Une)]TJ -36.771 -1.4036 Td [(application)]TJ /F8 1 Tf 5.0723 0 Td [(f)]TJ /F2 1 Tf 0.84337 0 Td [(de)]TJ /F8 1 Tf 1.241 0 Td [(X)]TJ /F2 1 Tf 1.1446 0 Td [(dans)]TJ /F8 1 Tf 2.247 0 Td [(E)]TJ /F2 1 Tf 1.0422 0 Td [(p)-24(ourra)-246(alors)-216(^)469(etre)-240(repr)30(\266)469(esen)24(t)24(\266)475(ee)-246(par)-240(ses)-246(co)-36(ordonn)24(\266)469(ees)-240(\()]TJ /F8 1 Tf 21.602 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 55.836 75.021 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.464 52.575 Td [(;)-168(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.81 75.021 Td [(2)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.849 52.575 Td [(;)-168(:)-162(:)-168(:)-168(;)-168(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.336 75.021 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 52.575 Td [(\))]TJ -38.777 -1.1988 Td [(o)-24(\265)524(u)-331(les)]TJ /F8 1 Tf 2.8373 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.397 73.306 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.994 51.376 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(son)24(t)-337(les)-337(co)-36(ordonn)24(\266)469(ees)-331(de)]TJ /F8 1 Tf 11.307 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)]TJ -17.482 -1.6205 Td [(Nous)-277(sa)30(v)24(ons)-283(que)]TJ /F8 1 Tf 7.3675 0 Td [(E)]TJ /F2 1 Tf 1.0663 0 Td [(p)-30(eut)-253(^)475(etre)-277(m)24(uni)-277(de)-271(di\256)24(\266)469(eren)24(tes)-277(normes)-277(qui)-271(son)24(t)-253(\266)475(equiv)48(alen)30(tes)-283(en)30(tre)-277(elles)-277(et)]TJ -8.4337 -1.2048 Td [(donc)-349(d)24(\266)469(e\257nissen)24(t)-349(toutes)-355(la)-355(m)30(^)469(eme)-349(top)-36(ologie)-349(sur)]TJ /F8 1 Tf 20.373 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(.)-499(L'espace)]TJ /F8 1 Tf 4.8133 0 Td [(E)]TJ /F2 1 Tf 1.1446 0 Td [(sera)-355(toujours)-355(m)30(uni)-355(de)-355(cette)]TJ -27.127 -1.1988 Td [(top)-30(ologie.)]TJ 0 -1.6205 Td [(Rapp)-30(elons)-451(qu'alors)-445(toute)-445(forme)-445(lin)24(\266)475(eaire)]TJ /F8 1 Tf 18.12 0 Td [(\256)]TJ /F2 1 Tf 1.0843 0 Td [(sur)]TJ /F8 1 Tf 1.7892 0 Td [(E)]TJ /F2 1 Tf 1.241 0 Td [(est)-445(con)24(tin)24(ue)-445(et)-445(v)54(\266)469(eri\257e)-445(une)-445(ma)-54(joration)]TJ /F6 1 Tf -22.235 -1.1988 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(A)]TJ /F6 1 Tf 0.74699 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(k)]TJ /F2 1 Tf 0.83735 0 Td [(p)-30(our)-331(un)]TJ /F8 1 Tf 3.8072 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(con)24(v)24(enable.)]TJ -11.434 -1.6205 Td [(On)-301(sait)-307(par)-301(ailleurs)-307(que)-301(les)-307(v)24(ecteurs)-307(\265)499(a)-301(co)-36(ordonn)24(\266)469(ees)-301(rationnelles)-307(formen)24(t)-301(une)-307(partie)-307(d)24(\266)475(enom-)]TJ 0 -1.2048 Td [(brable)-433(dense)-433(de)]TJ /F8 1 Tf 7.4157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(.)-740(Il)-427(s'ensuit)-433(que)-439(tout)-433(ouv)24(ert)-433(de)]TJ /F8 1 Tf 14.681 0 Td [(E)]TJ /F2 1 Tf 1.2289 0 Td [(est)-433(r)30(\266)469(eunion)-433(de)-433(b)-30(oules)-433(ouv)18(ertes)-433(de)]TJ -24.12 -1.1988 Td [(ra)30(y)24(on)-349(rationnel)-343(et)-349(de)-343(cen)24(tre)-349(\265)500(a)-343(co)-30(ordonn)18(\266)475(ees)-349(rationnelles.)-481(Comme)-343(ces)-349(b)-30(oules)-343(son)18(t)-343(en)-349(quan-)]TJ 0 -1.1988 Td [(tit)30(\266)469(e)-391(d)30(\266)469(enom)24(brable)-385(elles)-391(su\261sen)24(t)-385(p)-30(our)-385(engendrer)-391(une)-385(tribu)-385(qui)-385(p)-30(oss)24(\265)475(ede)-391(tous)-385(les)-391(ouv)24(erts)-391(et)]TJ 0 -1.2048 Td [(qui)-331(est)-337(donc)-337(la)-337(tribu)-331(des)-337(b)-30(or)24(\266)475(eliens)-337(de)]TJ /F8 1 Tf 16.53 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 29.238 Td [(IV.8.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 35.051 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084335 Td [(\266)]TJ 0 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ -0.0060244 -0.084335 Td [(eme.)]TJ /F12 1 Tf 2.7169 0 Td [(Av)24(ec)-349(les)-349(notations)-349(pr)30(\266)469(ec)24(\266)469(eden)30(tes,)-349(p)-30(our)]TJ /F8 1 Tf 16.404 0 Td [(f)]TJ /F12 1 Tf 0.94579 0 Td [(application)-349(de)]TJ /F8 1 Tf 6.5241 0 Td [(X)]TJ /F12 1 Tf 1.253 0 Td [(dans)]TJ /F8 1 Tf 2.3554 0 Td [(E)]TJ /F12 1 Tf 0.79518 0 Td [(,)]TJ -38.886 -1.1988 Td [(les)-307(\266)469(enonc)30(\266)469(es)-331(suiv)48(an)30(ts)-337(son)24(t)-307(\266)469(equiv)54(alen)24(ts)-331(:)]TJ /F1 1 Tf 0.49398 -1.6205 Td [(\(i\))]TJ /F8 1 Tf 1.8795 0 Td [(f)]TJ /F12 1 Tf 0.93374 0 Td [(est)-337(mesurable)-331(\(lorsque)]TJ /F8 1 Tf 10.108 0 Td [(E)]TJ /F12 1 Tf 1.1265 0 Td [(est)-343(m)30(uni)-343(de)-331(sa)-337(tribu)-337(des)-331(b)-30(or)24(\266)475(eliens\))-337(;)]TJ /F1 1 Tf -14.367 -1.6205 Td [(\(ii\))]TJ /F12 1 Tf 2.1988 0 Td [(p)-30(our)-337(toute)-331(forme)-337(lin)24(\266)475(eaire)]TJ /F8 1 Tf 11.289 0 Td [(\256)]TJ /F12 1 Tf 0.64458 0 Td [(,)]TJ /F8 1 Tf 0.94578 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(est)-337(mesurable)-331(;)]TJ /F1 1 Tf -17.476 -1.6205 Td [(\(iii\))]TJ /F12 1 Tf 2.5181 0 Td [(c)24(haque)-337(co)-30(ordonn)24(\266)469(ee)]TJ /F8 1 Tf 8.5904 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.017 41.271 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.452 28.991 Td [(est)-337(mesurable.)]TJ -12.12 -1.6205 Td [(De)-337(m)24(^)475(eme)-337(les)-307(\266)475(enonc)24(\266)469(es)-331(suiv)48(an)30(ts)-337(son)24(t)-307(\266)469(equiv)54(alen)24(ts)-331(:)]TJ /F1 1 Tf -0.11446 -2.0783 Td [(\(iv\))]TJ /F8 1 Tf 2.488 0 Td [(f)]TJ /F12 1 Tf 0.93374 0 Td [(est)-337(mesurable)-331(et)]TJ /F9 1 Tf 7.4578 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F8 1 Tf 0.66265 0 Td [(dm)]TJ /F12 1 Tf 1.7349 0 Td [(est)-337(\257ni)-337(;)]TJ /F1 1 Tf -15.06 -2.1325 Td [(\(v\))]TJ /F12 1 Tf 2.1687 0 Td [(p)-30(our)-337(toute)-331(forme)-337(lin)24(\266)475(eaire)]TJ /F8 1 Tf 11.289 0 Td [(\256)]TJ /F12 1 Tf 0.64458 0 Td [(,)]TJ /F8 1 Tf 0.94578 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)475(egrable)-337(;)]TJ /F1 1 Tf -17.446 -1.6205 Td [(\(vi\))]TJ /F12 1 Tf 2.488 0 Td [(c)24(haque)-337(co)-30(ordonn)24(\266)469(ee)]TJ /F8 1 Tf 8.5904 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.017 30.607 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.452 21.539 Td [(est)-337(in)24(t)24(\266)469(egrable.)]TJ -12.12 -1.6205 Td [(En)-301(outre,)-319(lorsque)-301(les)-283(\266)475(enonc)24(\266)475(es)]TJ /F1 1 Tf 12.867 0 Td [(\(iv\))]TJ /F12 1 Tf 1.8193 0 Td [(,)]TJ /F1 1 Tf 0.59036 0 Td [(\(v\))]TJ /F12 1 Tf 1.8072 0 Td [(et)]TJ /F1 1 Tf 1.1325 0 Td [(\(vi\))]TJ /F12 1 Tf 2.1265 0 Td [(son)24(t)-307(vrais,)-313(on)-307(dit)-307(que)-307(la)-307(fonction)-307(v)24(ectorielle)]TJ /F8 1 Tf -20.343 -1.1988 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)]TJ /F1 1 Tf 1.5663 0 Td [(in)36(t)18(\266)548(egrable)]TJ /F12 1 Tf 4.994 0 Td [(.)-445(Il)-331(y)-337(a)-337(alors)-337(un)-337(unique)-337(v)24(ecteur)]TJ /F8 1 Tf 13.988 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(qui)-337(v)54(\266)469(eri\257e,)-331(p)-30(our)-337(toute)-337(forme)]TJ /F8 1 Tf 12.711 0 Td [(\256)]TJ /F12 1 Tf 0.64458 0 Td [(,)]TJ /F9 1 Tf -21.602 -0.29518 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\256)]TJ /F9 1 Tf 0.63856 0.80723 Td [(\241)]TJ /F8 1 Tf 0.46386 -0.80723 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62651 -0.80723 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)]TJ -24.795 -2.747 Td [(Ce)-337(v)24(ecteur)]TJ /F8 1 Tf 5.006 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(s'app)-30(elle)-337(l')]TJ /F1 1 Tf 4.6446 0 Td [(in)30(t)24(\266)548(egrale)]TJ /F12 1 Tf 4.6928 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(f)]TJ /F12 1 Tf 0.93374 0 Td [(et)-331(se)-337(note)-337(encore)]TJ /F9 1 Tf 7.6747 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.39156 0 Td [(.)]TJ /F1 1 Tf -31.012 -2.3855 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6205 Td [(Comme)-319(toute)-319(forme)-313(lin)24(\266)469(eaire)]TJ /F8 1 Tf 12.524 0 Td [(\256)]TJ /F2 1 Tf 0.95783 0 Td [(sur)]TJ /F8 1 Tf 1.6566 0 Td [(E)]TJ /F2 1 Tf 1.1145 0 Td [(\(de)-319(dimension)-319(\257nie\))-319(est)-319(con)24(tin)24(ue)-319(donc)-313(mesurable,)-325(on)]TJ -16.253 -1.1988 Td [(conclut)-337(que,)-337(si)]TJ /F8 1 Tf 6.6446 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(est)-337(mesurable,)]TJ /F8 1 Tf 6.5723 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(l'est)-307(\266)469(egalemen)24(t.)-445(Cela)-337(prouv)24(e)]TJ /F1 1 Tf 12.765 0 Td [(\(i\))]TJ /F6 1 Tf 1.2169 0 Td [(\))]TJ /F1 1 Tf 1 0 Td [(\(ii\))]TJ /F2 1 Tf 1.5361 0 Td [(.)]TJ -32.741 -1.6205 Td [(P)30(our)-313(c)24(haque)-307(indice)]TJ /F8 1 Tf 8.5843 0 Td [(i)]TJ /F2 1 Tf 0.6506 0 Td [(l'application)]TJ /F8 1 Tf 5.6988 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.819 10.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.482 7.4906 Td [(qui)-307(asso)-24(cie)-313(\265)500(a)-313(tout)-307(v)24(ecteur)]TJ /F8 1 Tf 11.392 0 Td [(u)]TJ /F2 1 Tf 0.87952 0 Td [(sa)-313(co)-30(ordonn)18(\266)475(ee)]TJ /F8 1 Tf 6.4337 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.328 10.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.404 7.4906 Td [(est)-313(une)]TJ -36.072 -1.2048 Td [(forme)-331(lin)24(\266)469(eaire)-337(et)-337(l'on)-337(a)]TJ /F8 1 Tf 10.259 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.009 8.7797 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.693 6.2858 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.216 8.7797 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.651 6.2858 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)-337(il)-331(est)-343(donc)-307(\266)475(eviden)18(t)-337(que)]TJ /F1 1 Tf 10.819 0 Td [(\(ii\))]TJ /F2 1 Tf 1.8675 0 Td [(implique)]TJ /F1 1 Tf 4.0843 0 Td [(\(iii\))]TJ /F2 1 Tf 1.8554 0 Td [(.)]TJ /F5 1 Tf -33.042 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(64)]TJ ET endstream endobj 227 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 226 0 R /ProcSet 2 0 R >> /Contents 225 0 R >> endobj 226 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 228 0 obj << /Length 17854 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(En\257n,)-337(supp)-30(osons)-337(les)]TJ /F8 1 Tf 9.1807 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.466 88.823 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.669 62.22 Td [(mesurables)-337(et)-331(prouv)24(ons)-337(que)]TJ /F8 1 Tf 12.44 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(est)-337(mesurable.)]TJ -23.711 -1.7711 Td [(Munissons)]TJ /F8 1 Tf 4.9458 0 Td [(E)]TJ /F2 1 Tf 1.1988 0 Td [(de)-403(la)-409(norme)]TJ /F6 1 Tf 5.7169 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(k)]TJ /F2 1 Tf 0.89759 0 Td [(=)]TJ /F9 1 Tf 1.1687 0.95181 Td [(X)]TJ /F6 1 Tf 1.6145 -0.95181 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.621 86.288 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.127 60.448 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-656(Nous)-403(a)24(v)24(ons)-403(remarqu)24(\266)469(e)-403(que)-409(les)-403(b)-30(oules)-409(ouv)24(ertes)]TJ -18.072 -1.3434 Td [(engendraien)24(t)-433(la)-427(tribu)-433(des)-433(b)-30(or)30(\266)469(eliens,)-457(il)-433(su\261t)-433(donc)-433(de)-427(mon)24(trer)-433(que)-433(l'image)-433(r)24(\266)475(ecipro)-30(que)-433(de)]TJ 0 -1.1988 Td [(la)-421(b)-30(oule)-421(ouv)18(erte)-421(de)-421(cen)24(tre)]TJ /F8 1 Tf 12.12 0 Td [(a)]TJ /F2 1 Tf 0.95181 0 Td [(et)-421(de)-421(ra)24(y)24(on)]TJ /F8 1 Tf 5.5181 0 Td [(r)]TJ /F2 1 Tf 0.90362 0 Td [(est)-427(mesurable.)-710(Or,)-451(si)]TJ /F8 1 Tf 10.024 0 Td [(a)]TJ /F2 1 Tf 0.95181 0 Td [(a)-421(p)-30(our)-427(co)-30(ordonn)24(\266)469(ees)]TJ -30.47 -1.3494 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.9397 80.719 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6988 56.557 Td [(;)-162(:)-168(:)-168(:)-168(;)-168(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.517 80.719 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.994 56.557 Td [(\),)-349(cette)-349(image)-349(r)24(\266)475(ecipro)-30(que)-349(est)-349(l'ensem)24(ble)-349(des)]TJ /F8 1 Tf 19.428 0 Td [(x)]TJ /F2 1 Tf 0.91566 0 Td [(qui)-343(v)48(\266)475(eri\257en)24(t)]TJ /F9 1 Tf 5.5843 0.95181 Td [(X)]TJ /F6 1 Tf 1.6145 -0.95181 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.81 80.719 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.633 56.557 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.62048 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.75 80.719 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.084 56.557 Td [(j)]TJ /F8 1 Tf 0.57831 0 Td [(<)-295(r)]TJ /F2 1 Tf 1.5542 0 Td [(;)]TJ -38.886 -1.6988 Td [(le)-337(r)30(\266)469(esultat)-337(d)24(\266)469(ecoule)-337(alors)-337(de)-331(la)-337(mesurabilit)24(\266)475(e)-343(de)]TJ /F9 1 Tf 20.145 0.95181 Td [(X)]TJ /F6 1 Tf 1.6145 -0.95181 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.862 78.288 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.41 54.858 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.845 78.288 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.271 54.858 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)]TJ -25.217 -2.1386 Td [(Prouv)30(ons)]TJ /F1 1 Tf 4.3735 0 Td [(\(iv\))]TJ /F6 1 Tf 1.8253 0 Td [(\))]TJ /F1 1 Tf 1 0 Td [(\(v\))]TJ /F2 1 Tf 1.5 0 Td [(.)-433(Remarquons)-301(tout)-301(d'ab)-30(ord)-301(que)-301(la)-295(con)24(tin)24(uit)24(\266)469(e)-301(de)-301(l'application)]TJ /F8 1 Tf 26.765 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)-277(k)]TJ /F8 1 Tf 1.7771 0 Td [(x)]TJ /F6 1 Tf 0.57228 0 Td [(k)]TJ /F2 1 Tf -38.663 -1.1988 Td [(asure)-325(que)]TJ /F6 1 Tf 4.4578 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F2 1 Tf 0.81928 0 Td [(est)-331(mesurable)-325(d)30(\265)469(es)-325(que)]TJ /F8 1 Tf 9.8434 0 Td [(f)]TJ /F2 1 Tf 0.92169 0 Td [(l'est.)-445(Cela)-325(justi\257e)-325(l'utilisation)-325(de)-319(l'in)18(t)30(\266)469(egrale)-331(et)-325(non)]TJ -17.145 -1.7108 Td [(de)-271(l'in)24(t)24(\266)469(egrale)-283(sup)0(\266)475(erieure)-283(dans)-271(l')24(\266)469(enonc)30(\266)469(e)]TJ /F1 1 Tf 16.675 0 Td [(\(iv\))]TJ /F2 1 Tf 1.8253 0 Td [(.)-421(Supp)-30(osons)-277(donc)]TJ /F8 1 Tf 7.9036 0 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(mesurable)-277(et)]TJ /F9 1 Tf 5.7831 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F2 1 Tf 1.6687 0 Td [(\257ni.)]TJ -37.494 -1.6566 Td [(Si)]TJ /F8 1 Tf 1.2048 0 Td [(\256)]TJ /F2 1 Tf 1.012 0 Td [(est)-379(une)-373(forme)-367(lin)24(\266)469(eaire)-373(sur)]TJ /F8 1 Tf 11.633 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(,)-379(on)-373(a)-373(vu)-373(que)]TJ /F8 1 Tf 6.3072 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(est)-379(mesurable.)-560(P)30(ar)-373(ailleurs)-367(on)-373(a)-373(une)]TJ -23.066 -1.6566 Td [(ma)-54(joration)]TJ /F6 1 Tf 5.1988 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.6506 0 Td [(6)]TJ /F8 1 Tf 1.1506 0 Td [(A)]TJ /F6 1 Tf 0.75301 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(k)]TJ /F2 1 Tf 0.89157 0 Td [(v)54(alable)-391(p)-30(our)-391(tout)-391(v)24(ecteur)]TJ /F8 1 Tf 11.62 0 Td [(u)]TJ /F2 1 Tf 0.57831 0 Td [(.)-614(En)-391(particulier)]TJ /F9 1 Tf 7.4157 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ -38.777 -2.3253 Td [(est)-337(ma)-60(jor)24(\266)469(e)-337(par)]TJ /F8 1 Tf 6.7048 0 Td [(A)]TJ /F9 1 Tf 0.92169 1.3615 Td [(Z)]TJ /F6 1 Tf 1 -1.3615 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-331(donc)-337(est)-343(\257ni)-337(et)]TJ /F8 1 Tf 7.5602 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)475(egrable.)]TJ -23.741 -2.1385 Td [(L'implication)]TJ /F1 1 Tf 6.1265 0 Td [(\(v\))]TJ /F6 1 Tf 1.5 0 Td [(\))]TJ /F1 1 Tf 1 0 Td [(\(vi\))]TJ /F2 1 Tf 2.1566 0 Td [(d)24(\266)475(ecoule)-343(de)]TJ /F8 1 Tf 4.8916 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.759 59.935 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.108 42.033 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.966 59.935 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.066 42.033 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(puisque)]TJ /F8 1 Tf 3.6446 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.526 59.935 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.38 42.033 Td [(est)-343(une)-331(forme)-337(lin)24(\266)475(eaire.)]TJ -25.048 -1.6265 Td [(En\257n,)-271(si)-253(les)]TJ /F8 1 Tf 5.1807 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.741 57.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.584 40.406 Td [(son)24(t)-253(in)24(t)24(\266)469(egrables,)-271(elles)-253(son)24(t)-253(en)-253(particulier)-253(mesurables)-253(et)-253(donc)]TJ /F8 1 Tf 25.91 0 Td [(f)]TJ /F2 1 Tf 0.84337 0 Td [(est)-259(mesurable.)]TJ -33.006 -1.6566 Td [(P)30(ar)-361(ailleurs)-367(on)-361(a)-361(la)-361(ma)-60(joration)]TJ /F6 1 Tf 13.982 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(k)]TJ /F11 1 Tf 0.8253 0 Td [(6)]TJ /F9 1 Tf 1.1024 0.95181 Td [(X)]TJ /F6 1 Tf 1.6145 -0.95181 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.457 55.237 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.114 38.75 Td [(j)-240(\242)-240(k)]TJ /F8 1 Tf 1.5361 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.802 55.237 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.452 38.75 Td [(k)]TJ /F2 1 Tf 0.86145 0 Td [(qui)-361(implique)-367(que)]TJ /F9 1 Tf 7.7289 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-361(est)]TJ -37.187 -2.3193 Td [(ma)-54(jor)18(\266)475(e)-337(par)]TJ /F9 1 Tf 5.1446 0.94579 Td [(X)]TJ /F6 1 Tf 1.6145 -0.94579 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.681 51.918 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.386 36.43 Td [(k)]TJ /F9 1 Tf 0.66867 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.647 51.918 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.157 36.43 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(donc)-331(est)-343(\257ni,)-337(ce)-337(qui)-331(prouv)24(e)]TJ /F1 1 Tf 12.873 0 Td [(\(vi\))]TJ /F6 1 Tf 1.8193 0 Td [(\))]TJ /F1 1 Tf 1 0 Td [(\(iv\))]TJ /F2 1 Tf 1.8253 0 Td [(.)]TJ -32.5 -2.4518 Td [(Supp)-24(osons)]TJ /F1 1 Tf 4.994 0 Td [(\(v\))]TJ /F2 1 Tf 1.8976 0 Td [(\(et)-403(donc)]TJ /F1 1 Tf 4.0723 0 Td [(\(iv\))]TJ /F2 1 Tf 2.2229 0 Td [(et)]TJ /F1 1 Tf 1.2289 0 Td [(\(vi\))]TJ /F2 1 Tf 1.8193 0 Td [(\))-403(v)54(\266)469(eri\257)24(\266)475(ee.)-644(Si)]TJ /F8 1 Tf 5.994 0 Td [(a)]TJ /F2 1 Tf 0.92771 0 Td [(est)-403(un)-397(v)24(ecteur)-397(de)]TJ /F8 1 Tf 8.0964 0 Td [(E)]TJ /F2 1 Tf 1.1988 0 Td [(de)-397(co)-30(ordonn)24(\266)469(ees)]TJ /F8 1 Tf -32.452 -1.7108 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3879 45.961 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.1928 32.268 Td [(,)-313(p)-30(our)-313(a)24(v)24(oir)]TJ /F9 1 Tf 5.3916 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\256)]TJ /F9 1 Tf 0.64458 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62651 -0.80723 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-313(p)-30(our)-313(toutes)-319(les)-313(formes)-319(lin)30(\266)469(eaires)]TJ /F8 1 Tf 14.518 0 Td [(\256)]TJ /F2 1 Tf 0.95783 0 Td [(il)-313(faut)-313(et)-313(il)-313(su\261t)]TJ -32.048 -1.7108 Td [(\(par)-313(lin)24(\266)469(earit)24(\266)475(e\))-313(que)-313(l'on)-313(ait)-313(cette)-283(\266)469(egalit)24(\266)475(e)-319(p)-24(our)-313(un)-313(syst)24(\265)475(eme)-307(g)24(\266)469(en)30(\266)469(erateur)-313(de)-307(l'espace)-313(des)-313(formes)]TJ 0 -1.2048 Td [(lin)24(\266)475(eaires.)-692(Les)-415(applications)]TJ /F8 1 Tf 12.054 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.69 41.788 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.699 29.352 Td [(formen)24(t)-415(un)-421(tel)-415(syst)24(\265)475(eme)-415(g)30(\266)469(en)30(\266)469(erateur,)-439(ces)-385(\266)469(egalit)24(\266)469(es)-385(\266)469(equiv)54(alen)24(t)]TJ -13.367 -1.6566 Td [(donc)-331(\265)500(a)]TJ /F8 1 Tf 3.2229 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13 39.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6928 27.695 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.517 39.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.572 27.695 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(ce)-337(qui)-331(\257xe)-337(une)-337(unique)-337(v)54(aleur)-337(p)-30(our)]TJ /F8 1 Tf 15.982 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)]TJ ET 512.64 329.86 m 519.84 329.86 l 519.84 338.29 l 512.64 338.29 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 25.31 Td [(Remarques.)]TJ 0.81325 -2.1687 Td [(1.)]TJ /F2 1 Tf 1.5602 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(in)30(t)24(\266)469(egrable)-337(on)-331(retiendra)-337(la)-331(form)24(ule)-331(:)]TJ /F8 1 Tf 15.765 0 Td [(\256)]TJ /F9 1 Tf 0.64458 1.4096 Td [(\265)]TJ 0.90362 -0.048195 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.38554 1.4096 Td [(\266)]TJ /F2 1 Tf 1.0181 -1.4096 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\256)]TJ /F9 1 Tf 0.64458 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39156 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62048 -0.80723 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)]TJ /F1 1 Tf -37.211 -2.2289 Td [(2.)]TJ /F2 1 Tf 1.6205 0 Td [(Dans)-373(la)-361(propri)24(\266)469(et)24(\266)475(e)]TJ /F1 1 Tf 8.0422 0 Td [(\(iv\))]TJ /F2 1 Tf 2.1867 0 Td [(du)-367(th)24(\266)469(eor)24(\265)469(eme)-361(pr)24(\266)475(ec)24(\266)469(eden)24(t)-367(l')30(\266)469(equiv)54(alence)-373(des)-367(normes)-361(mon)24(tre)-367(que)]TJ -12.663 -1.1988 Td [(la)-331(v)48(alidit)30(\266)469(e)-337(de)-337(cette)-337(propri)24(\266)475(et)24(\266)469(e)-337(ne)-331(d)24(\266)469(ep)-24(end)-337(pas)-337(de)-337(la)-331(norme)-337(c)24(hoisie.)]TJ 0 -2.253 Td [(L'ensem)24(ble)-361(des)-361(fonctions)-367(in)24(t)30(\266)469(egrables)-367(de)]TJ /F8 1 Tf 17.657 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(dans)]TJ /F8 1 Tf 2.3675 0 Td [(E)]TJ /F2 1 Tf 1.1566 0 Td [(se)-361(note)]TJ /F6 1 Tf 3.4518 0 Td [(L)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.672 25.581 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.367 17.461 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)-168(E)]TJ /F2 1 Tf 2.0904 0 Td [(\).)-530(Il)-361(est)-367(imm)24(\266)469(ediat)-367(\(en)]TJ -29.548 -1.1988 Td [(\266)469(etudian)24(t)-319(les)-325(co)-36(ordonn)24(\266)469(ees)-319(des)-325(fonctions\))-331(de)-319(v)24(oir)-325(qu'il)-325(s'agit)-319(d'un)-331(espace)-319(v)24(ectoriel)-325(sur)-325(lequel)]TJ 0.030121 -1.1988 Td 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Tf 0.59639 0 Td [(k)]TJ /F12 1 Tf 0.83133 0 Td [(est)-343(in)30(t)24(\266)469(egrable)-331(et)-337(l'on)-337(a)]TJ /F9 1 Tf -10.12 -0.86145 Td [(\260)]TJ 0 -0.59639 Td [(\260)]TJ 0.72289 1.1084 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.8494 Td [(\260)]TJ 0 -0.59639 Td [(\260)]TJ /F11 1 Tf 0.83133 -0.25301 Td [(6)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F2 1 Tf -8.3614 -3.5723 Td [(65)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 230 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 229 0 R /ProcSet 2 0 R >> /Contents 228 0 R >> endobj 229 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 231 0 obj << /Length 13613 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ 0 -2.4036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6145 Td [(On)-385(sait)-379(que)-385(l'application)]TJ /F8 1 Tf 11.343 0 Td [(u)]TJ /F6 1 Tf 0.92771 0 Td [(7!)-361(k)]TJ /F8 1 Tf 1.8614 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(k)]TJ /F2 1 Tf 0.87952 0 Td [(est)-385(con)24(tin)24(ue)-385(donc)-385(mesurable.)-590(La)-385(mesurabilit)24(\266)475(e)-385(de)]TJ /F6 1 Tf 21.982 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F2 1 Tf -38.663 -1.7169 Td [(en)-421(d)30(\266)469(ecoule.)-704(P)24(ar)-421(ailleurs)-421(l'in)24(t)24(\266)475(egrabilit)24(\266)475(e)-427(de)]TJ /F8 1 Tf 18.789 0 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(assure)-421(par)-421(d)24(\266)469(e\257nition)-421(que)]TJ /F9 1 Tf 11.452 1.3675 Td [(Z)]TJ /F6 1 Tf 1.006 -1.3675 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.8193 0 Td [(est)-421(\257ni,)]TJ -35.849 -1.7108 Td [(l'in)24(t)24(\266)475(egrabilit)24(\266)469(e)-337(de)]TJ /F6 1 Tf 7.4759 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F2 1 Tf 0.83735 0 Td [(en)-331(r)24(\266)475(esulte.)]TJ -9.4096 -1.6145 Td [(La)-397(ma)-60(joration)-397(de)-397(la)-397(norme)-397(de)-397(l'in)24(t)24(\266)469(egrale)-397(est)-403(plus)-391(d)24(\266)469(elicate)-397(\265)500(a)-391(obtenir.)-632(On)-397(p)-30(eut)-403(p)-24(our)-397(cela)]TJ 0 -1.2048 Td 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Tf 0.38554 0.8494 Td [(\260)]TJ 0 -0.60241 Td [(\260)]TJ /F2 1 Tf 0.83735 -0.24699 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F9 1 Tf 0.64458 0.80723 Td [(\241)]TJ 0.6265 0.55422 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F2 1 Tf 0.73494 -0.80723 Td [(=)]TJ /F9 1 Tf 1.0602 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(\256)]TJ /F9 1 Tf 0.64458 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F8 1 Tf 0.6265 -0.80723 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 1.3615 Td [(Z)]TJ /F6 1 Tf 1 -1.3615 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)]TJ -35.735 -2.0723 Td [(Nous)-355(donnerons)-361(plus)-355(loin)-355(une)-355(autre)-361(preuv)24(e)-355(de)-355(cette)-361(ma)-60(joration)-355(qui)-355(ne)-355(fait)-361(pas)-361(in)30(terv)24(enir)]TJ 0 -1.2048 Td [(le)-337(lemme)-337(de)-331(Hahn-Banac)24(h.)]TJ ET 512.64 527.5 m 519.84 527.5 l 519.84 535.93 l 512.64 535.93 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 33.168 Td [(IV.9.)-236(Op)-15(\266)537(erations)-376(classiques)-376(sur)-381(les)-371(mesures.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 38.141 Td [(En)-487(utilisan)30(t)-493(la)-487(prop)-30(osition)]TJ /F1 1 Tf 12.482 0 Td [(IV.2.2.)]TJ /F2 1 Tf 3.8976 0 Td [(caract)24(\266)469(erisan)24(t)-487(les)-493(in)30(t)24(\266)469(egrales)-487(on)-487(obtien)24(t)-487(les)-493(r)30(\266)469(esultats)]TJ -16.38 -1.1988 Td [(suiv)54(an)24(ts)-331(concernan)24(t)-337(certaines)-331(op)-6(\266)475(erations)-337(sur)-337(les)-337(mesures)-337(et)-337(les)-331(in)24(t)24(\266)469(egrales)-331(sup)-6(\266)475(erieures.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 28.274 Td [(IV.9.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 33.894 Td [(1.)-289(Com)30(binaison)-385(lin)24(\266)548(eaire.)]TJ /F2 1 Tf -2.9458 -1.6205 Td [(Soit)-337(\()]TJ /F8 1 Tf 2.4458 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))-343(un)-337(espace)-343(mesurable.)-457(Consid)30(\266)469(erons)-343(une)-337(famille)-337(\257nie)-343(de)-337(mesures)]TJ /F8 1 Tf 28.518 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.198 45.978 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.94 32.274 Td [(sur)-343(\()]TJ /F8 1 Tf 2.0723 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(\))]TJ -38.777 -1.1988 Td [(et)-481(une)-481(famille)-481(de)-481(r)24(\266)475(eels)-487(p)-24(ositifs)]TJ /F8 1 Tf 14.271 0 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.888 44.254 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21 31.075 Td [(index)24(\266)475(ee)-487(par)-481(le)-487(m)30(^)469(eme)-481(ensem)24(ble.)-891(Alors,)-518(l'application)]TJ /F8 1 Tf -15.669 -1.6566 Td [(f)]TJ /F6 1 Tf 0.98795 0 Td [(7!)]TJ /F9 1 Tf 1.3916 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\270)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.172 41.883 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 10.404 30.78 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)-271(dm)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.422 41.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.307 29.418 Td [(est)-403(clairemen)24(t)-403(p)-30(ositiv)24(emen)30(t)-409(homog)24(\265)475(ene)-409(et)-403(d)24(\266)475(enom)24(brablemen)24(t)-403(additiv)30(e)]TJ -8.9759 -1.6566 Td 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Tf 0.9759 0 Td [(est)-343(la)]TJ /F1 1 Tf 2.6747 0 Td [(densit)18(\266)548(e)]TJ /F2 1 Tf 3.8916 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(\256)-6(m)]TJ /F2 1 Tf 1.8554 0 Td [(par)-337(rapp)-30(ort)-337(\265)500(a)]TJ /F8 1 Tf 6.259 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ /F5 1 Tf -35.741 -2.8614 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.307 0 Td [(66)]TJ ET endstream endobj 233 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 232 0 R /ProcSet 2 0 R >> /Contents 231 0 R >> endobj 232 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R >> endobj 234 0 obj << /Length 15478 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10018(IV.)-385(Construction)-391(de)-385(l'in)30(t)24(\266)548(egrale)]TJ /F2 1 Tf 0 -2.4036 Td [(Une)-325(fonction)-325(mesurable)]TJ /F8 1 Tf 10.645 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(est)-325(alors)-325(in)24(t)24(\266)469(egrable)-319(p)-30(our)-325(cette)-325(mesure)]TJ /F8 1 Tf 16.741 0 Td [(\256)-6(m)]TJ /F2 1 Tf 1.8434 0 Td [(si)-325(et)-325(seulemen)30(t)-325(si)]TJ /F8 1 Tf 7.7771 0 Td [(\256)-6(f)]TJ /F2 1 Tf -37.922 -1.6566 Td [(est)-337(in)24(t)24(\266)475(egrable)-337(p)-30(our)]TJ /F8 1 Tf 8.5723 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(et)-331(l'on)-337(a)]TJ /F9 1 Tf 3.9458 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(d\256)-6(m)]TJ /F2 1 Tf 3.0843 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\256)-6(f)-277(dm)]TJ /F2 1 Tf 2.8072 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 47.51 Td [(IV.9.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 56.954 Td [(3.)-289(Image)-385(directe.)]TJ /F2 1 Tf -2.9458 -1.6024 Td [(Soit)]TJ /F8 1 Tf 2.0301 0 Td [(')]TJ /F2 1 Tf 0.96386 0 Td [(une)-307(application)-313(mesurable)-313(d'un)-313(espace)-307(mesur)24(\266)475(e)-313(\()]TJ /F8 1 Tf 20.825 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td 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1 Tf 0.5 0 Td [(')]TJ /F2 1 Tf 0.98795 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(p)-30(our)]TJ /F8 1 Tf 8.5723 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(et)-337(l'on)-337(a)]TJ /F9 1 Tf 3.9458 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(d')]TJ /F2 1 Tf 1.9398 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(')-168(dm)]TJ /F2 1 Tf 2.2169 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 32.218 Td [(IV.9.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 38.623 Td [(4.)-289(Restriction.)]TJ /F2 1 Tf -2.9458 -1.5964 Td [(Soien)24(t)-337(\()]TJ /F8 1 Tf 3.4217 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))-343(un)-337(espace)-337(mesurable)-337(et)]TJ /F8 1 Tf 11.199 0 Td [(Y)]TJ /F2 1 Tf 1.1386 0 Td [(une)-337(partie)]TJ /F6 1 Tf 4.7892 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable)-343(de)]TJ /F8 1 Tf 6.4096 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(.)-451(Nous)-337(a)24(v)24(ons)-337(d)24(\266)469(e\257ni)]TJ -30.747 -1.2048 Td [(\(cf.)]TJ /F1 1 Tf 1.8976 0 Td [(I)-30(I)-36(I.2.1.)]TJ /F2 1 Tf 3.4819 0 Td [(\))-349(une)-343(tribu)-349(induite)]TJ /F6 1 Tf 8.5542 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.345 51.047 Td [(Y)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.843 35.822 Td [(sur)]TJ /F8 1 Tf 1.6867 0 Td [(Y)]TJ /F2 1 Tf 1.1506 0 Td [(don)24(t)-349(les)-319(\266)469(el)24(\266)475(emen)24(ts)-349(son)24(t)-349(les)-343(parties)-349(de)]TJ /F8 1 Tf 16.145 0 Td [(Y)]TJ /F2 1 Tf 1.1506 0 Td [(qui)-343(son)18(t)]TJ /F6 1 Tf -35.645 -1.1988 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurables.)]TJ -0.79518 -1.6024 Td [(Si)]TJ /F8 1 Tf 1.2229 0 Td [(g)]TJ /F2 1 Tf 0.90361 0 Td [(est)-391(une)-391(application)-397(de)]TJ /F8 1 Tf 10.181 0 Td [(Y)]TJ /F2 1 Tf 1.1928 0 Td [(dans)]TJ ET 0.504 w 253.66 404.64 m 262.3 404.64 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 21.223 33.021 Td [(R)]TJ /F2 1 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[(-)24(\266)475(etag)24(\266)469(ee)-445(et,)-475(par)-445(passage)-451(\265)499(a)-445(la)-445(limite)-445(simple,)-475(on)-445(conclut)-451(que)]TJ /F8 1 Tf 26.452 0 Td [(g)]TJ /F2 1 Tf 0.95783 0 Td [(est)]TJ /F6 1 Tf -37.934 -1.2048 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4052 41.883 Td [(Y)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5663 29.418 Td [(-mesurable)-331(si)-331(et)-331(seulemen)24(t)-331(si)]TJ /F9 1 Tf 12.873 0 Td [(e)]TJ /F8 1 Tf -0.0060234 0 Td [(g)]TJ /F2 1 Tf 0.83735 0 Td [(est)]TJ /F6 1 Tf 1.5602 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable)-331(\(on)-337(p)-30(eut)-331(aussi)-331(utiliser)-331(la)-325(d)24(\266)469(e\257nition)-331(des)]TJ -17.295 -1.1988 Td [(fonctions)-337(mesurables)-337(p)-30(our)-331(v)24(oir)-337(cela\).)]TJ 0 -1.6024 Td [(Soit)]TJ /F8 1 Tf 2.0241 0 Td [(m)]TJ /F2 1 Tf 1.1747 0 Td [(une)-307(mesure)-301(sur)-307(\()]TJ /F8 1 Tf 7.253 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\).)-439(On)-301(p)-30(eut)-307(encore)-301(utiliser)-307(la)-301(prop)-30(osition)]TJ /F1 1 Tf 17.711 0 Td [(IV.2.2.)]TJ /F2 1 Tf 3.7169 0 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.1627 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F9 1 Tf -37.717 -0.3494 Td [(Z)]TJ 1.006 -1.3614 Td [(e)]TJ /F8 1 Tf -0.0060248 0 Td [(g)-204(dm)]TJ /F2 1 Tf 2.0783 0 Td [(,)-325(on)-331(conclut)-325(que,)-331(p)-30(our)]TJ /F8 1 Tf 9.9699 0 Td [(g)]TJ /F2 1 Tf 0.83735 0 Td [(mesurable)-325(p)-30(ositiv)30(e)-331(sur)]TJ /F8 1 Tf 10.084 0 Td [(Y)]TJ /F2 1 Tf 0.8012 0 Td [(,)]TJ /F9 1 Tf 0.60241 1.3614 Td [(Z)]TJ 1.012 -1.3614 Td [(e)]TJ /F8 1 Tf -0.0060234 0 Td [(g)-198(dm)]TJ /F2 1 Tf 2.4036 0 Td [(est)-325(l'in)24(t)24(\266)469(egrale)-331(de)]TJ /F8 1 Tf 7.5121 0 Td [(g)]TJ /F2 1 Tf 0.83735 0 Td [(p)-30(our)]TJ -37.133 -1.7108 Td [(une)-379(mesure)-379(sur)-379(\()]TJ /F8 1 Tf 7.4819 0 Td [(Y)-60(;)]TJ /F6 1 Tf 1.0783 0 Td [(T)]TJ /F7 1 Tf 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0 0 Tm 31.494 13.954 Td [(.)-524(En)-361(outre)-361(on)-361(a)]TJ /F9 1 Tf 7.3133 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.328 18.668 Td [(Y)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.054 13.954 Td [(f)]TJ ET 488.12 162.03 m 488.12 174.06 l S BT /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.629 19.331 Td [(Y)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.994 13.954 Td [(dm)]TJ /F2 1 Tf 1.7229 0 Td [(=)]TJ /F9 1 Tf -38.386 -1.006 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\242)]TJ /F2 1 Tf 0.5 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.75 16.366 Td [(Y)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0723 11.581 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(,)-337(cette)-337(in)30(t)24(\266)469(egrale)-337(se)-337(notera)-337(encore)]TJ /F9 1 Tf 14.548 1.3675 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.595 15.271 Td [(Y)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.265 11.581 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1687 0 Td [(.)]TJ /F1 1 Tf -23.102 -2.7048 Td [(Remarque.)]TJ /F2 1 Tf 6.4759 0 Td [(La)-313(restriction)-319(de)-313(la)-313(mesure)-319(de)-313(Leb)-30(esgue)-313(\265)500(a)-313(une)-313(partie)-319(mesurable)]TJ /F8 1 Tf 27.867 0 Td [(A)]TJ /F2 1 Tf 1.0602 0 Td [(s'app)-30(elle)]TJ -35.404 -1.1988 Td [(encore)-337(la)-331(mesure)-343(de)-331(Leb)-36(esgue)-331(sur)]TJ /F8 1 Tf 14.964 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)]TJ 3.2048 -3.7892 Td [(67)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 236 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 235 0 R /ProcSet 2 0 R >> /Contents 234 0 R >> endobj 235 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R >> endobj 237 0 obj << /Length 229 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)]TJ /F5 1 Tf 0 -60.735 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(68)]TJ ET endstream endobj 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Td [(dm)]TJ /F12 1 Tf 1.6506 0 Td [(est)-253(ma)-60(jor)24(\266)469(ee)-253(\(resp.)-415(minor)18(\266)475(ee\))-253(\(par)-253(un)-253(nom)24(bre)-253(\257ni\))-253(et)-246(alors)]TJ /F8 1 Tf 24.542 0 Td [(f)]TJ /F12 1 Tf 0.84337 0 Td [(est)-259(in)30(t)24(\266)469(egrable)]TJ -33.373 -2.3193 Td [(et)-337(l'on)-331(a)-337(:)]TJ /F9 1 Tf 5.2229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.233 51.349 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.759 36.033 Td [(dm)]TJ /F6 1 Tf 1.6747 0 Td [(\241)168(!)]TJ /F9 1 Tf 1.8916 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F12 1 Tf 2.494 0 Td [(;)]TJ -14.488 -2.6325 Td [({)-331(ou)-337(bien)]TJ /F9 1 Tf 4.3916 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.043 47.581 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.922 33.4 Td [(dm)]TJ /F12 1 Tf 1.7349 0 Td [(tend)-337(v)24(ers)]TJ /F2 1 Tf 4.3434 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F12 1 Tf 1.3313 0 Td [(\(resp.)]TJ /F6 1 Tf 2.9036 0 Td [(\2411)]TJ /F12 1 Tf 1.7771 0 Td [(\))-337(et)-337(alors)]TJ /F8 1 Tf 4.2892 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(n'est)-337(pas)-337(in)24(t)24(\266)475(egrable.)]TJ /F1 1 Tf -24.675 -2.1747 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Supp)-24(osons)-481(tout)-475(d'ab)-36(ord)-475(la)-475(suite)-481(croissan)30(te)-481(partout)-481(et)-475(les)-481(fonctions)]TJ -8.8193 -1.6566 Td [(p)-30(ositiv)30(es.)-415(Alors)]TJ /F8 1 Tf 7 0 Td [(f)]TJ /F2 1 Tf 0.83133 0 Td [(est)-240(p)-30(ositiv)30(e)-240(et)-234(la)-234(propri)18(\266)475(et)24(\266)469(e)-234(de)-234(Bepp)-36(o)-234(Levi)-234(assure)]TJ /F9 1 Tf 20.62 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4398 0 Td [(=)-283(lim)]TJ /F9 1 Tf 2.6145 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.147 42.099 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.042 29.569 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ -38.386 -2.3193 Td [(sup)]TJ /F9 1 Tf 1.6687 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.155 38.78 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.2048 27.25 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-445(Le)-337(r)24(\266)475(esultat)-337(en)-337(d)24(\266)475(ecoule)-337(imm)18(\266)475(ediatemen)24(t.)]TJ -5.2711 -1.9699 Td [(Supp)-24(osons)-439(main)18(tenan)24(t)-439(la)-439(suite)-439(croissan)24(te)-439(presque)-439(partout.)-765(P)30(ar)-439(h)30(yp)-36(oth)24(\265)469(ese)-439(il)-439(existe)-439(une)]TJ 0 -1.1988 Td [(partie)-433(mesurable)-433(et)-427(n)24(\266)469(egligeable)]TJ /F8 1 Tf 14.289 0 Td [(A)]TJ /F2 1 Tf 1.1807 0 Td [(telle)-433(que,)-457(p)-30(our)]TJ /F8 1 Tf 6.988 0 Td [(x)-554(=)]TJ /F6 1 Tf 1.006 0 Td [(2)]TJ /F8 1 Tf 1.1084 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-457(la)-427(suite)-433(des)]TJ /F8 1 Tf 6.259 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.517 34.245 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.946 24.081 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-433(est)-433(d)24(\266)469(e\257nie,)]TJ -33.572 -1.1988 Td [(croissan)30(te)-373(et)-367(a)-367(donc)-367(une)-367(limite)-367(\()24(\266)475(ev)24(en)24(tuellemen)30(t)-367(in\257nie\).)-548(Cela)-373(prouv)30(e)-373(que)]TJ /F8 1 Tf 32.56 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(=)-331(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.129 32.461 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.867 22.882 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.612 32.53 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 22.882 Td [(est)]TJ -37.934 -1.1988 Td [(d)30(\266)469(e\257nie)-337(en)-331(dehors)-337(de)]TJ /F8 1 Tf 9.012 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(donc)-337(presque)-331(partout.)]TJ -10.096 -1.512 Td [(En)-277(outre,)-289(c)24(haque)]TJ /F8 1 Tf 7.6325 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.25 28.65 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.277 20.171 Td [(est)-277(\265)500(a)-277(v)54(aleurs)-283(\257nies)-277(en)-277(dehors)-283(d'un)-277(ensem)18(ble)-277(mesurable)-271(n)24(\266)469(egligeable)]TJ /F8 1 Tf 28.639 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.5 28.65 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 20.171 Td [(.)]TJ -38.886 -1.1988 Td [(La)-295(r)24(\266)475(eunion)]TJ /F8 1 Tf 4.988 0 Td [(B)]TJ /F2 1 Tf 1.1024 0 Td [(des)]TJ /F8 1 Tf 1.6867 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.845 26.935 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.705 18.973 Td [(est)-301(encore)-295(mesurable)-295(n)30(\266)469(egligeable)-295(et)-295(l'on)-295(v)24(oit)-295(que,)-307(quitte)-289(\265)500(a)-295(ann)24(uler)-295(les)]TJ -9.3735 -1.1988 Td [(fonctions)-367(sur)]TJ /F8 1 Tf 6.006 0 Td [(A)]TJ /F6 1 Tf 0.99398 0 Td [([)]TJ /F8 1 Tf 0.91566 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(,)-379(on)-373(p)-24(eut)-373(se)-373(ramener)-373(au)-367(cas)-373(o)-30(\265)530(u)-373(toutes)-373(les)-373(fonctions)]TJ /F8 1 Tf 22.946 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.655 25.219 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.404 17.774 Td [(son)18(t)-373(\265)500(a)-367(v)54(aleurs)]TJ -33.072 -1.2048 Td [(\257nies)-325(et)-325(la)-319(suite)-325(croissan)24(te)-319(partout.)-445(Alors)-325(la)-319(suite)-325(des)-325(fonctions)]TJ /F8 1 Tf 27.849 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.172 23.495 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.482 16.569 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.56 23.495 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 36.771 16.569 Td [(\241)]TJ /F8 1 Tf 0.97591 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.716 23.495 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.006 16.569 Td [(est)-325(une)-325(suite)]TJ -33.675 -1.6566 Td [(croissan)30(te)-403(partout)-409(de)-397(fonctions)-403(in)24(t)24(\266)475(egrables)-403(p)-30(ositiv)24(es)-397(a)24(v)24(ec)]TJ /F9 1 Tf 25.602 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.379 21.125 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.12 14.912 Td [(dm)]TJ /F2 1 Tf 1.7892 0 Td [(=)]TJ /F9 1 Tf 1.1687 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.759 21.125 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.277 14.912 Td [(dm)]TJ /F6 1 Tf 1.6687 0 Td [(\241)]TJ /F9 1 Tf 1.0422 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 60.793 21.125 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.096 14.912 Td [(dm)]TJ /F2 1 Tf -37.765 -2.3193 Td [(et)]TJ /F8 1 Tf 1.1867 0 Td [(g)]TJ /F2 1 Tf 0.8253 0 Td [(=)-313(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.06 17.728 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.367 12.593 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.526 17.806 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.705 12.593 Td [(=)]TJ /F8 1 Tf 1.0904 0 Td [(f)]TJ /F6 1 Tf 0.83735 0 Td [(\241)]TJ /F8 1 Tf 1.012 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 21.664 17.806 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.584 12.593 Td [(.)-512(Comme)]TJ /F9 1 Tf 4.4759 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.845 17.806 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.169 12.593 Td [(dm)]TJ /F2 1 Tf 1.753 0 Td [(est)-361(\257nie,)-361(la)-355(suite)-355(des)]TJ /F9 1 Tf 9.3554 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.733 17.806 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.464 12.593 Td [(dm)]TJ /F2 1 Tf 1.753 0 Td [(est)-361(ma)-60(jor)24(\266)469(ee)-355(si)-361(et)]TJ -31.886 -2.3253 Td [(seulemen)30(t)-295(si)-289(la)-289(suite)-295(des)]TJ /F9 1 Tf 10.663 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.026 14.478 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.199 10.268 Td [(dm)]TJ /F2 1 Tf 1.6867 0 Td [(l'est,)]TJ /F8 1 Tf 2.3614 0 Td [(f)]TJ /F2 1 Tf 0.88554 0 Td [(est)-295(in)24(t)24(\266)475(egrable)-289(si)-295(et)-289(seulemen)24(t)-289(si)]TJ /F8 1 Tf 13.765 0 Td [(g)]TJ /F2 1 Tf 0.8012 0 Td [(l'est)-289(et)]TJ /F9 1 Tf 3.1928 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.655 14.478 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.096 10.268 Td [(dm)]TJ /F2 1 Tf -37.765 -2.3253 Td [(tend)-385(v)24(ers)]TJ /F9 1 Tf 4.4337 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5422 0 Td [(si)-385(et)-379(seulemen)24(t)-379(si)]TJ /F9 1 Tf 8.012 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.621 11.15 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.506 7.9424 Td [(dm)]TJ /F2 1 Tf 1.7831 0 Td [(tend)-385(v)24(ers)]TJ /F9 1 Tf 4.4337 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F2 1 Tf 2.0783 0 Td [(.)-584(Le)-385(r)30(\266)469(esultat)-385(d)30(\266)469(ecoule)-385(donc)]TJ -27.47 -1.6566 Td [(du)-337(cas)-331(pr)24(\266)469(ec)24(\266)475(eden)24(t.)]TJ 18.916 -2.3976 Td [(69)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 242 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 241 0 R /ProcSet 2 0 R >> /Contents 240 0 R >> endobj 241 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F12 35 0 R /F15 68 0 R >> endobj 243 0 obj << /Length 16761 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ /F2 1 Tf 0 -2.4036 Td [(Le)-271(cas)-271(de)-271(la)-271(suite)-271(d)24(\266)469(ecroissan)30(te)-277(presque)-271(partout)-271(se)-277(ram)30(\265)469(ene)-277(imm)24(\266)469(ediatemen)30(t)-277(au)-271(cas)-271(de)-271(la)-271(suite)]TJ 0 -1.1988 Td [(croissan)30(te)-337(en)-337(consid)30(\266)469(eran)30(t)-337(les)-337(fonctions)]TJ /F6 1 Tf 16.958 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.707 87.107 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.102 61.021 Td [(.)]TJ ET 512.64 728.1 m 519.84 728.1 l 519.84 736.52 l 512.64 736.52 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 58.268 Td [(Remarque.)]TJ /F2 1 Tf 6.8133 0 Td [(Dans)-475(l')30(\266)469(enonc)24(\266)475(e)-481(pr)30(\266)469(ec)24(\266)475(eden)24(t)-475(on)-475(v)24(oit)-475(que)-481(lorsque)]TJ /F9 1 Tf 20.572 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.948 83.176 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.916 58.268 Td [(dm)]TJ /F2 1 Tf 1.8735 0 Td [(est)-475(b)-30(orn)24(\266)475(ee,)]TJ /F8 1 Tf 5.4096 0 Td [(f)]TJ /F2 1 Tf 1.0663 0 Td [(est)]TJ -37.934 -1.7108 Td [(in)24(t)24(\266)475(egrable)-240(donc)-246(\257nie)-240(presque)-246(partout.)-421(P)30(ar)-240(cons)24(\266)469(equen)24(t)-246(la)-240(suite)-240(des)]TJ /F8 1 Tf 28.229 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.724 80.719 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.596 56.557 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-246(est)-246(presque)-240(partout)]TJ -30.223 -1.1988 Td [(con)30(v)24(ergen)18(te)-331(\(v)24(ers)-337(une)-331(limite)-337(\257nie\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 42.143 Td [(V.2.)-241(Le)-376(th)20(\266)542(eor)15(\265)542(eme)-381(de)-376(con)30(v)30(ergence)-381(domin)25(\266)537(ee)-381(\(de)-371(Leb)-35(esgue\).)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 48.653 Td [(Commen)-6(\273)439(cons)-415(par)-421(la)-415(v)24(ersion)-421(de)-415(ce)-421(th)24(\266)469(eor)24(\265)469(eme)-415(concernan)24(t)-415(les)-421(fonctions)-421(n)30(um)24(\266)469(eriques)-415(\(i.e.)-698(\265)500(a)]TJ 0 -1.1988 Td [(v)54(aleurs)-337(dans)]TJ ET 0.504 w 131.98 577.08 m 140.62 577.08 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 11.042 47.454 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 36.736 Td [(V.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7771 44.039 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084335 Td [(\266)]TJ -0.0060244 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ 0 -0.084335 Td [(eme.)]TJ /F12 1 Tf 3.1325 0 Td [(Soien)24(t)-481(une)-481(suite)-481(de)-481(fonctions)-481(n)24(um)24(\266)475(eriques)-481(in)24(t)24(\266)475(egrables)]TJ /F8 1 Tf 24.259 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.103 62.806 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 41.627 44.039 Td [(et)-481(une)]TJ -36.295 -1.1988 Td [(fonction)-337(n)24(um)24(\266)475(erique)]TJ /F8 1 Tf 8.753 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(d)24(\266)475(e\257nie)-337(presque)-337(partout.)-445(On)-337(fait)-337(les)-337(h)24(yp)-30(oth)24(\265)469(eses)-337(:)]TJ /F1 1 Tf -9.1928 -1.8614 Td [(\(i\))]TJ /F8 1 Tf 1.8795 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.724 58.426 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.741 40.979 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(presque)-337(partout)-337(;)]TJ /F1 1 Tf -11.542 -1.8675 Td [(\(ii\))]TJ /F12 1 Tf 2.1747 0 Td [(il)-319(y)-325(a)-325(une)-319(fonction)-331(in)30(t)24(\266)469(egrable)]TJ /F8 1 Tf 12.904 0 Td [(g)]TJ /F12 1 Tf 0.83735 0 Td [(qui)-319(domine)-331(les)]TJ /F8 1 Tf 6.6084 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.819 55.762 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 29.066 39.111 Td [(,)-325(c'est-)0(\265)500(a-dire)-325(que,)-331(p)-30(our)-319(tout)]TJ /F8 1 Tf 12.349 0 Td [(n)]TJ /F12 1 Tf 0.60241 0 Td [(,)-325(on)-319(a)]TJ /F6 1 Tf -36.687 -1.1988 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7241 54.038 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6446 37.912 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(presque)-337(partout.)]TJ -6.4699 -2.3735 Td [(Alors)]TJ /F8 1 Tf 2.6446 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(et)-337(l'on)-337(a)-331(:)]TJ /F9 1 Tf 12.211 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.362 50.642 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.253 35.539 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F6 1 Tf 1.6807 0 Td [(\241)168(!)]TJ /F2 1 Tf 1.8855 0 Td [(0)]TJ /F12 1 Tf 2.1747 0 Td [(et)]TJ /F9 1 Tf 2.1747 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.233 50.642 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.813 35.539 Td [(dm)]TJ /F6 1 Tf 1.6807 0 Td [(\241)168(!)]TJ /F9 1 Tf 1.8855 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F12 1 Tf 2.5 0 Td [(.)]TJ /F1 1 Tf -37.548 -2.747 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(T)78(out)-379(d'ab)-30(ord)-379(on)-379(p)-30(eut)-379(trouv)24(er)-379(des)-385(ensem)24(bles)-379(mesurables)-379(n)24(\266)475(egligeables)]TJ /F8 1 Tf -8.8193 -1.2048 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)]TJ /F8 1 Tf 0.59639 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.638 44.995 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2952 31.587 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(C)]TJ /F2 1 Tf 1.1024 0 Td [(tels)-313(que,)-325(en)-313(dehors)-319(de)]TJ /F8 1 Tf 9.7349 0 Td [(A)]TJ /F2 1 Tf 1.0602 0 Td [(les)]TJ /F8 1 Tf 1.4337 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.293 44.995 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.813 31.587 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-319(et)]TJ /F8 1 Tf 1.8554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-313(son)18(t)-319(d)30(\266)469(e\257nis)-319(et)]TJ /F8 1 Tf 7.0783 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.164 44.995 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.295 31.587 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-319(tend)-319(v)24(ers)]TJ /F8 1 Tf 5.012 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\),)]TJ -38.494 -1.1988 Td [(en)-289(dehors)-301(de)]TJ /F8 1 Tf 5.7169 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.897 43.271 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.645 30.388 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.802 43.271 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.681 30.388 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(et)]TJ /F8 1 Tf 1.8072 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(son)18(t)-295(d)30(\266)469(e\257nis)-301(et)]TJ /F6 1 Tf 6.994 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.75 43.271 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.223 30.388 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.56024 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\),)-301(en)-295(dehors)-295(de)]TJ /F8 1 Tf 6.6867 0 Td [(C)-367(g)]TJ /F2 1 Tf 1.5904 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(est)-295(d)24(\266)469(e\257ni)]TJ -34.572 -1.1988 Td [(et)-307(\257ni.)-445(La)-313(r)30(\266)469(eunion)]TJ /F8 1 Tf 8.2711 0 Td [(D)]TJ /F2 1 Tf 1.1627 0 Td [(de)-313(tous)-307(ces)-313(ensem)24(bles)-313(est)-313(mesurable)-313(n)30(\266)469(egligeable)-307(et,)-319(quitte)-307(\265)500(a)-313(mo)-30(di\257er)]TJ -9.4337 -1.1988 Td [(toutes)-403(les)-397(fonctions)-403(sur)-403(un)-403(ensem)24(ble)-397(n)24(\266)469(egligeable)-397(en)-397(les)-403(ann)24(ulan)24(t)-403(sur)]TJ /F8 1 Tf 30.554 0 Td [(D)]TJ /F2 1 Tf 0.85542 0 Td [(,)-415(on)-403(v)24(oit)-403(que)-403(l'on)]TJ -31.41 -1.2048 Td [(p)-30(eut)-403(supp)-30(oser)-403(toutes)-403(les)-403(fonctions)-403(d)24(\266)475(e\257nies)-409(partout,)-421(toutes)-409(les)-403(h)30(yp)-36(oth)24(\265)469(eses)-403(faites)-403(r)24(\266)475(ealis)24(\266)469(ees)]TJ 0 -1.1988 Td [(partout)-337(et)]TJ /F8 1 Tf 4.7831 0 Td [(g)]TJ /F2 1 Tf 0.84337 0 Td [(\265)500(a)-337(v)54(aleurs)-337(\257nies.)]TJ -5.6265 -1.8614 Td [(En)-361(particulier)-361(la)-361(ma)-60(joration)]TJ /F6 1 Tf 12.771 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.009 33.737 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.416 23.726 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.60241 0 Td [(6)]TJ /F8 1 Tf 1.1024 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-361(donne)-361(en)-361(passan)24(t)-367(\265)500(a)-361(la)-361(limite)]TJ /F6 1 Tf 13.566 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.60241 0 Td [(6)]TJ /F8 1 Tf 1.1024 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-367(et)]TJ -37.578 -1.1988 Td [(prouv)30(e)-253(que)-253(les)]TJ /F8 1 Tf 6.3373 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.405 32.021 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.952 22.527 Td [(et)]TJ /F8 1 Tf 1.0843 0 Td [(f)]TJ /F2 1 Tf 0.84337 0 Td [(son)24(t)-253(\265)499(a)-253(v)54(aleurs)-253(\257nies.)-415(P)24(ar)-246(ailleurs)]TJ /F8 1 Tf 14.181 0 Td [(f)]TJ /F2 1 Tf 0.8494 0 Td [(limite)-246(simple)-253(des)]TJ /F8 1 Tf 7.4277 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.129 32.021 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.62 22.527 Td [(est)-253(mesurable)]TJ -33.289 -1.6566 Td [(et)]TJ /F9 1 Tf 1.1687 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.7289 0 Td [(est)-343(ma)-54(jor)18(\266)475(ee)-337(par)]TJ /F9 1 Tf 7.1566 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)-204(dm)]TJ /F2 1 Tf 2.4096 0 Td [(donc)-337(est)-337(\257nie.)-451(Il)-331(s'ensuit)-337(que)]TJ /F8 1 Tf 12.97 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)475(egrable.)]TJ -29.687 -2.3735 Td [(Les)-283(ma)-60(jorations)-289(de)]TJ /F6 1 Tf 8.5241 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.56024 0 Td [(et)-289(des)]TJ /F6 1 Tf 2.8012 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27 26.254 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.41 18.497 Td [(j)]TJ /F2 1 Tf 0.56627 0 Td [(par)]TJ /F8 1 Tf 1.7349 0 Td [(g)]TJ /F2 1 Tf 0.79518 0 Td [(asuren)18(t)]TJ /F6 1 Tf 3.494 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.72289 0 Td [(\241)]TJ /F8 1 Tf 0.90361 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.638 26.254 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.94 18.497 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(.)-427(La)-295(suite)-283(des)-289(2)]TJ /F8 1 Tf 6.6506 0 Td [(g)]TJ /F6 1 Tf 0.63855 0 Td [(\241)-126(j)]TJ /F8 1 Tf 1.1807 0 Td [(f)]TJ /F6 1 Tf 0.72892 0 Td [(\241)]TJ /F8 1 Tf 0.90361 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.328 26.254 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.705 18.497 Td [(j)]TJ /F2 1 Tf 0.56024 0 Td [(est)]TJ -37.934 -1.2048 Td [(alors)-361(une)-355(suite)-361(de)-355(fonctions)-355(mesurables)-361(p)-30(ositiv)30(es)-361(qui)-355(tend)-361(simplemen)24(t)-361(v)24(ers)-355(2)]TJ /F8 1 Tf 33.602 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(.)-512(Le)-361(lemme)]TJ -34.114 -1.1988 Td [(de)-331(F)78(atou)-337(\(cf.)]TJ /F1 1 Tf 6.0422 0 Td [(IV.4.1.)]TJ /F2 1 Tf 3.4157 0 Td [(\))-337(p)-30(ermet)-337(donc)-337(d')24(\266)475(ecrire)]TJ /F9 1 Tf -5.0723 -0.3494 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(g)-204(dm)]TJ /F2 1 Tf 2.3614 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(lim)-168(inf)-78(\(2)]TJ /F8 1 Tf 3.6627 0 Td [(g)]TJ /F6 1 Tf 0.73494 0 Td [(\241)-222(j)]TJ /F8 1 Tf 1.2771 0 Td [(f)]TJ /F6 1 Tf 0.81927 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.802 20.366 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.163 14.382 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(lim)-174(inf)]TJ /F9 1 Tf 2.9398 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83735 -1.3614 Td [(\(2)]TJ /F8 1 Tf 0.88554 0 Td [(g)]TJ /F6 1 Tf 0.73494 0 Td [(\241)-228(j)]TJ /F8 1 Tf 1.2831 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.543 20.366 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.265 14.382 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.7349 0 Td [(.)]TJ -34.5 -2.988 Td [(En)-307(m)24(ultiplian)18(t)-313(par)]TJ /F6 1 Tf 8.4458 0 Td [(\241)]TJ /F2 1 Tf 0.78313 0 Td [(1)-307(on)-313(en)-307(tire)]TJ /F6 1 Tf 5.6084 0 Td [(\241)]TJ /F9 1 Tf 0.77711 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(g)-204(dm)]TJ /F11 1 Tf 2.3554 0 Td [(>)]TJ /F2 1 Tf 1.0602 0 Td [(lim)-168(sup)]TJ /F9 1 Tf 3.2289 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83133 -1.3614 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.77109 0 Td [(\241)]TJ /F8 1 Tf 0.95181 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.948 16.09 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.349 11.394 Td [(j)-174(\241)]TJ /F2 1 Tf 1.4096 0 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.7108 0 Td [(et,)-313(en)-313(a)-54(joutan)18(t)]TJ /F9 1 Tf -32.705 -0.95783 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(g)-204(dm)]TJ /F2 1 Tf 2.4277 0 Td [(qui)-355(est)-355(\257ni,)-361(0)]TJ /F11 1 Tf 6.1265 0 Td [(>)]TJ /F2 1 Tf 1.0843 0 Td [(lim)-174(sup)]TJ /F9 1 Tf 3.2349 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.83133 0 Td [(\241)]TJ /F8 1 Tf 1.012 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.362 12.771 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.861 9.075 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(.)-500(S'agissan)24(t)-355(d'une)-355(suite)-355(de)-349(r)24(\266)475(eels)-355(p)-30(ositifs)-355(on)]TJ -20.373 -2.3253 Td [(en)-331(tire)-337(que)]TJ /F9 1 Tf 5.0301 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.8253 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.966 9.4435 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.494 6.7497 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.7349 0 Td [(tend)-337(v)24(ers)-337(0.)]TJ /F5 1 Tf -11.343 -2.8614 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(70)]TJ ET endstream endobj 245 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 244 0 R /ProcSet 2 0 R >> /Contents 243 0 R >> endobj 244 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 246 0 obj << /Length 20596 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ /F2 1 Tf 0 -2.7651 Td [(On)-397(a)-391(alors)]TJ /F9 1 Tf 5.0783 0.8494 Td [(\257)]TJ 0 -0.59639 Td [(\257)]TJ 0.5 1.1084 Td [(Z)]TJ /F2 1 Tf 0.83735 -1.3614 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F6 1 Tf 0.86145 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.784 88.306 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.072 61.858 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F9 1 Tf 1.3976 0.8494 Td [(\257)]TJ 0 -0.59639 Td [(\257)]TJ /F11 1 Tf 0.71687 -0.25301 Td [(6)]TJ /F9 1 Tf 1.1566 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.86144 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.293 88.306 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.108 61.858 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F2 1 Tf 1.7952 0 Td [(donc)]TJ /F9 1 Tf 2.4518 1.3614 Td [(Z)]TJ /F2 1 Tf 0.83133 -1.3614 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F6 1 Tf 0.85542 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.948 88.306 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.958 61.858 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.7952 0 Td [(tend)-397(v)24(ers)-397(0)-397(et)]TJ /F9 1 Tf 6.5843 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.655 88.306 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.096 61.858 Td [(dm)]TJ /F2 1 Tf -37.765 -2.3253 Td [(tend)-337(v)24(ers)]TJ /F9 1 Tf 4.3434 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)]TJ ET 512.64 710.38 m 519.84 710.38 l 519.84 718.81 l 512.64 718.81 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 55.683 Td [(En)-325(application)-325(de)-325(cet)-295(\266)475(enonc)24(\266)469(e,)-331(donnons)-325(une)-325(nouv)30(elle)-331(preuv)30(e)-331(de)-319(la)-325(ma)-60(joration)]TJ /F9 1 Tf 33.663 0.8494 Td [(\260)]TJ 0 -0.59639 Td [(\260)]TJ 0.72289 1.1084 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F9 1 Tf 2.1687 0.8494 Td [(\260)]TJ 0 -0.59639 Td [(\260)]TJ /F11 1 Tf 0.83133 -0.25301 Td [(6)]TJ /F9 1 Tf -38.386 -0.96386 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.8072 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.4398 0 Td [(f)]TJ /F2 1 Tf 1.006 0 Td [(fonction)-415(in)24(t)24(\266)475(egrable)-415(\265)500(a)-409(v)54(aleurs)-415(dans)-409(un)-409(espace)-409(v)24(ectoriel)-415(norm)24(\266)469(e)]TJ /F8 1 Tf 27.361 0 Td [(E)]TJ /F2 1 Tf 1.2048 0 Td [(de)-409(di-)]TJ -36.584 -1.6566 Td [(mension)-337(\257nie.)]TJ 0 -1.8614 Td [(Fixons)-493(des)-500(v)24(ecteurs)-500(de)-493(base)]TJ /F8 1 Tf 13.241 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.241 71.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.867 49.84 Td [(dans)]TJ /F8 1 Tf 2.5 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(.)-933(On)-493(sait)-500(\(cf.)-927(th)24(\266)469(eor)24(\265)469(eme)]TJ /F1 1 Tf 11.94 0 Td [(I)-36(I)-36(I.5.9.)]TJ /F2 1 Tf 3.4819 0 Td [(\))-500(que)-500(c)24(haque)]TJ -33.253 -1.1988 Td [(co)-30(ordonn)18(\266)475(ee)]TJ /F8 1 Tf 5.4096 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.069 69.392 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.054 48.641 Td [(de)]TJ /F8 1 Tf 1.488 0 Td [(f)]TJ /F2 1 Tf 1.0843 0 Td [(est)-493(mesurable)-493(donc)-487(limite)-493(simple)-487(d'une)-493(suite)-493(de)-487(fonctions)-463(\266)469(etag)30(\266)469(ees)]TJ /F8 1 Tf -9.2952 -1.2048 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3103 67.668 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.3554 47.436 Td [(v)48(\266)475(eri\257an)24(t)]TJ /F6 1 Tf 4.0723 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.431 67.668 Td [(n;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.247 47.436 Td [(j)]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F6 1 Tf 1.3072 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.086 67.668 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.464 47.436 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-897(D)24(\266)469(e\257nissons)-487(alors)-487(une)-487(suite)-481(de)-487(fonctions)-487(v)24(ectorielles)]TJ /F8 1 Tf 24.801 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.129 67.668 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.048 47.436 Td [(par)]TJ /F8 1 Tf -37.717 -1.4337 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3103 65.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.3494 46.003 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-331(=)]TJ /F9 1 Tf 1.8253 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.069 65.616 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.295 46.003 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.19 65.616 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.44 46.003 Td [(.)-542(Il)-361(est)-373(clair)-361(que)-373(la)-361(suite)-367(des)]TJ /F8 1 Tf 12.892 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.793 65.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.717 46.003 Td [(con)24(v)24(erge)-367(simplemen)24(t)-367(v)24(ers)]TJ /F8 1 Tf 11.56 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(et)-367(donc)]TJ -35.91 -1.4036 Td [(que)]TJ /F6 1 Tf 1.8916 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.733 63.607 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.741 44.599 Td [(k)]TJ /F2 1 Tf 0.86747 0 Td [(con)30(v)24(erge)-367(simplemen)18(t)-361(v)24(ers)]TJ /F6 1 Tf 11.554 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F2 1 Tf 0.5 0 Td [(.)-536(Mais)-367(on)-367(a)-361(aussi)-367(les)-367(relations)-367(de)-361(domination)-367(par)]TJ -17.428 -1.3976 Td [(des)-283(fonctions)-283(in)24(t)24(\266)469(egrables)]TJ /F6 1 Tf 10.867 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.267 61.607 Td [(n;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.018 43.201 Td [(j)]TJ /F11 1 Tf 0.56024 0 Td [(6)]TJ /F6 1 Tf 1.0542 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.19 61.607 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.735 43.201 Td [(j)]TJ /F2 1 Tf 0.56024 0 Td [(et)]TJ /F6 1 Tf 1.1145 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.466 61.607 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.934 43.201 Td [(k)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F6 1 Tf 1.6145 -0.95181 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.259 61.607 Td [(n;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.199 43.201 Td [(j)-120(\242)-120(k)]TJ /F8 1 Tf 1.2952 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.302 61.607 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.289 43.201 Td [(k)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F6 1 Tf 1.6084 -0.95181 Td [(k)]TJ /F8 1 Tf 0.50603 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.095 61.607 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 36.036 43.201 Td [(k)-120(\242)-120(j)]TJ /F8 1 Tf 1.2952 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.121 61.607 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.157 43.201 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-427(Le)-283(th)18(\266)475(eor)18(\265)475(eme)]TJ -33.102 -2.0121 Td [(pr)30(\266)469(ec)24(\266)469(eden)30(t)-415(assure)-409(alors)]TJ /F9 1 Tf 10.169 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.017 58.728 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.024 41.189 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F6 1 Tf 1.8012 0 Td [(!)]TJ /F9 1 Tf 1.4036 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.8072 0 Td [(et,)-433(p)-30(our)-409(c)24(haque)]TJ /F8 1 Tf 7.3916 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(,)]TJ /F9 1 Tf 0.70482 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.078 58.728 Td [(n;i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.62 41.189 Td [(dm)]TJ /F6 1 Tf 1.8012 0 Td [(!)]TJ /F9 1 Tf 1.4036 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.56 58.728 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.819 41.189 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)]TJ -38.886 -2.3253 Td [(P)30(ar)-337(suite)-331(on)-337(a)]TJ /F9 1 Tf 6.494 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.043 55.4 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.018 38.864 Td [(dm)]TJ /F6 1 Tf 1.6747 0 Td [(!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5 0 Td [(et)]TJ /F9 1 Tf 1.1687 0.8494 Td [(\260)]TJ 0 -0.59638 Td [(\260)]TJ 0.72289 1.1084 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.112 55.4 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.548 38.864 Td [(dm)]TJ /F9 1 Tf 1.3976 0.8494 Td [(\260)]TJ 0 -0.59638 Td [(\260)]TJ /F6 1 Tf 0.83735 -0.25301 Td [(!)]TJ /F9 1 Tf 1.2771 0.8494 Td [(\260)]TJ 0 -0.59638 Td [(\260)]TJ 0.72289 1.1084 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F9 1 Tf 2.1627 0.8494 Td [(\260)]TJ 0 -0.59638 Td [(\260)]TJ /F2 1 Tf 0.55422 -0.25301 Td [(.)]TJ -27.169 -2.3193 Td [(P)30(our)-277(prouv)24(er)-277(la)-271(ma)-60(joration)-277(annonc)24(\266)469(ee)-277(il)-271(su\261t)-283(donc)-271(de)-277(prouv)30(er)-283(la)-271(ma)-60(joration)-277(analogue)-277(p)-30(our)]TJ 0 -1.6566 Td [(c)24(haque)]TJ /F8 1 Tf 3.3675 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.138 49.711 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7229 34.888 Td [(,)-379(c'est-)0(\265)500(a-dire)]TJ /F9 1 Tf 5.8133 0.8494 Td [(\260)]TJ 0 -0.60241 Td [(\260)]TJ 0.72289 1.1145 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.379 49.711 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.446 34.888 Td [(dm)]TJ /F9 1 Tf 1.4036 0.8494 Td [(\260)]TJ 0 -0.60241 Td [(\260)]TJ /F11 1 Tf 0.89157 -0.24699 Td [(6)]TJ /F9 1 Tf 1.1145 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.112 49.711 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.38 34.888 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-554(Or)]TJ /F8 1 Tf 2.3675 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.914 49.711 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.205 34.888 Td [(\(comme)-373(les)-367(co)-36(ordonn)24(\266)469(ees)]TJ /F8 1 Tf 10.988 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.629 49.711 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.735 34.888 Td [(\))-373(ne)]TJ -37.404 -1.6566 Td [(prend)-295(qu'un)-289(nom)24(bre)-295(\257ni)-295(de)-289(v)54(aleurs)]TJ /F8 1 Tf 15.398 0 Td [(u)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.483 47.34 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.753 33.232 Td [(,)]TJ /F8 1 Tf 0.57229 0 Td [(u)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32.776 47.34 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.349 33.232 Td [(,)]TJ /F8 1 Tf 0.27711 0 Td [(:)-168(:)-168(:)]TJ /F2 1 Tf 1.3373 0 Td [(,)]TJ /F8 1 Tf 0.57229 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.362 47.34 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.572 33.232 Td [(.)-433(Si)-289(l'on)-295(note)]TJ /F8 1 Tf 5.9157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.56 47.34 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.018 33.232 Td [(l'ensem)24(ble)-289(des)-295(p)-30(oin)24(ts)-289(o)-30(\265)530(u)]TJ /F8 1 Tf -28.687 -1.1988 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3103 45.625 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6145 32.033 Td [(prend)-265(la)-265(v)48(aleur)]TJ /F8 1 Tf 6.7108 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.897 45.625 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.657 32.033 Td [(on)-265(obtien)30(t)-271(des)-265(ensem)24(bles)-265(mesurables)-271(\(puisque)]TJ /F8 1 Tf 19.928 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.172 45.625 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.867 32.033 Td [(est)-271(mesurable\))-265(deux)]TJ -30.536 -1.4036 Td [(\265)499(a)-277(deux)-277(disjoin)24(ts)-277(et)-283(l'on)-277(a)]TJ /F8 1 Tf 10.825 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.802 43.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.452 30.629 Td [(=)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F2 1 Tf 1.6084 -0.95181 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.595 43.616 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 42.566 60.752 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.771 30.629 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.974 43.616 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.837 30.629 Td [(.)-427(P)30(ar)-277(ailleurs,)-295(dans)-277(la)-277(somme)-283(pr)30(\266)469(ec)24(\266)469(eden)30(te)-283(un)-283(terme)]TJ -17.506 -2.012 Td [(au)-415(plus)-409(et)-415(non)-421(n)30(ul,)-439(par)-415(suite)]TJ /F6 1 Tf 13.349 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.129 40.737 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.199 28.617 Td [(k)]TJ /F2 1 Tf 0.91566 0 Td [(=)]TJ /F9 1 Tf 1.1867 0.95181 Td [(X)]TJ /F2 1 Tf 1.6145 -0.95181 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.026 40.737 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 50.157 56.728 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.566 28.617 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.121 40.737 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.127 28.617 Td [(k)]TJ /F2 1 Tf 0.5 0 Td [(.)-686(On)-415(a)-415(alors)]TJ /F9 1 Tf 6.1024 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.388 40.737 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.922 28.617 Td [(dm)]TJ /F2 1 Tf 1.8072 0 Td [(=)]TJ /F9 1 Tf 1.1928 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.888 40.737 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.042 28.617 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.966 40.737 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 26.292 Td [(et)]TJ /F9 1 Tf 1.253 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.25 37.418 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1084 26.292 Td [(k)]TJ /F8 1 Tf 0.66265 0 Td [(dm)]TJ /F2 1 Tf 1.8253 0 Td [(=)]TJ /F9 1 Tf 1.2048 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.509 37.418 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.922 26.292 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.31 37.418 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.873 26.292 Td [(k)]TJ /F2 1 Tf 0.5 0 Td [(.)-710(La)-427(ma)-54(joration)-427(souhait)24(\266)469(ee)-421(r)24(\266)475(esulte)-421(alors)-427(de)-421(l'in)24(\266)469(egalit)24(\266)475(e)-427(de)]TJ -14.042 -1.6566 Td [(con)30(v)24(exit)24(\266)475(e)-343(v)54(\266)469(eri\257)30(\266)469(ee)-337(par)-337(toute)-331(norme.)]TJ ET 512.64 293.29 m 519.84 293.29 l 519.84 301.71 l 512.64 301.71 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 21.906 Td [(Le)-271(th)24(\266)469(eor)24(\265)469(eme)-271(de)-265(con)24(v)24(ergence)-271(domin)24(\266)475(ee)-277(a)-271(aussi)-271(une)-271(v)24(ersion)-271(p)-30(our)-271(les)-277(fonctions)-271(v)24(ectorielles)-271(\(i.e.)]TJ 0 -1.1988 Td [(\265)499(a)-331(v)48(aleurs)-337(dans)-331(un)-337(espace)-337(v)24(ectoriel)-331(de)-337(dimension)-337(\257nie\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 14.419 Td [(V.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7771 17.286 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084337 Td [(\266)]TJ -0.0060244 -0.084337 Td [(eor)]TJ 1.8012 0.084337 Td [(\265)]TJ 0 -0.084337 Td [(eme.)]TJ /F12 1 Tf 2.8735 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0964 0 Td [(E)]TJ /F12 1 Tf 1.1928 0 Td [(un)-397(espace)-397(v)24(ectoriel)-397(norm)18(\266)475(e)-403(de)-397(dimension)-403(\257nie,)-415(une)-397(suite)]TJ -14.554 -1.1988 Td [(de)-403(fonctions)-403(in)24(t)24(\266)469(egrables)]TJ /F8 1 Tf 10.843 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.845 22.806 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.608 16.087 Td [(\265)500(a)-409(v)54(aleurs)-409(dans)]TJ /F8 1 Tf 6.759 0 Td [(E)]TJ /F12 1 Tf 1.1988 0 Td [(et)-409(une)-403(fonction)]TJ /F8 1 Tf 7.1325 0 Td [(f)]TJ /F12 1 Tf 1.006 0 Td [(\265)500(a)-403(v)54(aleurs)-409(dans)]TJ /F8 1 Tf 6.753 0 Td [(E)]TJ /F12 1 Tf 1.2048 0 Td [(d)30(\266)469(e\257nie)]TJ -36.331 -1.1988 Td [(presque)-331(partout.)-451(On)-337(fait)-337(les)-337(h)30(yp)-36(oth)24(\265)469(eses)-337(:)]TJ /F1 1 Tf 0.49398 -1.8675 Td [(\(i\))]TJ /F8 1 Tf 1.8795 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.724 18.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.741 13.021 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(presque)-337(partout)-337(;)]TJ /F1 1 Tf -11.542 -1.8614 Td [(\(ii\))]TJ /F12 1 Tf 2.2048 0 Td [(il)-337(y)-337(a)-337(une)-337(fonction)-343(in)24(t)24(\266)475(egrable)]TJ /F8 1 Tf 12.994 0 Td [(g)]TJ /F12 1 Tf 0.8494 0 Td [(qui)-337(domine)-343(les)]TJ /F8 1 Tf 6.6566 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.069 15.754 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 29.247 11.159 Td [(,)-337(c'est-)0(\265)500(a-dire)-343(que,)-337(p)-30(our)-337(tout)]TJ /F8 1 Tf 12.422 0 Td [(n)]TJ /F12 1 Tf 0.93373 0 Td [(on)-337(a)]TJ /F6 1 Tf -37.271 -1.2048 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0431 14.038 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8614 9.9545 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(presque)-331(partout.)]TJ -6.9157 -2.3735 Td [(Alors)]TJ /F8 1 Tf 2.6446 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(et)-337(l'on)-337(a)-331(:)]TJ /F9 1 Tf 12.102 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.81927 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.526 10.633 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.367 7.581 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F6 1 Tf 1.6747 0 Td [(\241)162(!)]TJ /F2 1 Tf 1.8916 0 Td [(0)]TJ /F12 1 Tf 2.0602 0 Td [(et)]TJ /F9 1 Tf 2.0602 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.397 10.633 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.928 7.581 Td [(dm)]TJ /F6 1 Tf 1.6747 0 Td [(\241)168(!)]TJ /F9 1 Tf 1.8916 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F12 1 Tf 2.494 0 Td [(.)]TJ /F2 1 Tf -18.741 -3.6928 Td [(71)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 248 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 247 0 R /ProcSet 2 0 R >> /Contents 246 0 R >> endobj 247 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 249 0 obj << /Length 15908 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ 0 -2.4036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.5241 Td [(T)78(oute)-421(forme)-415(lin)24(\266)475(eaire)]TJ /F8 1 Tf 9.4217 0 Td [(\256)]TJ /F2 1 Tf 1.0663 0 Td [(sur)]TJ /F8 1 Tf 1.753 0 Td [(E)]TJ /F2 1 Tf 1.2169 0 Td [(est)-421(con)30(tin)18(ue)-415(et)-421(satisfait)-415(\265)500(a)-415(une)-421(ma)-54(joration)]TJ /F6 1 Tf 18.946 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.69277 0 Td [(6)]TJ /F8 1 Tf 1.1928 0 Td [(A)]TJ /F6 1 Tf 0.75301 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(k)]TJ /F2 1 Tf 0.5 0 Td [(;)]TJ -38.886 -1.1988 Td [(par)-331(suite)-337(les)-337(fonctions)]TJ /F8 1 Tf 9.8855 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.112 84.926 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.729 59.497 Td [(son)24(t)-337(in)24(t)24(\266)475(egrables,)-337(con)24(v)24(ergen)24(t)-337(simplemen)24(t)-337(v)24(ers)]TJ /F8 1 Tf 19.843 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(et)-331(satisfon)18(t)]TJ -34.319 -1.1988 Td [(\265)499(a)-415(la)-415(relation)-421(de)-415(domination)]TJ /F6 1 Tf 12.645 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(\256)]TJ /F6 1 Tf 0.63856 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.466 83.211 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.434 58.298 Td [(j)]TJ /F11 1 Tf 0.6988 0 Td [(6)]TJ /F8 1 Tf 1.1928 0 Td [(A)]TJ /F6 1 Tf 0.74699 0 Td [(k)]TJ /F8 1 Tf 0.50602 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.44 83.211 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.608 58.298 Td [(k)]TJ /F11 1 Tf 0.92169 0 Td [(6)]TJ /F8 1 Tf 1.1928 0 Td [(Ag)]TJ /F2 1 Tf 1.2651 0 Td [(.)-692(La)-421(v)24(ersion)-421(n)24(um)24(\266)469(erique)-421(du)-415(th)18(\266)475(eor)18(\265)475(eme)]TJ -22.657 -1.7108 Td [(p)-30(ermet)-409(donc)-403(de)-403(conclure)-409(que)]TJ /F8 1 Tf 13.313 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 1 0 Td [(est)-409(in)24(t)24(\266)469(egrable)-403(et)]TJ /F9 1 Tf 7.5904 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\256)]TJ /F6 1 Tf 0.63855 0 Td [(\261)]TJ /F8 1 Tf 0.50603 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.379 80.762 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.723 56.587 Td [(dm)]TJ /F6 1 Tf 1.7952 0 Td [(!)]TJ /F9 1 Tf 1.3976 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)-662(Il)-403(s'ensuit)]TJ -33.892 -2.3253 Td [(que)]TJ /F8 1 Tf 1.9036 0 Td [(f)]TJ /F2 1 Tf 0.9759 0 Td [(est)-379(in)24(t)24(\266)475(egrable)-379(et)]TJ /F9 1 Tf 7.5 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.612 77.435 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.91 54.262 Td [(dm)]TJ /F2 1 Tf 1.7771 0 Td [(tend)-385(v)24(ers)]TJ /F9 1 Tf 4.4337 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5361 0 Td [(\(il)-385(su\261t)-379(de)-373(consid)24(\266)475(erer)-385(les)-379(co)-30(ordonn)24(\266)469(ees)]TJ -22.325 -1.7108 Td [(de)-331(ces)-337(expressions\).)]TJ 0 -2.0361 Td [(En)-397(outre,)-421(on)-397(a)]TJ /F9 1 Tf 6.9578 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(\256)]TJ /F6 1 Tf 0.63855 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.86747 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.121 72.073 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.801 50.515 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dm)]TJ /F2 1 Tf 1.7892 0 Td [(=)]TJ /F9 1 Tf 1.1626 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(\261)]TJ /F2 1 Tf 0.5 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F6 1 Tf 0.86747 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.207 72.073 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.946 50.515 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F6 1 Tf 1.7831 0 Td [(!)]TJ /F2 1 Tf 1.3855 0 Td [(0.)-644(Fixons)-397(alors)-403(des)-397(v)24(ecteurs)]TJ -26.614 -1.6566 Td [(de)-367(base)]TJ /F8 1 Tf 3.6325 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.491 69.702 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.127 48.858 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(,)-379(le)-373(r)30(\266)469(esultat)-373(s'applique)-367(en)-367(particulier)-373(\265)499(a)-367(c)24(hacune)-367(des)-373(formes)]TJ /F8 1 Tf 26.151 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.819 69.702 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.711 48.858 Td [(obten)24(ue)-367(en)]TJ -34.38 -1.3434 Td [(en)30(v)24(o)24(y)24(an)30(t)-379(un)-373(v)24(ecteur)]TJ /F8 1 Tf 9.259 0 Td [(u)]TJ /F2 1 Tf 0.93976 0 Td [(sur)-379(sa)-373(co)-36(ordonn)24(\266)475(ee)]TJ /F8 1 Tf 8.2771 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.888 67.78 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.711 47.515 Td [(.)-560(P)30(ar)-373(ailleurs)-373(on)-367(a)]TJ /F6 1 Tf 8.5422 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(k)]TJ /F11 1 Tf 0.84337 0 Td [(6)]TJ /F9 1 Tf 1.1205 0.95181 Td [(X)]TJ /F6 1 Tf 1.6084 -0.95181 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.448 67.78 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.078 47.515 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)-253(\242)-246(k)]TJ /F8 1 Tf 1.5542 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.741 67.78 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.783 47.515 Td [(k)]TJ /F2 1 Tf 0.5 0 Td [(,)-379(il)]TJ -37.952 -2.012 Td [(s'ensuit)-385(que)]TJ /F9 1 Tf 5.5843 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.85542 0 Td [(\241)]TJ /F8 1 Tf 1.0301 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.172 64.9 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.337 45.503 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.7831 0 Td [(est)-385(ma)-60(jor)24(\266)469(e)-385(par)]TJ /F9 1 Tf 6.8554 0.95181 Td [(X)]TJ /F6 1 Tf 1.6145 -0.95181 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(e)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.957 64.9 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.56 45.503 Td [(k)]TJ /F9 1 Tf 0.66265 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.034 64.9 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.404 45.503 Td [(\261)]TJ /F2 1 Tf 0.5 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F6 1 Tf 0.85542 0 Td [(\241)]TJ /F8 1 Tf 1.0301 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.19 64.9 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.217 45.503 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43975 0 Td [(dm)]TJ /F2 1 Tf 1.7831 0 Td [(donc)-385(tend)-391(v)24(ers)-385(0.)]TJ ET 512.64 523.98 m 519.84 523.98 l 519.84 532.4 l 512.64 532.4 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 41.756 Td [(Remarques.)]TJ 0.81325 -1.5241 Td [(1.)]TJ /F2 1 Tf 1.8012 0 Td [(L')24(\266)469(equiv)54(alence)-463(des)-451(normes)-457(mon)24(tre)-457(que)-457(si)-457(la)-457(condition)-451(de)-457(domination)-451(du)-457(th)18(\266)475(eor)18(\265)475(eme)]TJ -2.6145 -1.1988 Td [(pr)30(\266)469(ec)24(\266)469(eden)30(t)-337(est)-337(v)48(alable)-337(p)-30(our)-331(une)-337(norme)-337(elle)-337(est)-337(v)48(alable)-337(p)-30(our)-331(toute)-337(autre)-337(norme.)]TJ /F1 1 Tf 0.81325 -1.5241 Td [(2.)]TJ /F2 1 Tf 1.7651 0 Td [(Les)-433(th)24(\266)469(eor)24(\265)469(emes)-439(pr)30(\266)469(ec)24(\266)469(eden)30(ts)-433(son)18(t)-433(v)48(alables)-439(p)-30(our)-433(des)-439(suites)-439(de)-439(fonctions.)-746(Nous)-439(allons)]TJ -2.5783 -1.1988 Td [(en)-391(tirer)-391(des)-391(cons)24(\266)469(equences)-397(p)-30(our)-391(des)-391(fonctions)-391(d)24(\266)475(ep)-30(endan)24(t)-397(d'un)-391(param)24(\265)469(etre)-391(v)48(arian)30(t)-397(dans)-391(un)]TJ 0 -1.1988 Td [(espace)-325(m)24(\266)475(etrique)-325(a\257n)-331(de)-325(se)-325(ramener)-325(au)-331(cas)-325(des)-325(suites.)-445(Il)-325(est)-331(cep)-36(endan)24(t)-325(imp)-36(ortan)24(t)-325(de)-325(noter)]TJ 0 -1.2048 Td [(que)-337(cela)-337(ne)-331(se)-337(g)24(\266)475(en)24(\266)475(eralisera)-337(pas)-337(\265)500(a)-337(un)-337(espace)-331(non)-337(m)24(\266)469(etrique.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 24.706 Td [(V.3.)-241(In)30(t)25(\266)537(egrales)-381(d)25(\266)537(ep)-35(endan)25(t)-376(d'un)-381(param)20(\265)537(etre.)]TJ 0 -2.392 Td [(V.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7771 26.75 Td [(1.)-289(Hyp)-30(oth)18(\265)548(eses)-385(et)-391(notations.)]TJ /F2 1 Tf -2.4458 -1.5181 Td [(Dans)-331(cette)-337(section)-331(on)-331(\257xe)-337(outre)-337(l'espace)-331(mesur)24(\266)475(e)]TJ /F8 1 Tf 21.253 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-331(un)-337(espace)-331(m)24(\266)475(etrique)]TJ ET 0.504 w 390.38 299.88 m 436.61 299.88 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.855 25.232 Td [(T)]TJ /F2 1 Tf 1.0602 0 Td [(et)-331(une)-331(fonction)]TJ /F8 1 Tf -32.584 -1.1988 Td [(f)]TJ /F2 1 Tf 0.99398 0 Td [(d)24(\266)475(e\257nie)-403(sur)-403(une)-403(partie)-403(de)]TJ /F8 1 Tf 11.289 0 Td [(T)]TJ /F6 1 Tf 0.99398 0 Td [(\243)]TJ /F8 1 Tf 1.0422 0 Td [(X)]TJ /F2 1 Tf 1.3072 0 Td [(\265)499(a)-397(v)48(aleurs)-403(dans)-397(un)-403(espace)-403(v)24(ectoriel)-397(de)-403(dimension)-403(\257nie)]TJ /F8 1 Tf -15.627 -1.2048 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(.)-608(On)-391(supp)-30(ose)-391(que,)-403(p)-30(our)-391(tout)]TJ /F8 1 Tf 13.277 0 Td [(t)]TJ /F2 1 Tf 0.74699 0 Td [(dans)]TJ /F8 1 Tf 2.3916 0 Td [(T)]TJ /F2 1 Tf 0.72891 0 Td [(,)-397(la)-391(fonction)]TJ /F8 1 Tf 5.759 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.25 32.452 Td [(t)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.241 22.828 Td [(:)]TJ /F8 1 Tf 0.6506 0 Td [(x)]TJ /F6 1 Tf 0.93976 0 Td [(7!)]TJ /F8 1 Tf 1.3675 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))-391(est)-391(d)30(\266)469(e\257nie)-391(presque)]TJ -30.235 -1.1988 Td [(partout)-337(et)-337(in)30(t)24(\266)469(egrable)-337(et)-331(on)-337(d)24(\266)475(e\257nit)-337(une)-331(fonction)]TJ /F8 1 Tf 20.849 0 Td [(F)]TJ /F2 1 Tf 1.1145 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(T)]TJ /F2 1 Tf 1.0602 0 Td [(dans)]TJ /F8 1 Tf 2.3373 0 Td [(E)]TJ /F2 1 Tf 1.1325 0 Td [(par)-337(:)]TJ /F8 1 Tf -15.886 -1.6566 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.509 28.366 Td [(t)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.536 19.973 Td [(dm)]TJ /F2 1 Tf 1.6807 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(.)]TJ -26.223 -1.9759 Td [(On)-319(dira)-319(alors)-325(que)-319(la)-319(fonction)]TJ /F8 1 Tf 12.867 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(est)]TJ /F1 1 Tf 1.5482 0 Td [(domin)18(\266)548(ee)]TJ /F2 1 Tf 4.506 0 Td [(sur)-319(une)-319(partie)]TJ /F8 1 Tf 6.4096 0 Td [(U)]TJ /F2 1 Tf 1.1084 0 Td [(de)]TJ /F8 1 Tf 1.3193 0 Td [(T)]TJ /F2 1 Tf 1.0422 0 Td [(s'il)-325(y)-319(a)-319(une)-319(fonction)]TJ /F8 1 Tf 8.9337 0 Td [(g)]TJ /F2 1 Tf -38.651 -1.2048 Td [(in)24(t)24(\266)475(egrable)-337(sur)]TJ /F8 1 Tf 6.3193 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(\265)500(a)-331(v)48(aleurs)-337(dans)]TJ ET 232.27 210.67 m 240.91 210.67 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 19.434 16.792 Td [(R)]TJ /F2 1 Tf 1.0602 0 Td [(telle)-331(que)-343(:)]TJ /F12 1 Tf -12.741 -1.1988 Td [(p)-30(our)-337(presque)-337(tout)]TJ /F8 1 Tf 8.1807 0 Td [(x)]TJ /F12 1 Tf 0.90964 0 Td [(dans)]TJ /F8 1 Tf 2.3374 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(et)-337(p)-30(our)-337(tout)]TJ /F8 1 Tf 5.6988 0 Td [(t)]TJ /F12 1 Tf 0.6988 0 Td [(dans)]TJ /F8 1 Tf 2.3374 0 Td [(U)]TJ /F12 1 Tf 0.79518 0 Td [(,)-331(on)-337(ait)-337(:)]TJ /F6 1 Tf 4.5542 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.71687 0 Td [(,)]TJ /F2 1 Tf -36.458 -1.5241 Td [(\()-337(on)-331(pr)24(\266)469(ecisera)-331(parfois)-337(que)]TJ /F8 1 Tf 11.41 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(est)-337(domin)24(\266)475(ee)-337(par)]TJ /F8 1 Tf 7.2892 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\).)]TJ -20.145 -1.741 Td [(L'ob)-54(jectif)-331(des)-331(th)24(\266)469(eor)24(\265)469(emes)-331(qui)-325(suiv)24(en)30(t)-331(est)-331(de)-331(donner)-331(des)-325(renseignemen)24(ts)-325(sur)-331(la)-331(fonction)]TJ /F8 1 Tf 37.554 0 Td [(F)]TJ /F2 1 Tf 1.1084 0 Td [(\265)500(a)]TJ -38.663 -1.1988 Td [(partir)-331(de)-337(renseignemen)30(ts)-337(analogues)-331(sur)-337(les)-337(fonctions)-337(de)]TJ /F8 1 Tf 24.386 0 Td [(t)]TJ /F2 1 Tf 0.6988 0 Td [(d)30(\266)469(e\257nies)-337(par)]TJ /F8 1 Tf 5.3434 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(en)-337(\257xan)24(t)]TJ /F8 1 Tf 4.1687 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(.)]TJ -36.096 -1.5241 Td [(On)-409(prendra)-415(bien)-409(garde)-409(\265)499(a)-409(distinguer)-409(le)-409(r)0(^)500(ole)-415(des)-415(deux)-409(v)54(ariables)]TJ /F8 1 Tf 27.928 0 Td [(x)]TJ /F2 1 Tf 0.9759 0 Td [(et)]TJ /F8 1 Tf 1.247 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(.)-668(La)-415(v)54(ariable)]TJ /F8 1 Tf 6.3072 0 Td [(x)]TJ /F2 1 Tf 0.98193 0 Td [(qui)]TJ -37.801 -1.1988 Td [(se)-463(d)24(\266)475(eplace)-469(dans)-469(l'espace)-463(mesur)24(\266)475(e)]TJ /F8 1 Tf 14.795 0 Td [(X)]TJ /F2 1 Tf 1.3735 0 Td [(est)-469(la)-463(v)48(ariable)-463(d'in)24(t)24(\266)469(egration,)-493(la)-469(v)54(ariable)]TJ /F8 1 Tf 18.313 0 Td [(t)]TJ /F2 1 Tf 0.8253 0 Td [(doit)-433(^)469(etre)]TJ -35.307 -1.6566 Td [(consid)30(\266)469(er)24(\266)469(ee)-337(comme)-331(un)-337(param)18(\265)475(etre)-337(dans)-331(l'in)24(t)24(\266)469(egrale)]TJ /F9 1 Tf 21.392 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)]TJ /F5 1 Tf -27.675 -2.8614 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(72)]TJ ET endstream endobj 251 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 250 0 R /ProcSet 2 0 R >> /Contents 249 0 R >> endobj 250 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R >> endobj 252 0 obj << /Length 16781 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ 0 -2.4036 Td [(Remarque.)]TJ /F2 1 Tf 6.2108 0 Td [(Un)-277(\266)469(enonc)30(\266)469(e)-307(du)-307(t)24(yp)-30(e)-307(")]TJ /F12 1 Tf 8.988 0 Td [(p)-30(our)-307(presque)-301(tout)]TJ /F8 1 Tf 8.0904 0 Td [(x)]TJ /F12 1 Tf 0.57229 0 Td [(,)]TJ /F8 1 Tf 0.58434 0 Td [(P)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\)")-307(s')24(\266)475(ecrit)-307(de)-301(fa)-6(\273)439(con)-301(formalis)24(\266)475(ee)-307(:)]TJ -14.349 -1.1988 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(9)]TJ /F8 1 Tf 0.55422 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(n)30(\266)469(egligeable)0(\))-1006(\()]TJ /F6 1 Tf 6.5602 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(x)-391(=)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F8 1 Tf 1.3916 0 Td [(P)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(.)]TJ -26.657 -1.7711 Td [(En)-295(particulier,)-307(bien)-301(que)-301(cela)-301(n'apparaisse)-301(pas)-301(dans)-295(l')24(\266)469(enonc)30(\266)469(e)-301(en)-295(langage)-301(couran)24(t,)-307(il)-301(con)30(tien)18(t)]TJ 0 -1.1988 Td [(un)-277(quan)24(ti\257cateur)]TJ /F6 1 Tf 7.7229 0 Td [(9)]TJ /F2 1 Tf 0.55422 0 Td [(.)-427(P)30(our)-283(cette)-277(raison)-277(l'expression)-277("p)-30(our)-277(presque)-277(tout)-283(x")-277(ne)-277(p)-30(eut)-277(pas,)-295(sans)]TJ -8.2771 -1.1988 Td [(pr)30(\266)469(ecaution,)-307(^)469(etre)-307(\266)475(ec)24(hang)24(\266)469(ee)-337(a)30(v)24(ec)-337(une)-337(expression)-331("p)-30(our)-337(tout)]TJ /F8 1 Tf 25.367 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(".)-445(P)30(ar)-337(exemple,)-331(l')24(\266)475(enonc)24(\266)469(e)]TJ /F12 1 Tf -16.042 -1.1988 Td [(p)-30(our)-331(presque)-337(tout)]TJ /F8 1 Tf 8.1807 0 Td [(x)]TJ /F12 1 Tf 0.57229 0 Td [(,)-331(p)-30(our)-337(tout)]TJ /F8 1 Tf 5.1446 0 Td [(t)]TJ /F12 1 Tf 0.36145 0 Td [(,)]TJ /F8 1 Tf 0.94578 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F6 1 Tf 0.66868 0 Td [(6)]TJ /F2 1 Tf [(=)-277(0)]TJ -27.922 -1.7711 Td [(n'est)-331(pas)-313(\266)469(equiv)54(alen)24(t)-331(\265)500(a)-337(l')24(\266)475(enonc)24(\266)475(e)]TJ /F12 1 Tf 9.3795 -1.1988 Td [(p)-30(our)-337(tout)]TJ /F8 1 Tf 4.5301 0 Td [(t)]TJ /F12 1 Tf 0.36144 0 Td [(,)-337(p)-30(our)-337(presque)-331(tout)]TJ /F8 1 Tf 8.7952 0 Td [(x)]TJ /F12 1 Tf 0.57229 0 Td [(,)]TJ /F8 1 Tf 0.94578 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-162(x)]TJ /F2 1 Tf 1.3735 0 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(6)]TJ /F2 1 Tf [(=)-277(0)-337(;)]TJ -27.614 -1.7711 Td [(lorsque)]TJ /F8 1 Tf 3.3795 0 Td [(T)]TJ /F2 1 Tf 1 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(=)]TJ /F3 1 Tf 1.0602 0 Td [(R)]TJ /F2 1 Tf 1.012 0 Td [(a)30(v)24(ec)-289(la)-295(mesure)-289(de)-289(Leb)-36(esgue,)-301(la)-289(fonction)]TJ /F8 1 Tf 17.307 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(t)]TJ /F6 1 Tf 0.49398 0 Td [(\241)]TJ /F8 1 Tf 0.90963 0 Td [(x)]TJ /F2 1 Tf 0.86145 0 Td [(v)54(\266)469(eri\257e)-289(le)-289(second)]TJ -32.386 -1.1988 Td [(\266)469(enonc)30(\266)469(e)-337(mais)-337(pas)-337(le)-337(premier.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 38.736 Td [(V.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7771 46.436 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084335 Td [(\266)]TJ -0.0060244 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ 0 -0.084335 Td [(eme.)]TJ /F12 1 Tf 2.6627 0 Td [(Av)30(ec)-295(les)-295(notations)-295(de)]TJ /F1 1 Tf 9.4819 0 Td [(V.3.1.)]TJ /F12 1 Tf 2.9819 0 Td [(,)-301(on)-295(supp)-30(ose)-289(que)]TJ /F8 1 Tf 7.4639 0 Td [(T)]TJ /F12 1 Tf 1.0181 0 Td [(est)-295(une)-295(partie)-295(d'un)]TJ -31 -1.1988 Td [(espace)-331(m)24(\266)469(etrique)]TJ /F9 1 Tf 7.4819 0.25301 Td [(e)]TJ /F8 1 Tf -0.16867 -0.25301 Td [(T)]TJ /F12 1 Tf 1.0602 0 Td [(et)-337(on)-331(\257xe)-337(un)-337(p)-30(oin)24(t)]TJ /F8 1 Tf 8.4759 0 Td [(a)]TJ /F12 1 Tf 0.86144 0 Td [(de)]TJ /F9 1 Tf 1.5 0.25301 Td [(e)]TJ /F8 1 Tf -0.16265 -0.25301 Td [(T)]TJ /F12 1 Tf 1.0542 0 Td [(adh)24(\266)469(eren)24(t)-337(\265)500(a)]TJ /F8 1 Tf 4.9759 0 Td [(T)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -25.801 -1.7651 Td [(On)-337(fait)-331(les)-337(h)24(yp)-30(oth)18(\265)475(eses)-337(:)]TJ /F1 1 Tf 0.49398 -1.7711 Td [(\(i\))]TJ /F12 1 Tf 1.8795 0 Td [(p)-30(our)-337(presque)-331(tout)]TJ /F8 1 Tf 8.1807 0 Td [(x)]TJ /F12 1 Tf 0.90361 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3735 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(a)-337(une)-337(limite)]TJ /F8 1 Tf 5.5602 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(lorsque)]TJ /F8 1 Tf 3.4277 0 Td [(t)]TJ /F12 1 Tf 0.69277 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(;)]TJ /F1 1 Tf -34.211 -1.8916 Td [(\(ii\))]TJ /F12 1 Tf 2.1988 0 Td [(il)-337(y)-337(a)-331(un)-337(v)24(oisinage)]TJ /F8 1 Tf 8.3133 0 Td [(!)]TJ /F12 1 Tf 0.99398 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(a)]TJ /F12 1 Tf 0.86747 0 Td [(dans)]TJ /F9 1 Tf 2.506 0.25301 Td [(e)]TJ /F8 1 Tf -0.16867 -0.25301 Td [(T)]TJ /F12 1 Tf 1.0602 0 Td [(tel)-331(que)]TJ /F8 1 Tf 3.3072 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(soit)-337(domin)30(\266)469(ee)-337(sur)]TJ /F8 1 Tf 7.5181 0 Td [(!)]TJ /F6 1 Tf 0.87952 0 Td [(\\)]TJ /F8 1 Tf 0.89156 0 Td [(T)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -31.524 -2.2771 Td [(Alors)-307(la)-307(fonction)]TJ /F8 1 Tf 7.5422 0 Td [(`)]TJ /F12 1 Tf 0.72289 0 Td [(est)-313(in)24(t)24(\266)475(egrable)-307(et)]TJ /F8 1 Tf 7.2952 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.69277 0 Td [(tend)-313(v)24(ers)]TJ /F9 1 Tf 4.2892 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(`)]TJ /F2 1 Tf 0.42169 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.6988 0 Td [(lorsque)]TJ /F8 1 Tf -36.072 -1.7108 Td [(t)]TJ /F12 1 Tf 0.69277 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(\(en)-337(restan)24(t)-337(dans)]TJ /F8 1 Tf 7.4337 0 Td [(T)]TJ /F12 1 Tf 0.72289 0 Td [(\).)]TJ /F1 1 Tf -14.054 -2.2771 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Comme)]TJ /F9 1 Tf 3.9217 0.25301 Td [(e)]TJ /F8 1 Tf -0.16265 -0.25301 Td [(T)]TJ /F2 1 Tf 1.1446 0 Td [(est)-427(m)24(\266)469(etrique)-421(et)]TJ /F8 1 Tf 7.1988 0 Td [(a)]TJ /F2 1 Tf 0.95181 0 Td [(est)-433(adh)24(\266)475(eren)24(t)-427(\265)499(a)]TJ /F8 1 Tf 6.8133 0 Td [(T)]TJ /F2 1 Tf 0.72289 0 Td [(,)-445(on)-427(p)-30(eut)-427(trouv)24(er)-427(une)]TJ -29.41 -1.2048 Td [(suite)-289(d')24(\266)475(el)18(\266)475(emen)24(ts)]TJ /F8 1 Tf 7.2289 0 Td [(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.491 46.073 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.753 32.34 Td [(de)]TJ /F8 1 Tf 1.2892 0 Td [(T)]TJ /F2 1 Tf 1.012 0 Td [(qui)-289(tend)-295(v)24(ers)]TJ /F8 1 Tf 5.9036 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-433(Mieux,)]TJ /F8 1 Tf 4.006 0 Td [(!)]TJ /F2 1 Tf 0.92169 0 Td [(\266)469(etan)24(t)-289(un)-295(v)24(oisinage)-295(de)]TJ /F8 1 Tf 9.5 0 Td [(a)]TJ /F2 1 Tf 0.52409 0 Td [(,)-301(on)-289(p)-30(eut,)-301(quitte)]TJ -32.108 -1.1988 Td [(\265)499(a)-307(\266)475(eliminer)-337(un)-337(nom)24(bre)-337(\257ni)-337(d'indices,)-337(supp)-30(oser)-331(que)-337(les)]TJ /F8 1 Tf 23.289 0 Td [(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.474 44.349 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.861 31.141 Td [(son)24(t)-337(dans)]TJ /F8 1 Tf 4.488 0 Td [(!)]TJ /F6 1 Tf 0.87952 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(T)]TJ /F2 1 Tf 0.72891 0 Td [(.)]TJ -31.512 -1.7651 Td [(P)30(osons)-337(alors,)-343(p)-30(our)-337(tout)]TJ /F8 1 Tf 10.548 0 Td [(n)]TJ /F2 1 Tf 0.59638 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.155 41.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.127 29.376 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7289 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.716 41.823 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.705 29.376 Td [(;)-168(x)]TJ /F2 1 Tf 1.0181 0 Td [(\).)-451(On)-337(a)-337(une)-337(suite)-337(de)-337(fonctions)-337(\(d)24(\266)475(e\257nies)-343(presque)]TJ -18.392 -1.2048 Td [(partout\))-367(in)24(t)24(\266)475(egrables)-373(et,)-373(p)-30(our)-367(presque)-367(tout)]TJ /F8 1 Tf 18.867 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)]TJ /F8 1 Tf 0.6506 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.078 40.107 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.458 28.171 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-373(tend)-367(v)24(ers)]TJ /F8 1 Tf 5.1627 0 Td [(`)]TJ /F2 1 Tf 0.42168 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).Comme)]TJ /F8 1 Tf 4.3675 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(est)-367(domin)24(\266)469(ee)]TJ -33.958 -1.1988 Td [(sur)]TJ /F8 1 Tf 1.5904 0 Td [(!)]TJ /F2 1 Tf 0.65663 0 Td [(,)-271(il)-246(y)-253(a)-253(une)-246(fonction)]TJ /F8 1 Tf 8.4639 0 Td [(g)]TJ /F2 1 Tf 0.75904 0 Td [(in)24(t)24(\266)469(egrable)-246(sur)]TJ /F8 1 Tf 6.1506 0 Td [(X)]TJ /F2 1 Tf 1.1566 0 Td [(telle)-253(que,)-271(p)-30(our)-253(c)24(haque)]TJ /F8 1 Tf 9.6867 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)-265(on)-253(ait)]TJ /F6 1 Tf 3.5181 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.672 38.383 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.446 26.973 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ -38.777 -1.1988 Td [(presque)-265(partout.)-427(Le)-271(th)24(\266)469(eor)24(\265)469(eme)-265(de)-265(con)24(v)24(ergence)-265(domin)24(\266)475(ee)-271(s'applique,)-283(il)-265(a\261rme)-271(que)-271(la)-265(fonction)]TJ /F8 1 Tf 0 -1.6566 Td [(`)]TJ /F2 1 Tf 0.74699 0 Td [(est)-343(in)30(t)24(\266)469(egrable)-331(et)-337(que)]TJ /F8 1 Tf 9.2349 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.112 34.297 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.392 24.117 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.716 34.297 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.006 24.117 Td [(;)-168(x)]TJ /F2 1 Tf 1.0181 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(tend)-337(v)24(ers)]TJ /F8 1 Tf 5.0663 0 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\).)]TJ -32.819 -2.2771 Td [(Comme)-283(la)-283(con)24(v)24(ergence)-283(de)]TJ /F8 1 Tf 11.44 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.198 31.038 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.849 21.84 Td [(\))-289(v)24(ers)]TJ /F8 1 Tf 2.6867 0 Td [(I)]TJ /F2 1 Tf 0.80121 0 Td [(a)-283(lieu)-283(p)-30(our)-283(toute)-283(suite)-283(d')24(\266)469(el)24(\266)469(emen)30(ts)-283(de)]TJ /F8 1 Tf 15.994 0 Td [(!)]TJ /F6 1 Tf 0.78313 0 Td [(\\)]TJ /F8 1 Tf 0.78313 0 Td [(T)]TJ /F2 1 Tf 1.006 0 Td [(qui)-283(tend)]TJ -35.572 -1.3313 Td [(v)24(ers)]TJ /F8 1 Tf 2.0843 0 Td [(a)]TJ /F2 1 Tf 0.88554 0 Td [(et)-361(que)]TJ /F9 1 Tf 3.247 0.25301 Td [(e)]TJ /F8 1 Tf -0.16868 -0.25301 Td [(T)]TJ /F2 1 Tf 1.0783 0 Td [(est)-361(m)24(\266)469(etrique)-355(on)-361(p)-30(eut)-355(conclure)-361(que)]TJ /F8 1 Tf 15.41 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))-361(tend)-361(v)24(ers)]TJ /F8 1 Tf 5.1386 0 Td [(I)]TJ /F2 1 Tf 0.87349 0 Td [(lorsque)]TJ /F8 1 Tf 3.4518 0 Td [(t)]TJ /F2 1 Tf 0.71687 0 Td [(tend)-361(v)24(ers)]TJ /F8 1 Tf 4.3855 0 Td [(a)]TJ /F2 1 Tf -38.633 -1.1988 Td [(\(en)-337(restan)30(t)-337(dans)]TJ /F8 1 Tf 7.4337 0 Td [(T)]TJ /F2 1 Tf 0.72289 0 Td [(\).)]TJ ET 512.64 229.64 m 519.84 229.64 l 519.84 238.06 l 512.64 238.06 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 17.159 Td [(Remarques.)]TJ 0.81325 -1.7651 Td [(1.)]TJ /F2 1 Tf 1.6084 0 Td [(C'est)-367(l'h)24(yp)-30(oth)18(\265)475(ese)-361(de)-361(m)30(\266)469(etrisabilit)24(\266)475(e)-367(de)]TJ /F9 1 Tf 16.572 0.25301 Td [(e)]TJ /F8 1 Tf -0.16867 -0.25301 Td [(T)]TJ /F2 1 Tf 1.0843 0 Td [(qui)-361(p)-30(ermet)-361(de)-361(ramener)-361(ce)-361(probl)24(\265)475(eme)-361(de)-361(con-)]TJ -19.91 -1.2048 Td [(v)24(ergence)-373(\265)500(a)-379(un)-373(probl)24(\265)469(eme)-373(de)-373(suites)-379(et)-379(d'utiliser)-373(ainsi)-373(le)-379(th)24(\266)469(eor)24(\265)469(eme)-373(de)-373(con)24(v)24(ergence)-373(domin)24(\266)475(ee.)]TJ 0 -1.1988 Td [(En)-506(fait,)-554(p)-30(our)-506(faire)-512(la)-506(d)24(\266)469(emonstration)-512(pr)30(\266)469(ec)24(\266)469(eden)30(te)-512(on)-506(a)-506(b)-30(esoin)-512(que)]TJ /F8 1 Tf 30.108 0 Td [(a)]TJ /F2 1 Tf 1.0361 0 Td [(admette)-506(une)-512(base)]TJ -31.145 -1.1988 Td [(d)30(\266)469(enom)24(brable)]TJ ET 0.504 w 63.72 139.24 m 130.54 139.24 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.253 11.792 Td [(de)-331(v)24(oisinages.)-451(Cette)-337(h)24(yp)-30(oth)24(\265)469(ese)-331(est)-343(essen)24(tielle.)]TJ /F1 1 Tf -5.1084 -1.7711 Td [(2.)]TJ /F2 1 Tf 1.6024 0 Td [(Si)-355(l'on)-355(applique)-355(l')24(\266)469(enonc)30(\266)469(e)-355(pr)24(\266)469(ec)24(\266)475(eden)24(t)-355(au)-355(cas)]TJ /F8 1 Tf 18.711 0 Td [(T)]TJ /F2 1 Tf 1.0361 0 Td [(=)]TJ /F3 1 Tf 1.0904 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(,)]TJ /F9 1 Tf 0.8012 0.25301 Td [(e)]TJ /F8 1 Tf -0.16867 -0.25301 Td [(T)]TJ /F2 1 Tf 1.0361 0 Td [(=)]TJ ET 383.26 129.74 m 391.9 129.74 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 32.066 10.021 Td [(R)]TJ /F2 1 Tf 1.0783 0 Td [(et)]TJ /F8 1 Tf 1.1867 0 Td [(a)]TJ /F2 1 Tf 0.83735 0 Td [(=)-313(+)]TJ /F6 1 Tf 1.8675 0 Td [(1)]TJ /F2 1 Tf 1.3554 0 Td [(on)-355(retrouv)24(e)-355(le)]TJ -33.06 -1.1988 Td [(th)24(\266)469(eor)24(\265)469(eme)-331(de)-331(con)24(v)24(ergence)-337(domin)30(\266)469(ee)-337(p)-30(our)-337(les)-331(suites.)]TJ 0 -2.5361 Td [(En)-331(appliquan)24(t)-337(ce)-337(th)24(\266)469(eor)24(\265)469(eme)-331(au)-337(cas)]TJ /F9 1 Tf 15.361 0.25301 Td [(e)]TJ /F8 1 Tf -0.16868 -0.25301 Td [(T)]TJ /F2 1 Tf 1.006 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(T)]TJ /F2 1 Tf 1.0542 0 Td [(on)-337(p)-30(eut)-337(le)-337(reform)24(uler)-331(ainsi)-337(:)]TJ 0.60843 -2.3976 Td [(73)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 254 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 253 0 R /ProcSet 2 0 R >> /Contents 252 0 R >> endobj 253 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 255 0 obj << /Length 16691 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(V.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7771 62.22 Td [(3.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084335 Td [(\266)]TJ -0.0060244 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ 0 -0.084335 Td [(eme.)]TJ /F1 1 Tf 2.6747 0 Td [(\(de)-391(con)30(tin)30(uit)24(\266)548(e\))]TJ /F12 1 Tf -10.066 -1.759 Td [(Av)30(ec)-301(les)-301(notations)-301(de)]TJ /F1 1 Tf 9.506 0 Td [(V.3.1.)]TJ /F12 1 Tf 2.9819 0 Td [(,)-301(on)-301(supp)-30(ose)-295(que)]TJ /F8 1 Tf 7.4819 0 Td [(T)]TJ /F12 1 Tf 1.0241 0 Td [(est)-301(un)-295(espace)-301(m)24(\266)475(etrique)-301(et)-301(on)-295(\257xe)-301(un)-301(p)-30(oin)24(t)]TJ /F8 1 Tf -20.994 -1.1988 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(T)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -2.9157 -1.7651 Td [(On)-337(fait)-331(les)-337(h)24(yp)-30(oth)18(\265)475(eses)-337(:)]TJ /F1 1 Tf 0.49398 -1.759 Td [(\(i\))]TJ /F12 1 Tf 1.8795 0 Td [(p)-30(our)-337(presque)-331(tout)]TJ /F8 1 Tf 8.1807 0 Td [(x)]TJ /F12 1 Tf 0.90361 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-331(la)-337(fonction)]TJ /F8 1 Tf 5.5843 0 Td [(t)]TJ /F6 1 Tf 0.63855 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-337(con)30(tin)18(ue)-331(au)-337(p)-30(oin)24(t)]TJ /F8 1 Tf 9.5602 0 Td [(a)]TJ /F12 1 Tf 0.86747 0 Td [(;)]TJ /F1 1 Tf -34.548 -1.759 Td [(\(ii\))]TJ /F12 1 Tf 2.1988 0 Td [(il)-337(y)-337(a)-331(un)-337(v)24(oisinage)-337(de)]TJ /F8 1 Tf 9.6506 0 Td [(a)]TJ /F12 1 Tf 0.86144 0 Td [(dans)]TJ /F8 1 Tf 2.3373 0 Td [(T)]TJ /F12 1 Tf 1.0602 0 Td [(sur)-337(lequel)]TJ /F8 1 Tf 4.5361 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(est)-337(domin)24(\266)475(ee.)]TJ -21.753 -1.7651 Td [(Alors)-331(la)-337(fonction)]TJ /F8 1 Tf 7.6205 0 Td [(F)]TJ /F12 1 Tf 1.1145 0 Td [(est)-343(con)30(tin)18(ue)-331(au)-337(p)-30(oin)30(t)]TJ /F8 1 Tf 9.5663 0 Td [(a)]TJ /F12 1 Tf 0.53012 0 Td [(.)]TJ -18.831 -1.759 Td [(En)-325(particulier,)-331(si,)-331(p)-24(our)-331(presque)-325(tout)]TJ /F8 1 Tf 16.102 0 Td [(x)]TJ /F12 1 Tf 0.57229 0 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(t)]TJ /F6 1 Tf 0.63856 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3735 0 Td [(\))]TJ /F12 1 Tf 0.71687 0 Td [(est)-331(d)30(\266)469(e\257nie)-325(et)-331(con)30(tin)24(ue)-325(sur)]TJ /F8 1 Tf 11.53 0 Td [(T)]TJ /F12 1 Tf 1.0482 0 Td [(et)-331(si)]TJ /F8 1 Tf 2.1566 0 Td [(f)]TJ /F12 1 Tf 0.92168 0 Td [(est)]TJ -37.934 -1.1988 Td [(domin)30(\266)469(ee)-337(au)-337(v)24(oisinage)-337(de)-331(tout)-337(p)-30(oin)24(t)-337(de)]TJ /F8 1 Tf 17.066 0 Td [(T)]TJ /F12 1 Tf 1.0602 0 Td [(alors)]TJ /F8 1 Tf 2.3976 0 Td [(F)]TJ /F12 1 Tf 1.1145 0 Td [(est)-337(con)24(tin)24(ue)-337(sur)]TJ /F8 1 Tf 7.241 0 Td [(T)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 38.364 Td [(V.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7771 45.991 Td [(4.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084339 Td [(\266)]TJ -0.0060244 -0.084339 Td [(eor)]TJ 1.8012 0.084339 Td [(\265)]TJ 0 -0.084339 Td [(eme.)]TJ /F1 1 Tf 2.6747 0 Td [(\(de)-391(d)24(\266)548(eriv)66(ation)-391(sous)-385(l'in)30(t)24(\266)548(egrale\))]TJ /F12 1 Tf -10.066 -1.759 Td [(Av)30(ec)-343(les)-343(notations)-343(de)]TJ /F1 1 Tf 9.6747 0 Td [(V.3.1.)]TJ /F12 1 Tf 2.9819 0 Td [(,)-343(on)-337(supp)-30(ose)-343(que)]TJ /F8 1 Tf 7.6506 0 Td [(T)]TJ /F12 1 Tf 1.0602 0 Td [(est)-349(un)-337(ouv)18(ert)-343(de)]TJ /F3 1 Tf 7.4819 0 Td [(R)]TJ /F12 1 Tf 1.0602 0 Td [(ou)-343(un)-343(in)24(terv)60(alle)-343(de)]TJ /F3 1 Tf 8.5301 0 Td [(R)]TJ /F12 1 Tf -38.44 -1.2048 Td [(et)-337(on)-331(fait)-337(les)-337(h)24(yp)-30(oth)24(\265)469(eses)-337(:)]TJ /F1 1 Tf 0.49398 -1.759 Td [(\(i\))]TJ /F12 1 Tf 2.0361 0 Td [(p)-30(our)-415(presque)-415(tout)]TJ /F8 1 Tf 8.4217 0 Td [(x)]TJ /F12 1 Tf 0.98193 0 Td [(de)]TJ /F8 1 Tf 1.4157 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-427(la)-415(fonction)]TJ /F8 1 Tf 5.8434 0 Td [(t)]TJ /F6 1 Tf 0.77109 0 Td [(7!)]TJ /F8 1 Tf 1.4096 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F12 1 Tf 0.8012 0 Td [(est)-421(d)30(\266)469(e\257nie)-415(et)-415(d)30(\266)469(eriv)54(able)-415(en)-415(tout)]TJ -25.446 -1.7229 Td [(p)-30(oin)30(t)]TJ /F8 1 Tf 2.6084 0 Td [(t)]TJ /F12 1 Tf 0.6988 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(T)]TJ /F12 1 Tf 1.0602 0 Td [(a)30(v)24(ec)-337(p)-30(our)-337(d)30(\266)469(eriv)54(\266)469(ee)]TJ /F8 1 Tf 8.0723 0.68073 Td [(@)-54(f)]TJ ET 0.504 w 228.31 475.63 m 242.5 475.63 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.223 38.858 Td [(@)-54(t)]TJ /F2 1 Tf 1.1807 0.68674 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ /F1 1 Tf -17.392 -2.7169 Td [(\(ii\))]TJ /F12 1 Tf 2.1988 0 Td [(p)-30(our)-337(un)-331(ouv)18(ert)]TJ /F8 1 Tf 6.9277 0 Td [(!)]TJ /F12 1 Tf 0.98795 0 Td [(la)-337(fonction)]TJ /F8 1 Tf 5.0964 0.6747 Td [(@)-54(f)]TJ ET 247.61 443.16 m 261.79 443.16 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.837 36.141 Td [(@)-54(t)]TJ /F2 1 Tf 1.4639 0.68675 Td [(:)-277(\()]TJ /F8 1 Tf 0.94578 0 Td [(t;)-162(x)]TJ /F2 1 Tf 1.3735 0 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(7!)]TJ /F8 1 Tf 1.3976 0.6747 Td [(@)-60(f)]TJ ET 318.96 443.16 m 333.14 443.16 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.807 36.141 Td [(@)-54(t)]TJ /F2 1 Tf 1.1867 0.68675 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-337(domin)24(\266)475(ee)-337(sur)]TJ /F8 1 Tf 7.1807 0 Td [(!)]TJ /F6 1 Tf 0.88554 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(T)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -34.825 -2.7169 Td [(Alors,)-512(p)-24(our)]TJ /F8 1 Tf 5.6024 0 Td [(a)]TJ /F12 1 Tf 1.006 0 Td [(dans)]TJ /F8 1 Tf 2.4819 0 Td [(!)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.98193 0 Td [(T)]TJ /F12 1 Tf 0.72892 0 Td [(,)-506(la)-475(fonction)]TJ /F8 1 Tf 6.0361 0 Td [(x)]TJ /F6 1 Tf 1.0843 0 Td [(7!)]TJ /F8 1 Tf 1.6325 0.6747 Td [(@)-54(f)]TJ ET 309.02 410.68 m 323.21 410.68 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.976 33.424 Td [(@)-54(t)]TJ /F2 1 Tf 1.1867 0.68674 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a;)-174(x)]TJ /F2 1 Tf 1.5482 0 Td [(\))]TJ /F12 1 Tf 0.86144 0 Td [(est)-475(in)24(t)24(\266)469(egrable,)-512(la)-469(fonction)]TJ /F8 1 Tf 12.054 0 Td [(F)]TJ /F12 1 Tf 1.253 0 Td [(est)]TJ -37.934 -1.4819 Td [(d)30(\266)469(eriv)54(able)-337(au)-337(p)-30(oin)30(t)]TJ /F8 1 Tf 8.253 0 Td [(a)]TJ /F12 1 Tf 0.86747 0 Td [(et)-331(l'on)-337(a)-337(:)]TJ /F8 1 Tf 4.4699 -1.6687 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.198 44.9 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.988 30.961 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1.1205 -0.68073 Td [(@)-54(f)]TJ ET 283.82 373.03 m 298.01 373.03 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.867 30.274 Td [(@)-54(t)]TJ /F2 1 Tf 1.1867 0.68675 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a;)-174(x)]TJ /F2 1 Tf 1.5482 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -25.295 -2.8916 Td [(En)-397(particulier,)-415(si)]TJ /F8 1 Tf 8.0181 0.6747 Td [(@)-60(f)]TJ ET 159.55 338.47 m 173.74 338.47 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.47 27.382 Td [(@)-54(t)]TJ /F12 1 Tf 1.5843 0.68675 Td [(est)-397(domin)24(\266)475(ee)-403(au)-397(v)24(oisinage)-403(de)-397(tout)-397(p)-30(oin)24(t)-397(de)]TJ /F8 1 Tf 19.133 0 Td [(T)]TJ /F12 1 Tf 1.1205 0 Td [(alors)]TJ /F8 1 Tf 2.4639 0 Td [(F)]TJ /F12 1 Tf 1.1747 0 Td [(est)-403(d)24(\266)469(eriv)54(able)]TJ -33.614 -1.5361 Td [(sur)]TJ /F8 1 Tf 1.7289 0 Td [(T)]TJ /F12 1 Tf 1.1084 0 Td [(et)-391(la)-385(d)24(\266)469(eriv)54(\266)469(ee)-385(\(par)-397(rapp)-30(ort)-391(\265)500(a)]TJ /F8 1 Tf 12.645 0 Td [(t)]TJ /F12 1 Tf 0.36145 0 Td [(\))-391(de)-385(l'in)24(t)24(\266)469(egrale)]TJ /F8 1 Tf 6.8614 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.35542 0 Td [(\))]TJ /F12 1 Tf 0.77711 0 Td [(est)-391(l'in)18(t)24(\266)475(egrale)-391(de)-391(la)-385(d)24(\266)475(eriv)48(\266)475(ee)-391(\(par)]TJ -25.012 -1.1988 Td [(rapp)-30(ort)-337(\265)500(a)]TJ /F8 1 Tf 4.4759 0 Td [(t)]TJ /F12 1 Tf 0.36145 0 Td [(\))-337(de)]TJ /F8 1 Tf 2.0602 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-162(x)]TJ /F2 1 Tf 1.3735 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -9.6506 -2.1446 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Comme)-337(la)-331(d)24(\266)475(eriv)48(abilit)30(\266)469(e)-337(est)-337(une)-337(propri)24(\266)469(et)30(\266)469(e)-337(lo)-30(cale,)-337(on)-331(p)-30(eut)-337(se)-337(restreindre)]TJ -8.8193 -1.1988 Td [(\265)499(a)-331(un)-337(v)24(oisinage)-337(de)]TJ /F8 1 Tf 7.8976 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(ce)-337(qui)-331(p)-30(ermet,)-337(dans)-337(tous)-337(les)-331(cas)-337(de)-337(supp)-30(oser)-331(que)]TJ /F8 1 Tf 21.602 0 Td [(T)]TJ /F2 1 Tf 1.0602 0 Td [(est)-337(un)-337(in)24(terv)54(alle.)]TJ -31.422 -1.759 Td [(Soit)]TJ /F8 1 Tf 1.9759 0 Td [(a)]TJ /F2 1 Tf 0.78313 0 Td [(dans)]TJ /F8 1 Tf 2.2651 0 Td [(!)]TJ /F6 1 Tf 0.72289 0 Td [(\\)]TJ /F8 1 Tf 0.72891 0 Td [(T)]TJ /F2 1 Tf 0.72289 0 Td [(,)-271(notons)]TJ /F8 1 Tf 3.6988 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.267 29.538 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.488 20.232 Td [(l'ensem)24(ble)]TJ /F8 1 Tf 4.7349 0 Td [(T)]TJ /F2 1 Tf 0.98193 0 Td [(priv)54(\266)475(e)-259(de)]TJ /F8 1 Tf 3.6807 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-415(Il)-259(s'agit)-253(donc)-259(d')24(\266)475(etudier)-265(la)-253(limite)-259(p)-30(our)]TJ /F8 1 Tf 16.717 0 Td [(t)]TJ /F2 1 Tf -38.801 -1.7229 Td [(tendan)30(t)-240(v)24(ers)]TJ /F8 1 Tf 5.5602 0 Td [(a)]TJ /F2 1 Tf 0.76506 0 Td [(\(en)-240(restan)24(t)-234(dans)]TJ /F8 1 Tf 7.1386 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.931 27.073 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.801 18.509 Td [(\))-240(du)-234(taux)-240(d'accroissemen)24(t)]TJ /F8 1 Tf 11.355 0.6747 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ ET 372.38 224.2 m 434.95 224.2 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.717 17.822 Td [(t)]TJ /F6 1 Tf 0.58434 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 2.488 0.68675 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(h)]TJ /F2 1 Tf 0.57229 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ -38.777 -2.3916 Td [(o)-24(\265)524(u)-325(l'on)-331(a)-325(p)-30(os)24(\266)475(e)]TJ /F8 1 Tf 6.3916 0 Td [(h)]TJ /F2 1 Tf 0.57831 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.8434 0.68072 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-174(x)]TJ /F2 1 Tf 1.5422 0 Td [(\))]TJ ET 190.15 195.62 m 272.59 195.62 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.301 15.43 Td [(t)]TJ /F6 1 Tf 0.58434 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F6 1 Tf 3.3675 0.68675 Td [(\242)]TJ /F2 1 Tf 0.60241 0 Td [(On)-331(remarque)-325(que,)-331(p)-30(our)-325(presque)-331(tout)]TJ /F8 1 Tf 16.367 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-331(lorsque)]TJ /F8 1 Tf -35.464 -2.241 Td [(t)]TJ /F2 1 Tf 0.69277 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(\(en)-337(restan)24(t)-337(dans)]TJ /F8 1 Tf 7.4337 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.741 20.452 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.669 13.876 Td [(\))]TJ /F8 1 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Tf 8.6626 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(,)-349(le)-349(th)24(\266)475(eor)18(\265)475(eme)-343(de)-343(con)24(v)24(ergence)]TJ /F5 1 Tf -26.584 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(74)]TJ ET endstream endobj 257 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 256 0 R /ProcSet 2 0 R >> /Contents 255 0 R >> endobj 256 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 258 0 obj << /Length 17817 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ /F2 1 Tf 0 -2.7711 Td [(domin)30(\266)469(ee)-361(p)-30(ermet)-361(alors)-361(de)-355(conclure)-355(que)-361(la)-361(fonction)]TJ /F8 1 Tf 22.072 0 Td [(x)]TJ /F6 1 Tf 0.89156 0 Td [(7!)]TJ /F8 1 Tf 1.4337 0.6747 Td 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Tf 0.74096 0 Td [(son)18(t)-355(in)24(t)24(\266)469(egrables,)-361(la)-355(fonction)]TJ /F8 1 Tf 12.53 0 Td [(F)]TJ /F12 1 Tf 1.1386 0 Td [(est)]TJ /F8 1 Tf -37.934 -1.4819 Td [(n)]TJ /F12 1 Tf 0.93373 0 Td [(fois)-337(d)30(\266)469(eriv)54(able)-337(au)-337(p)-24(oin)24(t)]TJ /F8 1 Tf 10.066 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(et)-337(l'on)-337(a)-331(p)-30(our)]TJ /F2 1 Tf 6.3133 0 Td [(1)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(p)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(n)]TJ /F12 1 Tf 0.93373 0 Td [(:)]TJ /F8 1 Tf -9.8253 -1.9759 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.293 43.711 Td [(\()]TJ /F7 1 Tf 0.44827 0 Td [(p)]TJ /F4 1 Tf 0.58621 0 Td [(\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.157 30.135 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1.1205 -0.68675 Td [(@)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.069 44.607 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.97 30.81 Td [(f)]TJ ET 285.91 363.16 m 305.64 363.16 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.036 29.448 Td [(@)-60(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.759 42.556 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.687 30.135 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-168(x)]TJ /F2 1 Tf 1.5422 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -25.928 -2.7651 Td [(En)-222(particulier,)-246(si)-228(les)]TJ /F8 1 Tf 8.8434 0.68073 Td [(@)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.121 40.659 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.223 28.051 Td [(f)]TJ ET 169.42 330.19 m 189.14 330.19 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.289 26.683 Td [(@)-60(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.81 38.599 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 16.163 27.37 Td [(son)24(t)-228(domin)30(\266)469(ees)-222(au)-222(v)24(oisinage)-228(de)-222(tout)-228(p)-24(oin)24(t)-228(de)]TJ /F8 1 Tf 18.723 0 Td [(T)]TJ /F12 1 Tf 0.94579 0 Td [(alors)]TJ /F8 1 Tf 2.2892 0 Td [(F)]TJ /F12 1 Tf 1 0 Td [(est)-228(d)24(\266)469(eriv)54(able)]TJ /F8 1 Tf -33.789 -1.5361 Td [(n)]TJ /F12 1 Tf 0.96385 0 Td [(fois)-367(sur)]TJ /F8 1 Tf 3.5482 0 Td [(T)]TJ /F12 1 Tf 1.0904 0 Td [(et)-367(la)-361(d)24(\266)469(eriv)54(\266)469(ee)]TJ /F8 1 Tf 5.7651 0 Td [(p)]TJ /F12 1 Tf 0.5 0 Td [(-i)24(\265)469(eme)-361(\(par)-373(rapp)-30(ort)-367(\265)499(a)]TJ /F8 1 Tf 9.4458 0 Td [(t)]TJ /F12 1 Tf 0.36145 0 Td [(\))-367(de)-361(l'in)24(t)24(\266)469(egrale)]TJ /F8 1 Tf 6.7892 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F12 1 Tf 0.75301 0 Td [(est)-367(l'in)24(t)24(\266)469(egrale)-367(de)-367(la)]TJ -30.753 -1.1988 Td [(d)30(\266)469(eriv)54(\266)469(ee)]TJ /F8 1 Tf 3.3916 0 Td [(p)]TJ /F12 1 Tf 0.50603 0 Td [(-i)30(\265)469(eme)-331(\(par)-343(rapp)-30(ort)-337(\265)500(a)]TJ /F8 1 Tf 9.3133 0 Td [(t)]TJ /F12 1 Tf 0.36145 0 Td [(\))-337(de)]TJ /F8 1 Tf 2.0602 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-162(x)]TJ /F2 1 Tf 1.3735 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -18.386 -1.9398 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(C'est)-337(imm)24(\266)469(ediat)-337(par)-337(r)30(\266)469(ecurrence)-337(sur)]TJ /F8 1 Tf 15.307 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(.)]TJ ET 512.64 270.03 m 519.84 270.03 l 519.84 278.46 l 512.64 278.46 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 20.75 Td [(Remarque.)]TJ /F2 1 Tf 6.3133 0 Td [(En)-355(fait,)-355(dans)-349(le)-355(th)24(\266)469(eor)24(\265)469(eme)-349(pr)24(\266)475(ec)24(\266)469(eden)24(t,)-355(on)-349(p)-30(eut)-355(mon)24(trer)-349(qu'une)-355(h)24(yp)-30(oth)24(\265)469(ese)-349(de)]TJ -6.3133 -1.1988 Td [(domination)-331(p)-30(ortan)18(t)-331(sur)-343(les)-331(d)24(\266)469(eriv)54(\266)469(ees)]TJ /F8 1 Tf 15.807 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(-i)24(\265)475(emes)-337(uniquemen)18(t)-337(est)-337(su\261san)24(te)-337(p)-30(our)-337(conclure.)]TJ -16.41 -2.259 Td [(On)-367(p)-30(eut)-367(m)24(^)469(eme)-361(donner)-373(une)-367(v)24(ersion)-373(de)-361(ce)-373(th)24(\266)469(eor)24(\265)469(eme)-361(p)-30(our)-367(les)-373(fonctions)-367(de)]TJ /F8 1 Tf 32.217 0 Td [(q)]TJ /F2 1 Tf 0.8494 0 Td [(v)54(ariables.)-542(Les)]TJ -33.066 -1.1988 Td [(ordres)-397(de)-397(d)24(\266)469(eriv)54(ation)-397(seron)24(t)-397(rep)-6(\266)475(er)24(\266)469(es)-397(par)-397(un)]TJ /F8 1 Tf 19.127 0 Td [(q)]TJ /F2 1 Tf 0.48193 0 Td [(-uplet)]TJ /F8 1 Tf 2.9518 0 Td [(\256)]TJ /F2 1 Tf 1.0301 0 Td [(=)-385(\()]TJ /F8 1 Tf 1.5482 0 Td [(\256)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.517 22.814 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.56 16.093 Td [(;)-168(:)-168(:)-162(:)-168(;)-168(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.259 22.814 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.867 16.093 Td [(\))-397(o)-30(\265)530(u)]TJ /F8 1 Tf 2.2349 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.009 22.814 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.633 16.093 Td [(est)-403(l'ordre)-397(de)]TJ -33.301 -1.4337 Td [(d)30(\266)469(eriv)54(ation)-337(par)-337(rapp)-30(ort)-337(\265)500(a)-331(la)-337(v)54(ariable)]TJ /F8 1 Tf 15.825 0 Td [(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.793 20.762 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.012 14.659 Td [(,)-331(l'ordre)-337(de)-337(d)30(\266)469(eriv)54(ation)-337(total)-307(\266)475(etan)24(t)]TJ /F6 1 Tf 14.789 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(\256)]TJ /F6 1 Tf 0.63856 0 Td [(j)]TJ /F2 1 Tf 0.56024 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.517 20.762 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.084 14.659 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 9.4394 Td [(V.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7771 11.316 Td [(6.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084337 Td [(\266)]TJ -0.0060244 -0.084337 Td [(eor)]TJ 1.8012 0.084337 Td [(\265)]TJ 0 -0.084337 Td [(eme.)]TJ /F12 1 Tf 2.8072 0 Td [(Av)24(ec)-379(les)-379(notations)-379(de)]TJ /F1 1 Tf 9.8253 0 Td [(V.3.1.)]TJ /F12 1 Tf 2.9819 0 Td [(,)-385(on)-379(supp)-30(ose)-379(que)]TJ /F8 1 Tf 7.8072 0 Td [(T)]TJ /F12 1 Tf 1.1024 0 Td [(est)-379(un)-379(ouv)24(ert)-379(de)]TJ /F3 1 Tf -31.916 -1.1988 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 15.073 Td [(q)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 6.494 10.117 Td [(,)-337(on)-331(\257xe)-337(un)-337(en)24(tier)]TJ /F8 1 Tf 8.1205 0 Td [(n)]TJ /F11 1 Tf 0.8735 0 Td [(>)]TJ /F2 1 Tf 1.0602 0 Td [(1)]TJ /F12 1 Tf 0.83133 0 Td [(et)-337(on)-337(fait)-331(les)-337(h)24(yp)-30(oth)18(\265)475(eses)-337(:)]TJ /F1 1 Tf -11.554 -1.6265 Td [(\(i\))]TJ /F12 1 Tf 1.8494 0 Td [(p)-30(our)-319(presque)-319(tout)]TJ /F8 1 Tf 8.1325 0 Td [(x)]TJ /F12 1 Tf 0.89157 0 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-319(la)-319(fonction)]TJ /F8 1 Tf 5.5422 0 Td [(t)]TJ /F6 1 Tf 0.63855 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F12 1 Tf 0.70482 0 Td [(est)-325(de)-319(classe)]TJ /F8 1 Tf 5.6386 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.345 12.737 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.741 8.4906 Td [(sur)]TJ /F8 1 Tf 1.6566 0 Td [(T)]TJ /F12 1 Tf 0.72892 0 Td [(,)-319(de)-319(d)30(\266)469(eriv)54(\266)469(ees)]TJ -33.795 -1.9157 Td [(partielles)]TJ /F8 1 Tf 4.4096 0.6747 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.776 10.892 Td [(j)]TJ /F7 1 Tf 0.33621 0 Td [(\256)]TJ /F10 1 Tf 0.75 0 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.367 7.2497 Td [(f)]TJ ET 116.42 81.572 m 143.14 81.572 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.096 5.8882 Td [(@)-54(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.802 8.84 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.09 6.575 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(\()]TJ /F2 1 Tf 0.38554 0 Td [(1)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F6 1 Tf 1.0542 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F12 1 Tf 0.60241 0 Td [(\))-337(;)]TJ /F2 1 Tf 4.3133 -2.6867 Td [(75)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 260 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 259 0 R /ProcSet 2 0 R >> /Contents 258 0 R >> endobj 259 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 261 0 obj << /Length 16995 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ 0.1747 -2.9699 Td [(\(ii\))]TJ /F12 1 Tf 2.3916 0 Td [(sur)-433(un)-433(ouv)18(ert)]TJ /F8 1 Tf 6.5301 0 Td [(!)]TJ /F12 1 Tf 1.0843 0 Td [(de)]TJ /F8 1 Tf 1.4337 0 Td [(T)]TJ /F12 1 Tf 1.1566 0 Td [(et)-427(p)-30(our)-433(tout)]TJ /F8 1 Tf 5.988 0 Td [(\256)]TJ /F12 1 Tf 1.0723 0 Td [(\()]TJ /F2 1 Tf 0.39157 0 Td [(1)]TJ /F11 1 Tf 0.93976 0 Td [(6)]TJ /F6 1 Tf 1.2169 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F11 1 Tf 0.71687 0 Td [(6)]TJ /F8 1 Tf 1.2169 0 Td [(n)]TJ /F12 1 Tf 0.60241 0 Td [(\))-433(la)-427(d)24(\266)469(eriv)54(\266)469(ee)-433(d'ordre)]TJ /F8 1 Tf 9.0663 0 Td [(\256)]TJ 1.1928 0.68073 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.121 89.719 Td [(j)]TJ /F7 1 Tf 0.34483 0 Td [(\256)]TJ /F10 1 Tf 0.74138 0 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.06 62.334 Td [(f)]TJ ET 0.504 w 495.14 739.94 m 521.86 739.94 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.783 60.967 Td [(@)-60(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.147 87.659 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 61.653 Td [(:)]TJ -38.886 -2.3494 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(7!)]TJ /F8 1 Tf 1.3976 0.6747 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.948 86.357 Td [(j)]TJ /F7 1 Tf 0.34483 0 Td [(\256)]TJ /F10 1 Tf 0.74138 0 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.795 59.979 Td [(f)]TJ ET 109.51 711.79 m 136.22 711.79 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.5181 58.617 Td [(@)-54(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.974 84.297 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.512 59.304 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t;)-162(x)]TJ /F2 1 Tf 1.3735 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-337(domin)24(\266)469(ee)-337(sur)]TJ /F8 1 Tf 7.1867 0 Td [(!)]TJ /F6 1 Tf 0.87952 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(T)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -18.349 -2.9337 Td [(Alors,)-355(p)-30(our)]TJ /F8 1 Tf 5.3313 0 Td [(a)]TJ /F12 1 Tf 0.87952 0 Td [(dans)]TJ /F8 1 Tf 2.3554 0 Td [(!)]TJ /F12 1 Tf 0.66265 0 Td [(,)-355(les)-355(fonctions)]TJ /F8 1 Tf 6.3735 0 Td [(x)]TJ /F6 1 Tf 0.87952 0 Td [(7!)]TJ /F8 1 Tf 1.4277 0.68073 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.103 82.159 Td [(j)]TJ /F7 1 Tf 0.33621 0 Td [(\256)]TJ /F10 1 Tf 0.75 0 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.873 57.051 Td [(f)]TJ ET 277.78 676.8 m 304.49 676.8 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.602 55.683 Td [(@)-54(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.129 80.099 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.596 56.37 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a;)-174(x)]TJ /F2 1 Tf 1.5482 0 Td [(\))]TJ /F12 1 Tf 0.74096 0 Td [(son)24(t)-355(in)30(t)24(\266)469(egrables,)-361(la)-349(fonction)]TJ /F8 1 Tf 12.512 0 Td [(F)]TJ /F12 1 Tf 1.1325 0 Td [(est)-355(de)]TJ -36.584 -1.5361 Td [(classe)]TJ /F8 1 Tf 2.7892 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.75 79.064 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7831 54.834 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(!)]TJ /F12 1 Tf 0.99398 0 Td [(et)-337(l'on)-337(a)-331(p)-30(our)]TJ /F2 1 Tf 6.3133 0 Td [(1)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F6 1 Tf 1.0542 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F11 1 Tf 0.56024 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F12 1 Tf 0.93374 0 Td [(:)]TJ /F8 1 Tf -6.1747 -1.6265 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.44 76.659 Td [(j)]TJ /F7 1 Tf 0.34483 0 Td [(\256)]TJ /F10 1 Tf 0.74138 0 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.524 53.207 Td [(F)]TJ ET 213.84 630.86 m 242.71 630.86 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.337 51.846 Td [(@)-60(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.603 74.607 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.428 52.533 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1.1205 -0.68675 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.448 76.659 Td [(j)]TJ /F7 1 Tf 0.34483 0 Td [(\256)]TJ /F10 1 Tf 0.74138 0 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.819 53.207 Td [(f)]TJ ET 289.08 630.86 m 315.79 630.86 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.542 51.846 Td [(@)-60(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.474 74.607 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.536 52.533 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-168(x)]TJ /F2 1 Tf 1.5422 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -26.777 -3.1084 Td [(En)-361(particulier,)-367(si)-367(les)]TJ /F8 1 Tf 9.3795 0.6747 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.897 72.219 Td [(j)]TJ /F7 1 Tf 0.33621 0 Td [(\256)]TJ /F10 1 Tf 0.74138 0 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.343 50.099 Td [(f)]TJ ET 175.82 593.71 m 202.54 593.71 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.066 48.738 Td [(@)-60(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.922 70.159 Td [(\256)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.422 49.424 Td [(son)24(t)-367(domin)30(\266)469(ees)-361(au)-361(v)24(oisinage)-367(de)-361(tout)-367(p)-24(oin)24(t)-367(de)]TJ /F8 1 Tf 19.831 0 Td [(T)]TJ /F12 1 Tf 1.0843 0 Td [(alors)]TJ /F8 1 Tf 2.4277 0 Td [(F)]TJ /F12 1 Tf 1.1386 0 Td [(est)-367(de)]TJ -36.572 -1.5361 Td [(classe)]TJ /F8 1 Tf 2.7892 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.75 69.116 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7892 47.888 Td [(sur)]TJ /F8 1 Tf 1.6807 0 Td [(T)]TJ /F12 1 Tf 1.0542 0 Td [(et)-337(la)-337(d)24(\266)469(eriv)54(\266)469(ee)-337(d'ordre)]TJ /F8 1 Tf 9.1325 0 Td [(\256)]TJ /F12 1 Tf 0.98193 0 Td [(\(par)-343(rapp)-30(ort)-337(\265)500(a)]TJ /F8 1 Tf 6.6566 0 Td [(t)]TJ /F12 1 Tf 0.36145 0 Td [(\))-337(de)-337(l'in)24(t)24(\266)469(egrale)]TJ /F8 1 Tf 6.7048 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-343(l'in)24(t)24(\266)475(egrale)]TJ -33.289 -1.2048 Td [(de)-331(la)-337(d)30(\266)469(eriv)54(\266)469(ee)-337(d'ordre)]TJ /F8 1 Tf 9.2892 0 Td [(\256)]TJ /F12 1 Tf 0.9759 0 Td [(\(par)-343(rapp)-30(ort)-337(\265)500(a)]TJ /F8 1 Tf 6.6506 0 Td [(t)]TJ /F12 1 Tf 0.36145 0 Td [(\))-337(de)]TJ /F8 1 Tf 2.0602 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t;)-168(x)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -22.09 -2.1687 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(L'application)-493(du)-493(th)24(\266)475(eor)18(\265)475(eme)-493(pr)30(\266)469(ec)24(\266)469(eden)30(t)-493(successiv)24(emen)30(t)-493(\265)500(a)-493(c)24(hacune)-493(des)]TJ -8.8193 -1.1988 Td [(v)54(ariables)]TJ /F8 1 Tf 4.1988 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.155 61.771 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.343 43.316 Td [(;)-168(:)-168(:)-162(:)-168(;)-168(t)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.5 61.771 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.759 43.316 Td [(donne)-385(l'existence)-391(de)-391(la)-385(d)24(\266)475(eriv)48(\266)475(ee)-391(d'ordre)]TJ /F8 1 Tf 17.367 0 Td [(\256)]TJ /F2 1 Tf 1.0301 0 Td [(de)]TJ /F8 1 Tf 1.3916 0 Td [(F)]TJ /F2 1 Tf 1.1687 0 Td [(et)-385(son)-397(expression)-385(sous)]TJ -29.386 -1.1988 Td [(forme)-421(d'in)24(t)24(\266)475(egrale)-427(de)-421(la)-421(d)30(\266)469(eriv)54(\266)469(ee)-421(d'ordre)]TJ /F8 1 Tf 17.542 0 Td [(\256)]TJ /F2 1 Tf 1.0602 0 Td [(de)]TJ /F8 1 Tf 1.4217 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)-704(Le)-421(seul)-427(p)-30(oin)24(t)-421(nouv)30(eau)-421(\265)500(a)-421(mon)24(trer)-421(est)-427(la)]TJ -20.62 -1.2048 Td [(con)30(tin)18(uit)24(\266)469(e)-307(des)-307(d)24(\266)475(eriv)48(\266)475(ees)-307(de)]TJ /F8 1 Tf 11.404 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(.)-433(Cep)-30(endan)24(t)-307(celle-ci)-301(d)24(\266)469(ecoule,)-307(gr)0(^)499(ace)-313(au)-301(th)18(\266)475(eor)18(\265)475(eme)-307(de)-301(con)24(tin)24(uit)18(\266)475(e,)]TJ -12.187 -1.1988 Td [(de)-343(la)-349(con)30(tin)18(uit)24(\266)469(e)-349(des)-349(d)24(\266)475(eriv)48(\266)475(ees)-349(de)]TJ /F8 1 Tf 14.036 0 Td [(f)]TJ /F2 1 Tf 0.94578 0 Td [(par)-349(rapp)-30(ort)-349(\265)500(a)]TJ /F8 1 Tf 6.3012 0 Td [(t)]TJ /F2 1 Tf 0.70482 0 Td [(et)-349(de)-343(la)-349(relation)-349(de)-343(domination)-349(qu'elles)]TJ -21.988 -1.1988 Td [(v)54(\266)469(eri\257en)24(t.)]TJ ET 512.64 459.1 m 519.84 459.1 l 519.84 467.53 l 512.64 467.53 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 36.34 Td [(Remarque.)]TJ /F2 1 Tf 6.512 0 Td [(Nous)-397(laissons)-403(au)-397(lecteur)-403(le)-403(soin)-403(de)-397(donner)-403(une)-397(v)24(ersion)-403(de)-397(ce)-403(th)24(\266)469(eor)24(\265)469(eme)-397(en)]TJ -6.512 -1.1988 Td [(termes)-331(de)-337(fonctions)-331(di\256)24(\266)469(eren)24(tiables)-337(\(par)-337(rapp)-30(ort)-343(\265)499(a)]TJ /F8 1 Tf 21.873 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 25.379 Td [(V.4.)-241(Somme)-376(de)-381(fonctions)-376(in)25(t)25(\266)537(egrables.)]TJ /F15 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 28.256 Td [(Th)]TJ 1.4036 0.078314 Td [(\266)]TJ -0.0060239 -0.078314 Td [(eor)]TJ 1.8012 0.078314 Td [(\265)]TJ -0.0060244 -0.078314 Td [(eme.)]TJ /F12 1 Tf 2.7048 0 Td [(Soit)]TJ /F2 1 Tf 2.0663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.284 40.219 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.506 28.256 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.319 40.219 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 16.518 28.256 Td [(une)-349(famille)-343(d)30(\266)469(enom)24(brable)]TJ ET 259.13 336.02 m 325.94 336.02 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 27.608 28.256 Td [(de)-343(fonctions)-343(in)24(t)24(\266)469(egrables)-343(\265)500(a)-343(v)54(aleurs)-349(dans)]TJ -22.277 -1.2048 Td [(un)-337(espace)-331(v)24(ectoriel)-337(de)-331(dimension)-343(\257nie.)]TJ 0 -2.2349 Td [(Alors,)-337(on)-331(a)-337(dans)]TJ ET 153.22 306.5 m 161.86 306.5 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 12.819 24.816 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.379 35.297 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.542 24.816 Td [(l')30(\266)469(egalit)24(\266)469(e)]TJ /F9 1 Tf 4.7229 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.724 33.806 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 20.873 26.177 Td [(Z)]TJ /F6 1 Tf 1.006 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.724 35.297 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.199 24.816 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ 1 -0.40964 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.517 33.806 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.91 24.816 Td [(k)]TJ /F8 1 Tf 0.50603 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.086 35.297 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.235 24.816 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -33.006 -3.3916 Td [(En)-240(outre,)-265(si)-246(les)-246(quan)24(tit)30(\266)469(es)-240(pr)24(\266)469(ec)24(\266)475(eden)24(tes)-246(son)24(t)-246(\257nies,)-259(la)-246(famille)-240(des)]TJ /F9 1 Tf 26.741 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.034 30.443 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.898 21.424 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F12 1 Tf 0.63253 0 Td [(est)-246(sommable,)]TJ -33.072 -2.012 Td [(la)-451(famille)-451(des)]TJ /F8 1 Tf 6.4398 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.552 27.564 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.596 19.412 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.83735 0 Td [(est)-457(sommable)-451(presque)-451(partout,)-487(la)-451(somme)]TJ /F8 1 Tf 18.964 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-475(=)]TJ /F9 1 Tf 2.1145 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.155 26.073 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.645 19.412 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.009 27.564 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.47 19.412 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.83735 0 Td [(\(d)24(\266)469(e\257nie)]TJ -35.94 -2.1988 Td [(presque)-331(partout\))-337(est)-343(in)30(t)24(\266)469(egrable)-331(et)-337(l'on)-337(a)]TJ /F9 1 Tf 10.91 -0.75904 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.388 20.478 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 17.849 16.864 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.681 21.969 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.675 15.503 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ 1.006 -0.40964 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.75 20.478 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.886 15.503 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.603 21.969 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.711 15.503 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.7229 0 Td [(.)]TJ /F1 1 Tf -27.976 -3.4699 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Remarquons)-289(tout)-289(d'ab)-30(ord)-289(que)]TJ /F9 1 Tf 13.319 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.466 15.512 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.078 12.033 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.034 17.004 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.404 12.033 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F2 1 Tf 0.78916 0 Td [(est)-295(mesurable)-289(ce)-289(qui)-289(justi\257e)]TJ -27.211 -2.8916 Td [(l'utilisation)-313(du)-319(sym)24(b)-30(ole)]TJ /F9 1 Tf 10.542 1.3675 Td [(Z)]TJ /F2 1 Tf 1.3133 -1.3675 Td [(plut)-6(^)500(ot)-313(que)]TJ /F9 1 Tf 4.8253 1.3675 Td [(Z)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.94 14.676 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.789 9.1412 Td [(dans)-313(la)-319(premi)24(\265)469(ere)-289(\266)469(egalit)24(\266)475(e.)-445(Cette)-319(premi)24(\265)469(ere)-289(\266)469(egalit)24(\266)475(e)]TJ -18.458 -1.6506 Td [(est)-259(une)-253(cons)24(\266)469(equence)-253(imm)24(\266)469(ediate)-259(de)-253(l'additivit)30(\266)469(e)-259(d)30(\266)469(enom)24(brable)-253(de)-259(l'in)24(t)24(\266)475(egrale)-259(p)-30(our)-253(les)-259(fonctions)]TJ 0 -1.2048 Td [(mesurables)-337(p)-30(ositiv)30(es.)]TJ /F5 1 Tf 0 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(76)]TJ ET endstream endobj 263 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 262 0 R /ProcSet 2 0 R >> /Contents 261 0 R >> endobj 262 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 264 0 obj << /Length 15407 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8765(V.)-385(Les)-385(th)18(\266)548(eor)24(\265)548(emes)-385(de)-391(con)30(v)30(ergence)]TJ /F2 1 Tf 0 -2.7651 Td [(Lorsque)]TJ /F9 1 Tf 3.8012 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.224 86.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 10.747 63.22 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.224 88.306 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.072 61.858 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-373(est)-367(\257nie,)-379(la)-361(ma)-60(joration)]TJ /F9 1 Tf 11.151 0.8494 Td [(\260)]TJ 0 -0.59639 Td [(\260)]TJ 0.72289 1.1084 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.095 88.306 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.145 61.858 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.8494 Td [(\260)]TJ 0 -0.59639 Td [(\260)]TJ /F11 1 Tf 0.88554 -0.25301 Td [(6)]TJ /F9 1 Tf 1.1084 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.379 88.306 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.729 61.858 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F8 1 Tf 0.66867 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ -38.777 -2.8072 Td [(assure)-283(que)]TJ /F9 1 Tf 4.7771 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.612 82.788 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 11.717 59.9 Td [(\260)]TJ 0 -0.60241 Td [(\260)]TJ 0.72289 1.1145 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.94 84.288 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.265 59.051 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.8494 Td [(\260)]TJ 0 -0.60241 Td [(\260)]TJ /F2 1 Tf 0.83735 -0.24699 Td [(est)-259(\266)469(egalemen)24(t)-283(\257ni)-289(et)-283(donc)-289(que)-283(la)-283(famille)-289(des)]TJ /F9 1 Tf 19.036 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.095 84.288 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.229 59.051 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ -38.777 -2.1446 Td [(est)-337(sommable.)]TJ 0 -1.8614 Td [(Lorsque)]TJ /F9 1 Tf 3.753 1.3615 Td [(Z)]TJ 1 -0.41566 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.586 77.056 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.699 55.045 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.155 78.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.018 55.045 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-319(est)-319(\257ni)-313(on)-319(sait)-313(que)]TJ /F9 1 Tf 9.0361 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.983 77.056 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.042 55.045 Td [(k)]TJ /F8 1 Tf 0.50602 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.552 78.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.367 55.045 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F2 1 Tf 0.81325 0 Td [(est)-319(\257ni)-319(presque)-319(partout.)-439(Mais)]TJ -26.199 -2.1988 Td [(dire)-331(que)-337(cette)-331(quan)24(tit)24(\266)475(e)-337(est)-337(\257nie)-331(signi\257e)-331(que)-337(la)-331(famille)-337(des)]TJ /F8 1 Tf 25.253 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.466 75.409 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.41 52.846 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(est)-337(sommable.)-445(Celle-ci)-331(est)]TJ -27.036 -1.4036 Td [(donc)-313(sommable)-313(presque)-313(partout)-313(ce)-313(qui)-313(d)30(\266)469(e\257nit)-307(presque)-313(partout)-319(la)-307(somme)]TJ /F8 1 Tf 31.819 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.017 71.909 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.042 51.442 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.871 73.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.867 51.442 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\).)]TJ -38.494 -2.5422 Td [(En\257n,)-319(si)-319(l'on)-289(\266)469(ecrit)]TJ /F8 1 Tf 8.1205 0 Td [(I)]TJ /F2 1 Tf 0.83133 0 Td [(comme)-313(r)24(\266)475(eunion)-319(d'une)-319(suite)-319(croissan)30(te)-319(de)-313(parties)-319(\257nies)]TJ /F8 1 Tf 23.753 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.06 69.762 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.018 48.9 Td [(,)-319(on)-319(v)24(oit)-313(que)]TJ /F9 1 Tf -33.578 -0.39759 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.6293 66.34 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 13.012 92.523 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1627 47.557 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.957 67.84 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.988 47.557 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-331(tend)-331(presque)-331(partout)-331(v)24(ers)]TJ /F8 1 Tf 12.301 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-331(et)-325(on)-331(a)-325(la)-331(relation)-325(de)-325(domination)]TJ /F6 1 Tf 15.458 0 Td [(k)]TJ /F9 1 Tf 0.77711 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.716 66.34 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 80.217 92.523 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.765 47.557 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.034 67.84 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.59 47.557 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(k)]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F9 1 Tf -38.386 -1.9096 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.7759 62.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 44.695 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.353 63.745 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2651 44.695 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F2 1 Tf 0.77711 0 Td [(o)-30(\265)530(u)-277(le)-283(deuxi)24(\265)475(eme)-277(mem)24(bre)-277(est)-283(mesurable.)-427(Lorsque)]TJ /F9 1 Tf 20.777 1.3614 Td [(Z)]TJ 1 -0.40964 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.181 62.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.777 44.695 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.759 63.745 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.102 44.695 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(k)]TJ /F8 1 Tf 0.66868 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))-283(est)-283(\257ni,)-295(ce)]TJ -34.145 -2.1446 Td [(deuxi)30(\265)469(eme)-325(mem)24(bre)-331(est)-331(in)24(t)24(\266)475(egrable,)-331(le)-337(th)24(\266)475(eor)18(\265)475(eme)-331(de)-325(con)24(v)24(ergence)-331(domin)30(\266)469(ee)-331(s'applique)-325(et)-331(l'on)-331(a)]TJ /F9 1 Tf 0 -0.28915 Td [(Z)]TJ 1.1084 -0.41566 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 56.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 15.012 79.198 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1627 40.894 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.388 58.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.988 40.894 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))-349(=)]TJ /F9 1 Tf 1.9699 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.069 56.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 31.795 79.198 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 16.56 42.262 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.828 58.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.38 40.894 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-373(qui)-379(tend)-379(v)24(ers)]TJ /F9 1 Tf 6.9217 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-349(=)]TJ /F9 1 Tf 1.8614 1.3675 Td [(Z)]TJ 1 -0.41566 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.836 56.814 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.127 40.894 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.698 58.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.952 40.894 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\).)]TJ -38.494 -2.8554 Td [(Comme,)-451(par)-427(d)30(\266)469(e\257nition)-427(la)-427(limite)-421(de)]TJ /F9 1 Tf 16.139 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.56 52.728 Td [(i)]TJ /F10 1 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0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F13 39 0 R >> endobj 267 0 obj << /Length 229 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)]TJ /F5 1 Tf 0 -60.735 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(78)]TJ ET endstream endobj 269 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 268 0 R /ProcSet 2 0 R >> /Contents 267 0 R >> endobj 268 0 obj << /F1 12 0 R /F2 13 0 R /F5 16 0 R >> endobj 270 0 obj << /Length 10866 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F0 1 Tf 20.592 0 0 20.592 0 0 Tm 3.2133 32.159 Td [(Chapitre)-377(VI.)-1129(L'in)27(t)20(\266)538(egrale)-377(de)-377(Leb)-34(esgue)-377(sur)-377(I)164(R)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 50.997 Td 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Tf 0 -1.1988 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(ou)-337(une)-331(partie)-343(mesurable)-331(de)]TJ /F3 1 Tf 12.241 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\).)]TJ -14.018 -1.5602 Td [(Dans)-361(la)-367(pratique)-367(les)-361(parties)-367(de)]TJ /F3 1 Tf 13.88 0 Td [(R)]TJ /F2 1 Tf 1.0843 0 Td [(et)-361(les)-367(fonctions)-361(que)-367(nous)-361(utiliserons)-367(seron)24(t)-361(b)-30(or)24(\266)475(eliennes,)]TJ -14.964 -1.1988 Td [(d'ailleurs,)-295(quitte)-283(\265)500(a)-283(faire)-289(des)-289(mo)-30(di\257cations)-283(sur)-289(un)-289(ensem)24(ble)-283(n)24(\266)475(egligeable)-289(il)-283(est)-289(toujours)-289(p)-30(ossi-)]TJ 0 -1.2048 Td [(ble)-319(de)-325(se)-319(ramener)-325(\265)500(a)-319(ce)-325(cas.)-439(Il)-325(est)-325(donc)-325(p)-24(ossible)-325(de)-319(restreindre)-325(la)-319(mesure)-325(de)-325(Leb)-30(esgue)-325(p)-30(our)]TJ 0 -1.1988 Td [(ne)-331(consid)24(\266)475(erer)-337(que)-337(la)-337(mesure)-337(de)-337(Borel.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 21.671 Td [(VI.1.)-236(In)30(t)20(\266)537(egrale)-376(orien)30(t)20(\266)537(ee.)]TJ 0 -2.4573 Td [(VI.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 23.033 Td [(1.)]TJ /F2 1 Tf 1.753 0 Td [(Remarquons)-391(tout)-391(d'ab)-30(ord)-397(que)-391(les)-397(singletons)-391(son)24(t)-397(des)-391(parties)-391(de)-391(mesure)-397(n)24(ulle.)]TJ -4.6988 -1.1988 Td [(Il)-355(s'ensuit)-355(que)-361(la)-355(mo)-30(di\257cation)-361(d'une)-355(fonction)-361(sur)-361(un)-361(ensem)24(ble)-355(\257ni)-361(ou)-355(m)24(^)475(eme)-355(d)24(\266)469(enom)24(brable)]TJ 0 -1.2048 Td [(ne)-331(c)24(hange)-337(rien)-337(\265)500(a)-331(son)-337(in)24(t)24(\266)469(egrabilit)30(\266)469(e)-337(ni)-337(\265)500(a)-331(la)-337(v)54(aleur)-337(de)-331(son)-343(in)30(t)24(\266)469(egrale)-313(\266)475(ev)24(en)30(tuelle.)]TJ 0 -1.5602 Td [(Soit)]TJ /F8 1 Tf 2.0663 0 Td [(f)]TJ /F2 1 Tf 0.94578 0 Td [(une)-349(fonction)-349(mesurable)-349(de)]TJ /F3 1 Tf 11.873 0 Td [(R)]TJ /F2 1 Tf 1.0663 0 Td [(dans)]TJ ET 0.504 w 282.53 237.88 m 291.17 237.88 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 23.639 19.069 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-487(Si)]TJ /F8 1 Tf 1.9458 0 Td [(I)]TJ /F2 1 Tf 0.86144 0 Td [(est)-355(un)-349(in)24(terv)60(alle)-349(de)]TJ /F3 1 Tf 8.7349 0 Td [(R)]TJ /F2 1 Tf 1.0663 0 Td [(on)-349(dira)-349(que)]TJ /F8 1 Tf 5.3554 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(est)]TJ /F1 1 Tf -37.934 -1.1988 Td [(in)30(t)24(\266)548(egrable)-301(sur)]TJ /F8 1 Tf 7.1566 0 Td [(I)]TJ /F2 1 Tf 0.77711 0 Td [(lorsque)-265(la)-259(restriction)-265(de)]TJ /F8 1 Tf 10.367 0 Td [(f)]TJ /F2 1 Tf 0.86145 0 Td [(\265)500(a)]TJ /F8 1 Tf 0.75904 0 Td [(I)]TJ /F2 1 Tf 0.77711 0 Td [(est)-265(in)24(t)24(\266)475(egrable.)-421(Nous)-265(a)30(v)24(ons)-265(vu)-265(\(cf.)]TJ /F1 1 Tf 14.663 0 Td [(IV.9.4.)]TJ /F2 1 Tf 3.4157 0 Td [(\))]TJ -38.777 -1.1988 Td [(qu'il)-343(revien)18(t)-343(au)-349(m)24(^)469(eme)-343(de)-343(dire)-349(que)-349(que)-349(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.793 23.642 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Tf 5.994 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(sur)]TJ /F8 1 Tf 1.6867 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(d'extr)24(\266)469(emit)24(\266)469(es)]TJ /F8 1 Tf 5.7108 0 Td [(a)]TJ /F2 1 Tf -38.633 -1.759 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(b)]TJ /F2 1 Tf 0.75904 0 Td [(\(lorsqu'elle)-337(est)-337(d)24(\266)475(e\257nie\))-343(sans)-331(pr)24(\266)469(eciser)-331(si)]TJ /F8 1 Tf 17.078 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(ou)]TJ /F8 1 Tf 1.3916 0 Td [(b)]TJ /F2 1 Tf 0.76506 0 Td [(fait)-331(partie)-343(de)]TJ /F8 1 Tf 6.0301 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(ou)-337(non.)]TJ -28.904 -1.5602 Td [(P)30(our)-427(tout)]TJ /F8 1 Tf 4.7831 0 Td [(a)]TJ /F2 1 Tf 0.95181 0 Td [(dans)]TJ ET 161.28 99.428 m 169.92 99.428 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 13.494 7.4906 Td [(R)]TJ /F2 1 Tf 1.1446 0 Td [(notons)-427(alors)]TJ /F8 1 Tf 5.8133 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.155 10.504 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.982 7.4906 Td [(la)-427(fonction)-427(caract)24(\266)475(eristique)-427(de)-427(l'in)24(terv)54(alle)-421(])]TJ /F6 1 Tf 18.922 0 Td [(\241)-283(1)]TJ /F2 1 Tf 2.0602 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [([.)]TJ -38.608 -1.2048 Td [(Av)30(ec)-283(les)-283(notations)-283(pr)24(\266)469(ec)24(\266)475(eden)24(tes)]TJ /F8 1 Tf 13.476 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.81 8.7797 Td [(b)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.952 6.2858 Td [(\241)]TJ /F8 1 Tf 0.89157 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.724 8.7797 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.235 6.2858 Td [(est)-289(la)-277(fonction)-289(caract)30(\266)469(eristique)-283(de)-283([)]TJ /F8 1 Tf 14.59 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44579 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [([)-283(si)]TJ /F8 1 Tf 1.512 0 Td [(a)-283(>)]TJ /F6 1 Tf 1.8614 0 Td [(\2411)]TJ /F2 1 Tf 2.0602 0 Td [(et)]TJ -19.416 -2.3976 Td [(79)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 272 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 271 0 R /ProcSet 2 0 R >> /Contents 270 0 R >> endobj 271 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R >> endobj 273 0 obj << /Length 19027 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F2 1 Tf 0 -2.4036 Td [(de)-343(])]TJ /F8 1 Tf 1.6205 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [([)-343(si)]TJ /F8 1 Tf 1.6386 0 Td [(a)]TJ /F2 1 Tf 0.81928 0 Td [(=)]TJ /F6 1 Tf 1.0723 0 Td [(\2411)]TJ /F2 1 Tf 1.7831 0 Td [(.)-475(De)-343(toute)-349(fa)-6(\273)439(con)-343(l'in)24(t)24(\266)475(egrabilit)24(\266)469(e)-343(de)]TJ /F8 1 Tf 15.072 0 Td [(f)]TJ /F2 1 Tf 0.94579 0 Td [(sur)]TJ /F8 1 Tf 1.6867 0 Td [(I)]TJ /F2 1 Tf 0.86144 0 Td [(d'extr)24(\266)475(emit)24(\266)469(es)]TJ /F8 1 Tf 5.7108 0 Td [(a)]TJ /F2 1 Tf 0.87349 0 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(b)]TJ /F2 1 Tf 0.74096 0 Td [(\266)469(equiv)54(aut)]TJ -35.41 -1.9518 Td [(\265)499(a)-331(celle)-337(de)-337(\()]TJ /F8 1 Tf 4.7771 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.362 86.03 Td [(b)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.361 60.268 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.586 86.03 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.47 60.268 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(sur)]TJ /F3 1 Tf 1.6747 0 Td [(R)]TJ /F2 1 Tf 1.0602 0 Td [(et)]TJ /F9 1 Tf 1.1687 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.181 87.84 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.09 60.268 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F16 1 Tf 8.352 0 0 8.352 0 0 Tm 36.043 84.943 Td [(R)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.578 60.268 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.052 86.03 Td [(b)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.217 60.268 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.276 86.03 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.325 60.268 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)]TJ -27.59 -2.3012 Td [(P)30(ar)-319(extension,)-319(p)-30(our)]TJ /F8 1 Tf 8.8976 0 Td [(a)]TJ /F2 1 Tf 0.84337 0 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(b)]TJ /F2 1 Tf 0.74699 0 Td [(dans)]TJ ET 0.504 w 230.62 702.79 m 239.26 702.79 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 19.295 57.967 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-319(et)-319(quelque)-319(soit)-319(l'ordre)-319(de)]TJ /F8 1 Tf 11.753 0 Td [(a)]TJ /F2 1 Tf 0.84338 0 Td [(et)]TJ /F8 1 Tf 1.1506 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(,)-319(on)-319(d)30(\266)469(esignera)-319(par)-319(\()]TJ /F8 1 Tf 8.506 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(\))]TJ -38.777 -1.1988 Td [(n'imp)-30(orte)-277(quel)-283(in)24(terv)54(alle)-277(d'extr)24(\266)469(emit)24(\266)469(es)]TJ /F8 1 Tf 16.548 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(et)]TJ /F8 1 Tf 1.1084 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(.)-427(On)-277(dira)-283(que)]TJ /F8 1 Tf 6.1265 0 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(est)-283(in)24(t)24(\266)475(egrable)-283(sur)-283(\()]TJ /F8 1 Tf 8.1024 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44579 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))-283(lorsque)]TJ -35.404 -1.9518 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0776 78.228 Td [(b)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0542 54.816 Td [(\241)]TJ /F8 1 Tf 1.0843 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.543 78.228 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.247 54.816 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 1.0602 0 Td [(est)-463(in)24(t)24(\266)475(egrable)-463(sur)]TJ /F3 1 Tf 8.259 0 Td [(R)]TJ /F2 1 Tf 1.1807 0 Td [(et)-463(on)-463(notera)]TJ /F9 1 Tf 6.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.914 80.038 Td [(b)]TJ -0.62931 -2.8965 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.596 54.816 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.5542 0 Td [(l'in)24(t)24(\266)469(egrale)]TJ /F9 1 Tf 4.7711 1.3614 Td [(Z)]TJ /F16 1 Tf 8.352 0 0 8.352 0 0 Tm 52.362 77.142 Td [(R)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.982 54.816 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.371 78.228 Td [(b)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.705 54.816 Td [(\241)]TJ /F8 1 Tf 1.0843 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.836 78.228 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.898 54.816 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf -38.072 -2.6145 Td [(lorsque)-385(celle-ci)-391(est)-391(d)24(\266)475(e\257nie.)-608(On)-391(a)-385(alors)-391(de)-385(fa)-6(\273)439(con)-361(\266)469(eviden)24(te)]TJ /F9 1 Tf 25.289 1.3675 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.25 76.297 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.018 52.201 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4639 0 Td [(=)]TJ /F6 1 Tf 1.1446 0 Td [(\241)]TJ /F9 1 Tf 0.94579 1.3675 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.362 76.297 Td [(a)]TJ -0.63793 -2.8966 Td [(b)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.169 52.201 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)-608(En)]TJ -37.042 -2.6084 Td [(particulier)-367(on)-367(prendra)-373(garde)-367(au)-367(fait)-373(que,)-379(p)-30(our)]TJ /F8 1 Tf 20.663 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(\265)499(a)-367(v)48(aleurs)-373(p)-24(ositiv)24(es,)-379(on)-367(a)]TJ /F9 1 Tf 10.982 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.724 72.564 Td [(b)]TJ -0.63793 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.343 49.593 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F11 1 Tf 1.4277 0 Td [(>)]TJ /F2 1 Tf 1.1084 0 Td [(0)]TJ -38.663 -2.6145 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(a)]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F2 1 Tf 1.7651 0 Td [(mais)]TJ /F9 1 Tf 2.3373 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.034 68.823 Td [(b)]TJ -0.63793 -2.8965 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.705 46.979 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F11 1 Tf 1.3675 0 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(0)-331(si)]TJ /F8 1 Tf 1.8373 0 Td [(a)]TJ /F11 1 Tf 0.80723 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(.)]TJ -17.048 -3.0121 Td [(Ainsi)-313(les)-313(in)30(t)24(\266)469(egrales)]TJ /F9 1 Tf 8.4518 1.3615 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.155 64.512 Td [(b)]TJ -0.63793 -2.8965 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.187 43.967 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3976 0 Td [(et)]TJ /F9 1 Tf 1.1446 1.3615 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.828 64.512 Td [(a)]TJ -0.63793 -2.8965 Td [(b)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.331 43.967 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3976 0 Td [(corresp)-30(onden)24(t)-313(\265)499(a)-313(des)-313(in)24(t)24(\266)475(egrations)-313(sur)-313(le)-319(m)24(^)475(eme)]TJ -19.512 -1.759 Td [(in)24(terv)60(alle)-433(\(puisque)-439(\()]TJ /F8 1 Tf 8.9578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))-451(=)-445(\()]TJ /F8 1 Tf 2.4458 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\)\))-433(mais)-433(elle)-439(di\256)24(\265)475(eren)18(t)-433(par)-439(l'ordre)-433(dans)-433(lequel)-439(on)-433(a)-433(rang)24(\266)475(e)]TJ -14.211 -1.2048 Td [(les)-343(extr)30(\266)469(emit)24(\266)469(es)-343(de)-337(cet)-343(in)24(terv)54(alle.)-463(Elles)-343(d)24(\266)475(ep)-30(enden)24(t)-343(donc)-343(de)-343(l'orien)24(tation)-343(de)-337(l'in)24(terv)54(alle,)-343(c'est)]TJ 0 -1.1988 Td [(p)-30(ourquoi)-331(on)-337(parle)-337(d')]TJ /F1 1 Tf 8.9518 0 Td [(in)30(t)24(\266)548(egrale)-385(orien)24(t)24(\266)548(ee)]TJ /F2 1 Tf 8.747 0 Td [(.)]TJ -17.699 -2.1988 Td [(De)-337(m)24(^)475(eme)-337(on)-331(a)-337(clairemen)30(t)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 28.846 Td [(VI.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 34.581 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.9759 0 Td [(Si)]TJ /F8 1 Tf 1.1988 0 Td [(f)]TJ /F12 1 Tf 0.96386 0 Td [(est)-373(in)24(t)24(\266)469(egrable)-367(sur)]TJ /F2 1 Tf 7.988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F12 1 Tf 0.75904 0 Td [(et)-367(sur)]TJ /F2 1 Tf 2.9096 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(;)]TJ /F8 1 Tf 0.44579 0 Td [(c)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F12 1 Tf 0.75904 0 Td [(elle)-373(est)-373(in)24(t)24(\266)475(egrable)-367(sur)]TJ /F2 1 Tf -29.735 -1.2048 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(c)]TJ /F2 1 Tf 0.43374 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(et)-337(on)-331(a)-337(la)]TJ /F1 1 Tf 4.5 0 Td [(relation)-385(de)-385(Chasles)]TJ /F12 1 Tf 9.9157 0 Td [(:)]TJ /F9 1 Tf -5.8554 -0.88554 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.914 46.142 Td [(b)]TJ -0.63793 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.813 31.129 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3133 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.259 46.142 Td [(c)]TJ -0.63793 -2.9052 Td [(b)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.651 31.129 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56626 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.776 46.142 Td [(c)]TJ -0.63793 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.596 31.129 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dx)]TJ /F12 1 Tf 1.4277 0 Td [(.)]TJ /F2 1 Tf -27.807 -3.3072 Td [(Concernan)24(t)-337(l'in)24(t)24(\266)475(egrabilit)24(\266)475(e)-343(sur)-337(les)-337(in)30(terv)54(alles)-337(b)-30(orn)24(\266)475(es)-337(on)-337(a)-331(l'imp)-30(ortan)24(te)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 20.686 Td [(VI.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 24.798 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Si)]TJ /F8 1 Tf 1.1566 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-331(mesurable)-325(et)-331(b)-30(orn)30(\266)469(ee)-331(sur)-331(un)-331(in)30(terv)54(alle)-331(b)-30(orn)30(\266)469(e)]TJ /F2 1 Tf 20.928 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F12 1 Tf 0.71687 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F12 1 Tf 0.85542 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.2048 Td [(b)]TJ /F12 1 Tf 0.75904 0 Td [(\257nis\),)-343(alors)]TJ /F8 1 Tf 5.1867 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-343(in)30(t)24(\266)469(egrable)-331(sur)]TJ /F2 1 Tf 7.8855 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a;)-174(b)]TJ /F2 1 Tf 1.4036 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(et)-337(l'on)-337(a)-331(:)]TJ /F9 1 Tf -2.988 -1.3976 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ 0.5 1.1145 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.216 32.142 Td [(b)]TJ -0.62931 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.518 21.346 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F9 1 Tf 1.0904 0.8494 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ /F11 1 Tf 0.61446 -0.24699 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F6 1 Tf 1.0783 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F12 1 Tf -24.602 -2.1627 Td [(p)-30(our)-331(tout)]TJ /F8 1 Tf 4.5301 0 Td [(M)]TJ /F12 1 Tf 1.4157 0 Td [(ma)-54(joran)24(t)]TJ /F6 1 Tf 4.3373 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F12 1 Tf 0.61446 0 Td [(sur)]TJ /F2 1 Tf 1.6747 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -15.627 -1.9036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Notons)]TJ /F8 1 Tf 3.4217 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(l'in)24(terv)54(alle)-331(\()]TJ /F8 1 Tf 5.2771 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\),)-337(on)-331(sait)-331(que)-337(la)-331(mesure)-337(de)]TJ /F8 1 Tf 11.976 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(est)-337(sa)-331(longueur)]TJ /F6 1 Tf -32.596 -1.7108 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(b)]TJ /F6 1 Tf 0.68072 0 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-572(Si)]TJ /F8 1 Tf 2.0602 0 Td [(M)]TJ /F2 1 Tf 1.4578 0 Td [(ma)-54(jore)]TJ /F6 1 Tf 3.4036 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.6506 0 Td [(sur)]TJ /F8 1 Tf 1.7169 0 Td [(I)]TJ /F2 1 Tf 0.5241 0 Td [(,)-385(on)-373(a)-379(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.974 22.064 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.054 15.569 Td [(\242)-253(j)]TJ /F8 1 Tf 0.80723 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.62651 0 Td [(6)]TJ /F8 1 Tf 1.1265 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.853 22.064 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.307 15.569 Td [(,)-391(d'o)-30(\265)530(u)]TJ /F9 1 Tf 2.9337 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.5 20.978 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.253 15.569 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dx)]TJ /F11 1 Tf 1.4458 0 Td [(6)]TJ /F8 1 Tf 1.1265 0 Td [(M)]TJ /F9 1 Tf 1.247 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.672 22.064 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.759 15.569 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dx)]TJ /F2 1 Tf 1.4398 0 Td [(=)]TJ /F8 1 Tf -38.386 -2.3735 Td [(M)]TJ /F6 1 Tf 1.0783 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(b)]TJ /F6 1 Tf 0.66867 0 Td [(\241)]TJ /F8 1 Tf 1.012 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-506(Il)-355(s'ensuit)-355(que)]TJ /F9 1 Tf 7.3072 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.379 17.581 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.494 13.195 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.4458 0 Td [(est)-361(\257ni)-355(et)-355(donc)-355(que)]TJ /F8 1 Tf 8.8072 0 Td [(f)]TJ /F2 1 Tf 0.95181 0 Td [(est)-361(in)24(t)24(\266)475(egrable)-355(sur)]TJ /F8 1 Tf 7.9458 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-506(En)-355(outre)]TJ -34.506 -2.6145 Td [(on)-331(a)]TJ /F9 1 Tf 2.2229 0.8494 Td [(\257)]TJ 0 -0.59639 Td [(\257)]TJ 0.5 1.1084 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.957 16.737 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.4578 10.581 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F9 1 Tf 1.0904 0.8494 Td [(\257)]TJ 0 -0.59639 Td [(\257)]TJ /F2 1 Tf 0.60843 -0.25301 Td [(=)]TJ /F9 1 Tf 1.0602 0.8494 Td [(\257)]TJ 0 -0.59639 Td [(\257)]TJ 0.5 1.1084 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.017 13.84 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.843 10.581 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F9 1 Tf 1.0964 0.8494 Td [(\257)]TJ 0 -0.59639 Td [(\257)]TJ /F11 1 Tf 0.60843 -0.25301 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.448 13.84 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 21.735 10.581 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F11 1 Tf 1.3735 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F6 1 Tf 1.0783 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(a)]TJ /F6 1 Tf 0.5241 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)]TJ ET 512.64 125.31 m 519.84 125.31 l 519.84 133.74 l 512.64 133.74 l f BT /F5 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 3.8882 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(80)]TJ ET endstream endobj 275 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 274 0 R /ProcSet 2 0 R >> /Contents 273 0 R >> endobj 274 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F16 81 0 R >> endobj 277 0 obj << /Length 22046 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(VI.2.)-236(In)30(t)20(\266)537(egrales)-376(et)-376(primitiv)25(es.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 60.617 Td [(Dans)-337(cette)-343(section)-337(on)-343(consid)30(\265)469(ere)-343(un)-337(in)24(terv)54(alle)]TJ /F8 1 Tf 19.88 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(de)]TJ /F3 1 Tf 1.3374 0 Td [(R)]TJ /F2 1 Tf 1.0602 0 Td [(et)-343(une)-337(fonction)]TJ /F8 1 Tf 6.9337 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(in)24(t)24(\266)469(egrable)-337(sur)-343(tout)]TJ -31 -1.2048 Td [(segmen)24(t)-373(con)24(ten)24(u)-373(dans)]TJ /F8 1 Tf 10.048 0 Td [(I)]TJ /F2 1 Tf 0.89157 0 Td [(\(on)-373(dit)-379(alors)-379(que)]TJ /F8 1 Tf 7.747 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(est)-379(lo)-30(calemen)24(t)-373(in)24(t)24(\266)475(egrable)-373(sur)]TJ /F8 1 Tf 13.042 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\).)-566(On)-373(\257xe)-379(un)]TJ -33.217 -1.1988 Td [(p)-30(oin)30(t)]TJ /F8 1 Tf 2.6386 0 Td [(a)]TJ /F2 1 Tf 0.88554 0 Td [(de)]TJ /F8 1 Tf 1.3614 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-361(alors,)-373(p)-30(our)-361(tout)]TJ /F8 1 Tf 7.9337 0 Td [(t)]TJ /F2 1 Tf 0.71687 0 Td [(de)]TJ /F8 1 Tf 1.3614 0 Td [(I)]TJ /F2 1 Tf 0.87952 0 Td [(la)-355(fonction)-367(est)-361(in)24(t)24(\266)469(egrable)-355(sur)-367(\()]TJ /F8 1 Tf 13.367 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-361(ce)-361(qui)-361(p)-30(ermet)-361(de)]TJ -31 -1.8976 Td [(d)30(\266)469(e\257nir)-361(sur)]TJ /F8 1 Tf 4.8373 0 Td [(I)]TJ /F2 1 Tf 0.87349 0 Td [(une)-361(fonction)]TJ /F8 1 Tf 5.8012 0 Td [(F)]TJ /F2 1 Tf 1.1386 0 Td [(par)]TJ /F8 1 Tf 1.8072 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))-325(=)]TJ /F9 1 Tf 1.8072 1.3675 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.526 82.185 Td [(t)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.476 56.316 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)-518(Nous)-361(allons)-331(\266)475(etudier)-367(les)-355(propri)24(\266)469(et)24(\266)475(es)-361(de)]TJ -22.349 -1.7048 Td [(cette)-337(fonction.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 43.027 Td [(VI.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 51.581 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8494 0 Td [(Av)30(ec)-283(les)-283(notations)-283(pr)24(\266)475(ec)24(\266)469(eden)30(tes,)-289(la)-283(fonction)]TJ /F8 1 Tf 18.627 0 Td [(F)]TJ /F12 1 Tf 1.0602 0 Td [(est)-289(con)30(tin)24(ue)-283(sur)]TJ /F8 1 Tf 7.1325 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(.)]TJ /F1 1 Tf -38.886 -1.9036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Fixons)]TJ /F8 1 Tf 3.2349 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 25.397 70.875 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.524 49.677 Td [(dans)]TJ /F8 1 Tf 2.3313 0 Td [(I)]TJ /F2 1 Tf 0.84337 0 Td [(et)-331(prouv)30(ons)-331(la)-331(con)30(tin)18(uit)24(\266)469(e)-331(de)]TJ /F8 1 Tf 12.536 0 Td [(F)]TJ /F2 1 Tf 1.1084 0 Td [(au)-325(p)-30(oin)24(t)]TJ /F8 1 Tf 3.9879 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.802 70.875 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.139 49.677 Td [(.)-445(On)-331(p)-30(eut)]TJ -34.807 -1.2048 Td [(alors)-361(trouv)24(er)-361(un)-355(segmen)18(t)]TJ /F8 1 Tf 11.307 0 Td [(S)]TJ /F2 1 Tf 1.0301 0 Td [(con)30(ten)24(u)-361(dans)]TJ /F8 1 Tf 6.1084 0 Td [(I)]TJ /F2 1 Tf 0.8735 0 Td [(qui)-355(p)-30(oss)24(\265)475(ede)]TJ /F8 1 Tf 5.3916 0 Td [(a)]TJ /F2 1 Tf 0.88554 0 Td [(et)-361(est)-361(un)-355(v)24(oisinage)-367(\(dans)]TJ /F8 1 Tf 11.301 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-361(de)]TJ /F8 1 Tf -37.416 -1.1988 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.1466 67.435 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4759 47.274 Td [(;)-331(nous)-337(p)-24(ouv)18(ons)-337(donc)-337(faire)-337(la)-331(preuv)24(e)-337(en)-331(restreignan)24(t)]TJ /F8 1 Tf 22.729 0 Td [(F)]TJ /F2 1 Tf 1.1205 0 Td [(\265)500(a)-331(ce)-337(segmen)24(t.)]TJ -24.994 -1.6084 Td [(P)30(our)]TJ /F8 1 Tf 2.4699 0 Td [(x)]TJ /F2 1 Tf 0.94578 0 Td [(di\256)30(\266)469(eren)24(t)-373(de)]TJ /F8 1 Tf 5.3554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 20.698 65.142 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.283 45.665 Td [(\(donc)-373(p)-30(our)-373(presque)-373(tout)]TJ /F8 1 Tf 11.108 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\),)-385(la)-373(fonction)]TJ /F8 1 Tf 6.0964 0 Td [(t)]TJ /F6 1 Tf 0.6988 0 Td [(7!)]TJ /F8 1 Tf 1.3434 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.129 65.142 Td [(t)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.078 45.665 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-373(est)-373(con)24(tin)24(ue)-373(au)]TJ -31.711 -1.1988 Td [(p)-30(oin)30(t)]TJ /F8 1 Tf 2.6084 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 11.879 63.418 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.753 44.467 Td [(,)-331(ainsi)-337(donc)-337(que)-337(la)-331(fonction)]TJ /F8 1 Tf 12.175 0 Td [(t)]TJ /F6 1 Tf 0.64458 0 Td [(7!)]TJ /F2 1 Tf 1.2771 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.147 63.418 Td [(t)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.44 44.467 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.302 63.418 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.548 44.467 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)]TJ -23.163 -1.6024 Td [(P)30(ar)-397(ailleurs)-391(on)-391(a)-397(la)-391(relation)-397(de)-391(domination)]TJ /F6 1 Tf 19.428 0 Td [(j)]TJ /F2 1 Tf 0.28313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.284 61.125 Td [(t)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.669 42.864 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.552 61.125 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.819 42.864 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.65663 0 Td [(6)]TJ /F2 1 Tf 1.1566 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.983 61.125 Td [(S)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.108 42.864 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F2 1 Tf 0.6747 0 Td [(,)-409(v)54(alable)-397(p)-30(our)]TJ -33.024 -1.2048 Td [(tout)]TJ /F8 1 Tf 2.1446 0 Td [(x)]TJ /F2 1 Tf 0.89157 0 Td [(et)-313(tout)]TJ /F8 1 Tf 3.2952 0 Td [(t)]TJ /F2 1 Tf 0.6747 0 Td [(dans)]TJ /F8 1 Tf 2.3193 0 Td [(S)]TJ /F2 1 Tf 0.98795 0 Td [(\(remarquer)-319(que)]TJ /F8 1 Tf 7.0301 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.345 59.409 Td [(t)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.837 41.659 Td [(\241)]TJ /F8 1 Tf 0.95783 0 Td [(\302)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.379 59.409 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.223 41.659 Td [(ne)-313(p)-30(eut)-319(prendre)-319(que)-313(les)-319(v)48(aleurs)-319(0)-313(et)]TJ /F6 1 Tf 15.843 0 Td [(\247)]TJ /F2 1 Tf 0.78313 0 Td [(1)-313(et)]TJ -37.518 -1.1988 Td [(s'ann)30(ule)-355(en)-355(dehors)-361(de)]TJ /F8 1 Tf 9.7952 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(\))-355(a)24(v)24(ec)-355(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.647 57.685 Td [(S)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.898 40.461 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(j)]TJ /F2 1 Tf 0.63253 0 Td [(in)24(t)24(\266)475(egrable)-355(par)-355(h)24(yp)-36(oth)24(\265)475(ese.)-512(Le)-355(th)24(\266)469(eor)24(\265)469(eme)-355(de)-355(con)30(tin)24(uit)18(\266)475(e)]TJ -16.072 -1.1988 Td [(s'applique)-331(p)-30(our)-337(assurer)-337(que)]TJ /F8 1 Tf 12.337 0 Td [(F)]TJ /F2 1 Tf 1.1145 0 Td [(est)-337(con)24(tin)24(ue)-331(au)-337(p)-30(oin)24(t)]TJ /F8 1 Tf 9.5663 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.086 55.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.157 39.262 Td [(.)]TJ ET 512.64 468.03 m 519.84 468.03 l 519.84 476.46 l 512.64 476.46 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 37.352 Td [(Remarque.)]TJ /F2 1 Tf 6.3855 0 Td [(Dans)-367(la)-367(preuv)30(e)-373(on)-367(a)-367(utilis)24(\266)469(e)-367(que,)-379(p)-30(our)-367(c)18(haque)-361(p)-30(oin)24(t)]TJ /F8 1 Tf 22.843 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.974 53.237 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.741 37.352 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-373(on)-367(a)30(v)48(ait,)-379(p)-30(our)]TJ -32.295 -1.1988 Td [(presque)-421(tout)]TJ /F8 1 Tf 5.988 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-445(la)-421(con)30(tin)24(uit)18(\266)475(e)-427(en)]TJ /F8 1 Tf 8.0964 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 29.121 51.521 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.217 36.153 Td [(de)-421(la)-421(fonction)-427(\265)500(a)-415(in)24(t)24(\266)469(egrer.)-710(Cep)-24(endan)24(t)-427(il)-421(n'y)-421(a)-421(aucun)]TJ /F8 1 Tf -15.886 -1.1988 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-361(p)-30(our)-355(lequel)-361(on)-355(ait)-361(la)-355(con)30(tin)18(uit)24(\266)469(e)-355(en)-355(tout)-361(p)-30(oin)24(t)-355(de)]TJ /F8 1 Tf 22.199 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-512(La)-361(derni)24(\265)475(ere)-361(partie)-361(du)-355(th)18(\266)475(eor)18(\265)475(eme)-355(de)]TJ -23.289 -1.2048 Td [(con)30(tin)18(uit)24(\266)469(e)-337(ne)-331(s'applique)-337(donc)-337(pas.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 25.625 Td [(VI.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 30.72 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084337 Td [(\266)]TJ 0 -0.084337 Td [(eor)]TJ 1.8012 0.084337 Td [(\265)]TJ -0.0060244 -0.084337 Td [(eme.)]TJ /F12 1 Tf 2.6687 0 Td [(Av)30(ec)-307(les)-301(notations)-301(pr)24(\266)475(ec)24(\266)469(eden)30(tes,)-307(si)]TJ /F8 1 Tf 14.813 0 Td [(f)]TJ /F12 1 Tf 0.89759 0 Td [(est)-301(con)24(tin)24(ue)-301(en)-301(un)-301(p)-30(oin)24(t)]TJ /F8 1 Tf 10.783 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 61.172 43.745 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 43.494 30.72 Td [(de)]TJ /F8 1 Tf -38.163 -1.1988 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(alors)]TJ /F8 1 Tf 2.3976 0 Td [(F)]TJ /F12 1 Tf 1.1205 0 Td [(est)-337(d)30(\266)469(eriv)54(able)-337(au)-337(p)-30(oin)30(t)]TJ /F8 1 Tf 9.8133 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.44 42.03 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.657 29.521 Td [(et)-337(on)-337(a)]TJ /F8 1 Tf 3.7229 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.009 42.831 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.446 29.521 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.483 42.03 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.645 29.521 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 42.526 42.03 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.163 29.521 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ /F1 1 Tf -25.223 -2.6024 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(D'apr)30(\265)469(es)-397(la)-397(relation)-397(de)-397(Chasles)-397(on)-397(a)]TJ /F8 1 Tf 16.066 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 50.052 38.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.422 26.918 Td [(\))-385(=)]TJ /F9 1 Tf 1.9337 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.888 40.116 Td [(t)]TJ -0.63793 -2.8966 Td [(t)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 76.434 51.813 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.711 26.918 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(,)-409(d'o)-36(\265)530(u)]TJ /F8 1 Tf -36.584 -2.6566 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.64458 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.414 34.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.524 24.262 Td [(\))]TJ /F6 1 Tf 0.64458 0 Td [(\241)]TJ /F2 1 Tf 1.0301 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 20.897 34.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.048 24.262 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 24.026 34.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.235 24.262 Td [(\))-367(=)]TJ /F9 1 Tf 1.8916 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.802 36.314 Td [(t)]TJ -0.63793 -2.8965 Td [(t)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 39.964 46.499 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 20.476 25.069 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.64458 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.517 34.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.97 24.262 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62048 -0.80723 Td [(dx)]TJ /F2 1 Tf 1.0964 0 Td [(.)-590(Cep)-30(endan)24(t)-385(la)-385(con)24(tin)24(uit)24(\266)469(e)-385(de)]TJ /F8 1 Tf 13.241 0 Td [(f)]TJ /F2 1 Tf 0.98193 0 Td [(en)]TJ /F8 1 Tf 1.3855 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 63.034 34.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 22.4 Td [(assure)-403(que,)-433(p)-30(our)]TJ /F8 1 Tf 7.759 0 Td [(")-397(>)]TJ /F2 1 Tf 2.0422 0 Td [(0,)-427(il)-409(existe)]TJ /F8 1 Tf 5.0542 0 Td [(\264)-433(>)]TJ /F2 1 Tf 2.1084 0 Td [(0)-409(tel)-409(que,)-427(p)-30(our)]TJ /F6 1 Tf 7.0964 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.84337 0 Td [(\241)]TJ /F8 1 Tf 1.0482 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.681 31.84 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.373 22.4 Td [(j)]TJ /F8 1 Tf 0.6747 0 Td [(<)-403(\264)]TJ /F2 1 Tf 1.7108 0 Td [(,)-427(on)-409(ait)-409(la)-409(ma)-54(joration)]TJ /F6 1 Tf -29.428 -1.1988 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.68675 0 Td [(\241)]TJ /F8 1 Tf 1.0783 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.707 30.125 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.729 21.201 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.74699 0 Td [(<)-469(")]TJ /F2 1 Tf 1.7169 0 Td [(.)-789(En)-451(particulier,)-475(p)-30(our)]TJ /F6 1 Tf 10.494 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(t)]TJ /F6 1 Tf 0.66265 0 Td [(\241)]TJ /F8 1 Tf 1.0723 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.845 30.125 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.898 21.201 Td [(j)]TJ /F8 1 Tf 0.74699 0 Td [(<)-469(\264)]TJ /F2 1 Tf 1.7771 0 Td [(,)-481(cette)-451(ma)-54(joration)-457(est)-451(v)48(alable)-445(sur)]TJ -24.09 -1.8976 Td [(l'in)24(terv)54(alle)-331(\()]TJ /F8 1 Tf 5.2831 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.853 27.409 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.229 19.304 Td [(\))-337(et)-337(donc)]TJ /F9 1 Tf 4.2771 0.8494 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ 0.50603 1.1145 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.776 29.219 Td [(t)]TJ -0.63793 -2.8966 Td [(t)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 35.735 36.584 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 18.361 20.111 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 33.397 27.409 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.789 19.304 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62651 -0.80723 Td [(dx)]TJ /F9 1 Tf 1.0904 0.8494 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ /F11 1 Tf 0.61446 -0.24699 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(")]TJ /F6 1 Tf 0.46988 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(t)]TJ /F6 1 Tf 0.58434 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.293 27.409 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.699 19.304 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)]TJ -25.645 -2.2651 Td [(Autremen)24(t)-367(dit)]TJ /F8 1 Tf 6.6747 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.63253 0 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 23.922 24.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.169 17.039 Td [(\))]TJ /F6 1 Tf 0.62651 0 Td [(\241)]TJ /F2 1 Tf 1.0241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F6 1 Tf 0.60241 0 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.319 24.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.639 17.039 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 33.448 24.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.825 17.039 Td [(\))-361(est)-367(un)]TJ /F8 1 Tf 3.8133 0 Td [(o)]TJ /F2 1 Tf 0.48193 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F6 1 Tf 0.60241 0 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.638 24.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.94 17.039 Td [(\))-367(ce)-367(qui)-361(signi\257e)-361(bien)-367(que)]TJ /F8 1 Tf 11.181 0 Td [(F)]TJ /F2 1 Tf 1.1446 0 Td [(est)]TJ -37.934 -1.2048 Td [(d)30(\266)469(eriv)54(able)-337(en)]TJ /F8 1 Tf 5.5843 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.138 22.443 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.06 15.834 Td [(a)24(v)24(ec)-337(p)-30(our)-331(d)24(\266)469(eriv)54(\266)469(ee)]TJ /F8 1 Tf 7.9578 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.569 22.443 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.813 15.834 Td [(\).)]TJ ET 512.64 188.1 m 519.84 188.1 l 519.84 196.52 l 512.64 196.52 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 13.93 Td [(Remarque.)]TJ /F2 1 Tf 6.2169 0 Td [(De)-307(fa)-6(\273)439(con)-307(analogue)-307(en)-307(supp)-30(osan)24(t)-313(la)-307(con)30(tin)18(uit)24(\266)469(e)-307(\265)500(a)-307(droite)-313(\(resp.)-433(\265)499(a)-307(gauc)24(he\))-307(de)]TJ /F8 1 Tf 32.349 0 Td [(f)]TJ /F2 1 Tf -38.566 -1.2048 Td [(en)]TJ /F8 1 Tf 1.3313 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 10.052 18.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8072 12.726 Td [(on)-337(conclut)-337(\265)500(a)-331(la)-337(d)24(\266)475(eriv)48(abilit)30(\266)469(e)-337(\265)499(a)-331(droite)-343(\(resp.)-445(\265)499(a)-331(gauc)24(he\))-337(de)]TJ /F8 1 Tf 24.524 0 Td [(F)]TJ /F2 1 Tf 1.1205 0 Td [(en)]TJ /F8 1 Tf 1.3313 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 50.293 18.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.59 12.726 Td [(.)]TJ -30.259 -2.2048 Td [(Cette)-337(prop)-30(osition)-337(a)-337(l'imp)-24(ortan)18(t)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 6.2485 Td [(VI.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 7.4906 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Cor)24(ollaire.)]TJ /F12 1 Tf 6.5241 0 Td [(T)78(oute)-240(fonction)]TJ /F8 1 Tf 6.5361 0 Td [(f)]TJ /F12 1 Tf 0.84337 0 Td [(con)30(tin)18(ue)-240(sur)-246(un)-240(in)24(terv)54(alle)]TJ /F8 1 Tf 11.09 0 Td [(I)]TJ /F12 1 Tf 0.75903 0 Td [(admet)-240(une)-246(primitiv)30(e)]TJ -30.452 -1.2048 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(.)-445(En)-337(outre,)-337(si)]TJ /F8 1 Tf 6.1928 0 Td [(F)]TJ /F12 1 Tf 1.1145 0 Td [(est)-337(une)-337(primitiv)24(e)-337(de)]TJ /F8 1 Tf 9.0663 0 Td [(f)]TJ /F12 1 Tf 0.60241 0 Td [(,)-331(on)-337(a)]TJ /F2 1 Tf -0.25301 -2.3976 Td [(81)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 279 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 278 0 R /ProcSet 2 0 R >> /Contents 277 0 R >> endobj 278 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F19 276 0 R >> endobj 280 0 obj << /Length 22438 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F12 1 Tf 0 -3.006 Td [({)-668(p)-24(our)]TJ /F8 1 Tf 3.5301 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(b)]TJ /F12 1 Tf 0.76506 0 Td [(dans)]TJ /F8 1 Tf 2.3373 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(sur)]TJ /F2 1 Tf 7.8855 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(et)]TJ /F9 1 Tf 2.1687 1.3675 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.388 89.771 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.024 61.617 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ -34.867 -2.8133 Td [({)-668(si)]TJ /F8 1 Tf 2.1747 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-337(p)-30(ositiv)24(e)]TJ /F9 1 Tf 6.2651 1.3615 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.828 82.849 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.711 58.804 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)-277(sup)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.06 82.952 Td [(x)]TJ /F10 1 Tf 0.64655 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.928 58.804 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F2 1 Tf 1.0904 0 Td [(inf)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.181 83.228 Td [(x)]TJ /F10 1 Tf 0.64655 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.849 58.804 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)-283(longueur)-337(de)]TJ /F8 1 Tf 7.1687 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ -32.849 -2.2831 Td [({)-753(si)]TJ /F8 1 Tf 2.3012 0 Td [(f)]TJ /F12 1 Tf 0.96988 0 Td [(est)-379(in)24(t)24(\266)469(egrable)-373(sur)]TJ /F8 1 Tf 8.006 0 Td [(I)]TJ /F12 1 Tf 0.5241 0 Td [(,)]TJ /F8 1 Tf 0.66265 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.76506 0 Td [(admet)-379(une)-373(limite)-379(\257nie)-379(lorsque)]TJ /F8 1 Tf 13.59 0 Td [(x)]TJ /F12 1 Tf 0.94578 0 Td [(tend)-379(v)24(ers)]TJ /F8 1 Tf 4.4277 0 Td [(\256)]TJ /F2 1 Tf 0.99398 0 Td [(=)-349(inf)]TJ /F8 1 Tf 2.506 0 Td [(I)]TJ /F12 1 Tf 0.89759 0 Td [(et)]TJ -38.331 -1.7169 Td [(lorsque)]TJ /F8 1 Tf 3.4277 0 Td [(x)]TJ /F12 1 Tf 0.90361 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(\257)]TJ /F2 1 Tf 0.89759 0 Td [(=)-277(sup)]TJ /F8 1 Tf 2.7289 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(et)]TJ /F9 1 Tf 2.1687 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.345 77.125 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.669 54.804 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.81326 Td [(\243)]TJ /F8 1 Tf 0.41566 -0.81326 Td [(F)]TJ /F9 1 Tf 0.78313 0.81326 Td [(\244)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.819 79.288 Td [(\257)]TJ 0 -1.431 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.669 54.804 Td [(=)-457(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.543 77.495 Td [(x)]TJ /F10 1 Tf 0.64655 0 Td [(!)]TJ /F7 1 Tf 1.1379 0 Td [(\257)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.633 54.804 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.60844 0 Td [(\241)]TJ /F2 1 Tf 1.1928 0 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.069 77.573 Td [(x)]TJ /F10 1 Tf 0.65517 0 Td [(!)]TJ /F7 1 Tf 1.1379 0 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.928 54.804 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F1 1 Tf -34.06 -2.4699 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Remarquons)-409(tout)-409(d'ab)-30(ord)-409(que)]TJ /F8 1 Tf 13.795 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(,)-397(\266)475(etan)24(t)-415(con)30(tin)24(ue,)-427(est)-415(mesurable.)-668(En)]TJ -23.217 -1.1988 Td [(outre,)-361(sur)-355(tout)-355(segmen)24(t)-355(con)24(ten)24(u)-355(dans)]TJ /F8 1 Tf 16.783 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-355(elle)-361(est)-355(b)-30(orn)24(\266)475(ee)-355(et)-355(donc)-355(in)24(t)24(\266)475(egrable)-355(ce)-355(qui)-355(p)-24(ermet,)]TJ -17.301 -1.8373 Td [(apr)30(\265)469(es)-337(a)30(v)24(oir)-337(\257x)24(\266)469(e)]TJ /F8 1 Tf 6.9578 0 Td [(t)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.103 70.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.434 49.298 Td [(dans)]TJ /F8 1 Tf 2.3373 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-337(de)-331(d)24(\266)475(e\257nir)-337(une)-337(fonction)]TJ /F8 1 Tf 10.813 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.707 70.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.53 49.298 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(par)]TJ /F8 1 Tf 1.7771 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.914 70.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.934 49.298 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.526 72.142 Td [(t)]TJ -0.63793 -2.9052 Td [(t)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 74.53 96.572 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.759 49.298 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)]TJ -35.633 -2.253 Td [(La)-349(prop)-36(osition)-349(pr)24(\266)469(ec)24(\266)475(eden)24(te)-349(mon)24(tre)-349(clairemen)24(t)-355(que)]TJ /F8 1 Tf 21.934 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.94 67.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.711 47.045 Td [(est)-349(une)-355(primitiv)30(e)-355(de)]TJ /F8 1 Tf 9.1265 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)-493(P)30(ar)-349(ailleurs,)]TJ -33.102 -1.1988 Td [(si)]TJ /F8 1 Tf 1.0422 0 Td [(F)]TJ /F2 1 Tf 1.1506 0 Td [(est)-379(une)-373(primitiv)24(e)-373(quelconque)-379(de)]TJ /F8 1 Tf 14.428 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(on)-373(sait)-373(qu'elle)-373(di\256)24(\265)469(ere)-373(de)-373(celle-ci)-367(par)-373(une)-373(fonction)]TJ -17.59 -1.8374 Td [(constan)24(te;)-337(on)-331(a)-337(donc)]TJ /F8 1 Tf 9.3675 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.129 64.573 Td [(t)]TJ -0.63793 -2.9052 Td [(t)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 37.627 85.993 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.307 44.009 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3193 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(p)-30(our)-331(un)-337(r)24(\266)475(eel)]TJ /F8 1 Tf 5.6988 0 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(con)24(v)24(enable.)]TJ -26.349 -2.3072 Td [(Alors,)-349(p)-30(our)]TJ /F8 1 Tf 5.3193 0 Td [(a)]TJ /F2 1 Tf 0.87349 0 Td [(et)]TJ /F8 1 Tf 1.1747 0 Td [(b)]TJ /F2 1 Tf 0.77711 0 Td [(dans)]TJ /F8 1 Tf 2.3494 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-349(on)-349(vien)24(t)-349(de)-349(v)24(oir)-343(que)]TJ /F8 1 Tf 9.7771 0 Td [(f)]TJ /F2 1 Tf 0.94578 0 Td [(est)-349(in)24(t)24(\266)475(egrable)-349(sur)-349(\()]TJ /F8 1 Tf 8.3072 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(\))-343(et)-349(la)-349(relation)-343(de)]TJ -31.446 -1.9518 Td [(Chasles)-331(donne)]TJ /F8 1 Tf 6.5663 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.957 58.478 Td [(b)]TJ -0.63793 -2.8966 Td [(t)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 38.783 77.475 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.94 39.75 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F6 1 Tf 1.3133 0 Td [(\241)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.302 58.478 Td [(a)]TJ -0.63793 -2.8966 Td [(t)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 50.446 77.475 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.849 39.75 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.922 58.478 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.795 39.75 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)]TJ -29.669 -2.247 Td [(P)30(ar)-234(ailleurs,)-253(prenan)24(t)]TJ /F8 1 Tf 9.0241 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.302 53.452 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.657 37.503 Td [(tendan)24(t)-234(en)-234(d)30(\266)469(ecroissan)30(t)-234(v)18(ers)]TJ /F8 1 Tf 11.843 0 Td [(\256)]TJ /F2 1 Tf 0.92169 0 Td [(=)-283(inf)]TJ /F8 1 Tf 2.4398 0 Td [(I)]TJ /F2 1 Tf 0.75301 0 Td [(et)]TJ /F8 1 Tf 1.0663 0 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.379 53.452 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.886 37.503 Td [(tendan)30(t)-234(en)-234(croissan)24(t)-234(v)24(ers)]TJ /F8 1 Tf -28.554 -1.8976 Td [(\257)]TJ /F2 1 Tf 0.90964 0 Td [(=)-289(sup)]TJ /F8 1 Tf 2.741 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)-343(lorsque)]TJ /F8 1 Tf 4.0602 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(est)-343(p)-30(ositiv)24(e)-343(on)-343(a)]TJ /F9 1 Tf 7.5181 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.302 49.65 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.03 35.605 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3855 0 Td [(=)-295(lim)]TJ /F9 1 Tf 2.6265 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.155 52.547 Td [(b)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 61.024 73.246 Td [(n)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.517 49.65 Td [(a)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 60.289 69.186 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.048 35.605 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3855 0 Td [(=)-295(lim)]TJ /F9 1 Tf 2.6265 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.147 50.737 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.777 35.605 Td [(\))]TJ /F6 1 Tf 0.62048 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.681 50.737 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.645 35.605 Td [(\))]TJ /F9 1 Tf 0.39156 0.81325 Td [(\242)]TJ /F2 1 Tf -38.705 -2.7289 Td [(d'apr)24(\265)469(es)-295(la)-295(propri)18(\266)475(et)24(\266)469(e)-301(de)-295(Bepp)-36(o)-295(Levi.)-433(Mais,)-307(puisque)]TJ /F8 1 Tf 21.994 0 Td [(F)]TJ /F2 1 Tf 1.0783 0 Td [(est)-301(con)24(tin)24(ue)-295(et)-301(croissan)24(te)-295(\()]TJ /F8 1 Tf 11.572 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.328 48.806 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.319 33.689 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.89157 0 Td [(est)]TJ -37.934 -1.1988 Td [(p)-30(ositiv)30(e\),)]TJ /F8 1 Tf 4.3855 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.336 46.28 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.958 32.491 Td [(\))-349(tend)-355(v)24(ers)-439(inf)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.422 45.573 Td [(x)]TJ /F10 1 Tf 0.64655 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.633 32.491 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-301(=)-493(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.69 45.633 Td [(x)]TJ /F10 1 Tf 0.64655 0 Td [(!)]TJ /F7 1 Tf 1.1466 0 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.084 32.491 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-349(et)]TJ /F8 1 Tf 1.9157 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.991 46.28 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.886 32.491 Td [(\))-355(tend)-349(v)24(ers)-349(sup)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.155 45.297 Td [(x)]TJ /F10 1 Tf 0.64655 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.669 32.491 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-301(=)-481(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.491 45.564 Td [(x)]TJ /F10 1 Tf 0.65517 0 Td [(!)]TJ /F7 1 Tf 1.1379 0 Td [(\257)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.084 32.491 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\).)]TJ -38.494 -1.7892 Td [(Le)-337(r)30(\266)469(esultat)-337(annonc)24(\266)469(e)-337(lorsque)]TJ /F8 1 Tf 12.398 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-343(p)-24(ositiv)24(e)-337(en)-337(d)30(\266)469(ecoule.)]TJ -13.325 -1.6386 Td [(En)-355(particulier,)-361(lorsque)]TJ /F8 1 Tf 10.181 0 Td [(f)]TJ /F2 1 Tf 0.95181 0 Td [(est)-361(p)-30(ositiv)24(e)-355(et)-355(in)24(t)24(\266)469(egrable)-355(sur)]TJ /F8 1 Tf 12.861 0 Td [(I)]TJ /F2 1 Tf 0.86747 0 Td [(le)-361(r)24(\266)475(esultat)-361(pr)24(\266)475(ec)24(\266)469(eden)24(t)-355(mon)24(tre)-355(que)]TJ -24.861 -1.2048 Td [(les)-337(limites)-337(de)]TJ /F8 1 Tf 6.012 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))-337(lorsque)]TJ /F8 1 Tf 4.1506 0 Td [(x)]TJ /F2 1 Tf 0.90362 0 Td [(tend)-343(v)24(ers)]TJ /F8 1 Tf 4.3494 0 Td [(\256)]TJ /F2 1 Tf 0.9759 0 Td [(et)-337(lorsque)]TJ /F8 1 Tf 4.5904 0 Td [(x)]TJ /F2 1 Tf 0.90964 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(\257)]TJ /F2 1 Tf 0.95181 0 Td [(son)24(t)-337(\257nies.)]TJ -28.928 -1.6386 Td [(Lorsque)]TJ /F8 1 Tf 3.7831 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(est)-343(in)24(t)24(\266)469(egrable)-337(mais)-343(n'est)-343(plus)-343(supp)-30(os)24(\266)469(ee)-343(p)-30(ositiv)30(e,)-349(on)-343(p)-30(eut)-319(\266)469(ecrire)]TJ /F8 1 Tf 27.753 0 Td [(f)]TJ /F2 1 Tf 0.89157 0 Td [(=)]TJ /F8 1 Tf 1.0663 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.612 37.306 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.151 26.22 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.026 37.306 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 26.22 Td [(et)]TJ -38.331 -1.2048 Td [(en)-331(appliquan)24(t)-337(le)-337(r)30(\266)469(esultat)-337(pr)24(\266)469(ec)24(\266)475(eden)24(t)-337(\265)500(a)]TJ /F8 1 Tf 16.434 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 31.845 35.581 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.253 25.015 Td [(et)-337(\265)500(a)]TJ /F8 1 Tf 2 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.836 35.581 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.753 25.015 Td [(on)-337(obtien)30(t)-337(le)-337(r)24(\266)475(esultat)-343(annonc)24(\266)475(e.)]TJ ET 512.64 297.82 m 519.84 297.82 l 519.84 306.25 l 512.64 306.25 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 22.732 Td [(Remarque.)]TJ /F2 1 Tf 6.1687 0 Td [(Dans)-271(l')24(\266)475(enonc)24(\266)469(e)-271(pr)24(\266)469(ec)24(\266)475(eden)24(t,)-283(l'in)24(t)24(\266)469(egrabilit)24(\266)475(e)-277(de)]TJ /F8 1 Tf 18.367 0 Td [(f)]TJ /F2 1 Tf 0.86747 0 Td [(p)-24(ermet)-277(de)-271(conclure)-271(\265)500(a)-271(l'existence)]TJ -25.404 -1.2048 Td [(des)-361(limites)-361(de)]TJ /F8 1 Tf 6.3675 0 Td [(F)]TJ /F2 1 Tf 1.1386 0 Td [(aux)-367(extr)30(\266)469(emit)24(\266)469(es)-355(de)-361(l'in)24(terv)54(alle)]TJ /F8 1 Tf 13.12 0 Td [(I)]TJ /F2 1 Tf 0.87349 0 Td [(;)-373(cep)-30(endan)24(t)-367(la)-355(r)24(\266)475(ecipro)-30(que)-361(est)-367(incorrecte,)]TJ -21.5 -1.1988 Td [(l'existence)-500(de)-493(ces)-500(limites)-500(ne)-500(p)-30(ermet)-500(pas)-500(de)-493(conclure)-500(que)]TJ /F8 1 Tf 26.307 0 Td [(f)]TJ /F2 1 Tf 1.0964 0 Td [(est)-500(in)24(t)24(\266)469(egrable)-493(\(au)-506(sens)-493(de)]TJ -27.404 -1.4398 Td [(Leb)-30(esgue\).)-445(Le)-337(sym)24(b)-30(ole)]TJ /F9 1 Tf 10.373 0.81325 Td [(\243)]TJ /F8 1 Tf 0.42169 -0.81325 Td [(F)]TJ /F9 1 Tf 0.77711 0.81325 Td [(\244)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.793 27.892 Td [(\257)]TJ 0 -1.431 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.229 18.888 Td [(p)-30(eut)-337(a)30(v)24(oir)-337(un)-337(sens)-331(sans)-337(p)-30(our)-331(autan)24(t)-337(que)]TJ /F8 1 Tf 17.952 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(soit)-337(in)24(t)24(\266)475(egrable.)]TJ -31.777 -1.6446 Td [(On)-337(p)-24(ourra)-337(lire)-337(\265)500(a)-331(ce)-337(sujet)-337(la)-337(section)-331(sur)-343(les)-331(in)24(t)24(\266)469(egrales)-337(semi-con)30(v)24(ergen)18(tes.)]TJ 0 -2.2831 Td [(Ce)-379(corollaire)-379(p)-30(ermet)-379(d')24(\266)469(ev)54(aluer)-379(des)-379(in)24(t)24(\266)469(egrales)-379(\265)500(a)-379(partir)-379(de)-373(la)-379(connaissance)-385(d'un)-379(tableau)-379(de)]TJ 0 -1.1988 Td [(primitiv)30(es.)-451(Il)-331(a)-337(aussi)-331(les)-337(cons)24(\266)469(equences)-337(suiv)48(an)30(tes)-343(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 8.9017 Td [(VI.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 10.671 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084337 Td [(\266)]TJ 0 -0.084337 Td [(eor)]TJ 1.8012 0.084337 Td [(\265)]TJ -0.0060244 -0.084337 Td [(eme.)]TJ /F1 1 Tf 2.6627 0 Td [(\(d'in)30(t)24(\266)548(egration)-331(par)-331(parties\))]TJ /F12 1 Tf 13.38 0 Td [(Soien)24(t)]TJ /F8 1 Tf 2.9819 0 Td [(f)]TJ /F12 1 Tf 0.87952 0 Td [(et)]TJ /F8 1 Tf 1.1205 0 Td [(g)]TJ /F12 1 Tf 0.8012 0 Td [(des)-289(fonctions)-289(con)30(tin)-6(^)524(u-)]TJ -29.717 -1.1988 Td [(men)24(t)-337(d)30(\266)469(eriv)54(ables)-337(sur)-343(un)-331(segmen)18(t)]TJ /F2 1 Tf 14.169 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(.)-445(On)-337(a)-331(alors)-337(:)]TJ /F9 1 Tf -9.1205 -0.59036 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.086 12.357 Td [(b)]TJ -0.62931 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.741 7.5207 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.19 11.349 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.482 7.5207 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\241)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.534 12.357 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.524 7.5207 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.966 11.349 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.404 7.5207 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39156 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dx)]TJ /F12 1 Tf 1.4277 0 Td [(.)]TJ /F5 1 Tf -31.88 -3.6325 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(82)]TJ ET endstream endobj 282 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 281 0 R /ProcSet 2 0 R >> /Contents 280 0 R >> endobj 281 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F19 276 0 R >> endobj 283 0 obj << /Length 22162 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ 0 -2.4036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-337(fonction)]TJ /F8 1 Tf 5.3193 0 Td [(f)-108(g)]TJ /F2 1 Tf 1.4458 0 Td [(est)-337(une)-337(primitiv)24(e)-337(de)]TJ /F8 1 Tf 9.0663 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.759 89.633 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.861 62.22 Td [(g)]TJ /F2 1 Tf 0.73494 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.241 89.633 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.988 62.22 Td [(,)-337(on)-331(en)-337(d)30(\266)469(eduit)-331(:)]TJ /F8 1 Tf -28.639 -1.8916 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7289 1.3615 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.121 87.926 Td [(b)]TJ -0.63793 -2.9052 Td [(a)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 16.554 61.135 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.207 86.918 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.892 60.328 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6145 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.474 86.918 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.072 60.328 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62048 -0.80723 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3615 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.603 87.926 Td [(b)]TJ -0.62931 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.873 60.328 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.034 86.918 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.753 60.328 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3133 0 Td [(+)]TJ /F9 1 Tf 1 1.3615 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.017 87.926 Td [(b)]TJ -0.62931 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.849 60.328 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.112 86.918 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.59 60.328 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dx)]TJ /F2 1 Tf 1.4277 0 Td [(.)]TJ -39.205 -2.1446 Td [(Le)-337(th)24(\266)469(eor)24(\265)475(eme)-337(en)-331(d)24(\266)469(ecoule)-337(imm)24(\266)469(ediatemen)30(t.)]TJ ET 512.64 694.26 m 519.84 694.26 l 519.84 702.68 l 512.64 702.68 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 55.912 Td [(On)-337(utilise)-331(parfois)-343(cet)-307(\266)475(enonc)24(\266)475(e)-343(sous)-331(la)-337(forme)-331(g)24(\266)475(en)24(\266)475(eralis)24(\266)469(ee)-337(suiv)54(an)24(te)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 44.078 Td [(VI.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 52.84 Td [(5.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084339 Td [(\266)]TJ 0 -0.084339 Td [(eor)]TJ 1.8012 0.084339 Td [(\265)]TJ -0.0060244 -0.084339 Td [(eme.)]TJ /F12 1 Tf 3.0843 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.1566 0 Td [(I)]TJ /F12 1 Tf 0.98193 0 Td [(un)-463(in)24(terv)54(alle)-463(d'extr)30(\266)469(emit)24(\266)469(es)]TJ /F8 1 Tf 11.873 0 Td [(a)]TJ /F12 1 Tf 0.99397 0 Td [(et)]TJ /F8 1 Tf 1.2952 0 Td [(b)]TJ /F12 1 Tf 0.89156 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)-500(<)-493(b)]TJ /F12 1 Tf 2.7229 0 Td [(\),)]TJ /F8 1 Tf 1.1627 0 Td [(f)]TJ /F12 1 Tf 1.0602 0 Td [(et)]TJ /F8 1 Tf 1.2952 0 Td [(g)]TJ /F12 1 Tf 0.9759 0 Td [(des)]TJ -37.771 -1.2048 Td [(fonctions)-349(con)24(tin)24(ues)-355(sur)]TJ /F8 1 Tf 10.38 0 Td [(I)]TJ /F12 1 Tf 0.87349 0 Td [(et)-349(con)24(tin)-6(^)530(umen)24(t)-355(d)24(\266)469(eriv)54(ables)-355(en)-349(dehors)-361(d'un)-355(ensem)24(ble)]TJ /F8 1 Tf 22.741 0 Td [(E)]TJ /F12 1 Tf 1.1446 0 Td [(de)-349(p)-30(oin)24(ts)]TJ -35.139 -1.1988 Td [(isol)24(\266)469(es)-307(\(i.e.)-439(c)24(haque)-307(p)-30(oin)24(t)-307(de)]TJ /F8 1 Tf 11.898 0 Td [(E)]TJ /F12 1 Tf 1.1024 0 Td [(admet)-307(un)-307(v)24(oisinage)-313(qui)-307(ne)-307(p)-30(oss)30(\265)469(ede)-307(aucun)-313(autre)-307(p)-30(oin)24(t)-307(de)]TJ /F8 1 Tf 24.699 0 Td [(E)]TJ /F12 1 Tf 0.79518 0 Td [(\).)]TJ -38.494 -1.6325 Td [(On)-337(fait)-331(les)-337(h)24(yp)-30(oth)18(\265)475(eses)-337(suiv)48(an)30(tes)-343(:)]TJ /F1 1 Tf 0.81325 -1.6386 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(les)-337(fonctions)]TJ /F8 1 Tf 5.7108 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.784 68.09 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 15.139 47.165 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.19 68.09 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.181 47.165 Td [(g)]TJ /F12 1 Tf 0.8494 0 Td [(\(d)24(\266)469(e\257nies)-337(presque)-337(partout\))-337(son)24(t)-337(in)30(t)24(\266)469(egrables)-337(sur)]TJ /F8 1 Tf 20.464 0 Td [(I)]TJ /F12 1 Tf 0.85542 0 Td [(;)]TJ /F1 1 Tf -33.205 -1.6325 Td [(2.)]TJ /F8 1 Tf 1.5602 0 Td [(f)-108(g)]TJ /F2 1 Tf 1.1084 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(admet)-331(une)-337(limite)-337(\257nie)-331(lorsque)]TJ /F8 1 Tf 13.373 0 Td [(x)]TJ /F12 1 Tf 0.90964 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(a)]TJ /F12 1 Tf 0.86145 0 Td [(et)-331(lorsque)]TJ /F8 1 Tf 4.5904 0 Td [(x)]TJ /F12 1 Tf 0.90964 0 Td [(tend)-337(v)24(ers)]TJ /F8 1 Tf 4.3434 0 Td [(b)]TJ /F12 1 Tf 0.42771 0 Td [(.)]TJ -34.928 -1.9518 Td [(On)-337(a)-331(alors)]TJ /F9 1 Tf 10.03 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.414 63.961 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.765 43.581 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.509 62.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.506 43.581 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.81325 Td [(\243)]TJ /F8 1 Tf 0.42169 -0.81325 Td [(f)-108(g)]TJ /F9 1 Tf 1.1084 0.81325 Td [(\244)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.336 63.228 Td [(b)]TJ 0 -1.431 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.096 43.581 Td [(\241)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.216 63.961 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.5 43.581 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.638 62.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.38 43.581 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F12 1 Tf 1.4277 0 Td [(.)]TJ /F1 1 Tf -28.855 -2.4578 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Fixons)-295(un)-307(p)-30(oin)30(t)]TJ /F8 1 Tf 7.1988 0 Td [(c)]TJ /F2 1 Tf 0.72892 0 Td [(dans)]TJ /F8 1 Tf 2.3072 0 Td [(I)]TJ /F2 1 Tf 0.81928 0 Td [(et)-301(consid)30(\266)469(erons)-301(la)-301(fonction)]TJ /F8 1 Tf 11.355 0 Td [(\301)]TJ /F2 1 Tf 0.89156 0 Td [(d)24(\266)469(e\257nie)-301(sur)]TJ /F8 1 Tf 4.7771 0 Td [(I)]TJ /F2 1 Tf 0.81928 0 Td [(par)]TJ /F8 1 Tf -37.717 -1.8373 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-301(=)]TJ /F8 1 Tf 1.759 0 Td [(f)-108(g)]TJ /F2 1 Tf 1.1145 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.62048 0 Td [(\241)]TJ /F9 1 Tf 1.006 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.491 57.814 Td [(t)]TJ -0.63793 -2.9052 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.108 39.286 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.905 56.806 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.12 39.286 Td [(+)]TJ /F8 1 Tf 1.006 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.931 56.806 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.006 39.286 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\)\()]TJ /F8 1 Tf 0.77711 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)-481(Il)-343(s'agit)-349(d'une)-343(fonction)-349(con)24(tin)24(ue)-349(qui,)-343(en)-349(dehors)-349(de)]TJ /F8 1 Tf 22.91 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(,)]TJ -38.886 -1.8193 Td [(est)-361(d)30(\266)469(eriv)54(able)-355(de)-355(d)24(\266)475(eriv)48(\266)469(ee)-355(\()]TJ /F8 1 Tf 11.018 0 Td [(f)-108(g)]TJ /F2 1 Tf 1.1084 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.543 54.211 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.361 37.467 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.319 54.211 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 21.006 37.467 Td [(\241)]TJ /F8 1 Tf 1.012 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.371 54.211 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.898 37.467 Td [(g)]TJ /F2 1 Tf 0.8253 0 Td [(=)-313(0.)-512(Etudions)-355(ce)-355(qui)-355(se)-355(passe)-355(en)-355(un)-361(p)-30(oin)30(t)]TJ /F8 1 Tf 18.898 0 Td [(d)]TJ /F2 1 Tf 0.8735 0 Td [(de)]TJ /F8 1 Tf -38.163 -1.1988 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(,)-325(p)-30(our)-325(cela)-325(restreignons)-319(nous)-325(\265)500(a)-325(un)-325(in)30(terv)54(alle)]TJ /F8 1 Tf 19.422 0 Td [(J)]TJ /F2 1 Tf 0.96988 0 Td [(v)24(oisinage)-331(de)]TJ /F8 1 Tf 5.5964 0 Td [(d)]TJ /F2 1 Tf 0.84337 0 Td [(qui)-325(ne)-319(p)-30(oss)24(\265)475(ede)-331(pas)-325(d'autre)]TJ -27.627 -1.1988 Td [(p)-30(oin)30(t)-361(de)]TJ /F8 1 Tf 3.9819 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(.)-506(Sur)-355(c)24(haque)-355(in)24(terv)54(alle)]TJ /F8 1 Tf 10.349 0 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.069 49.969 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22 35.069 Td [(=)]TJ /F8 1 Tf 1.0904 0 Td [(J)]TJ /F6 1 Tf 0.6506 0 Td [(\\)]TJ /F2 1 Tf 0.66265 0 Td [(])]TJ /F6 1 Tf 0.51807 0 Td [(\241)-234(1)]TJ /F2 1 Tf 2.012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [([)-355(et)]TJ /F8 1 Tf 1.8193 0 Td [(J)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.319 49.969 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.247 35.069 Td [(=)]TJ /F8 1 Tf 1.0843 0 Td [(J)]TJ /F6 1 Tf 0.6506 0 Td [(\\)]TJ /F2 1 Tf 0.66867 0 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(d)]TJ /F2 1 Tf 0.5241 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)-355(la)-349(fonction)]TJ /F8 1 Tf 5.6446 0 Td [(\301)]TJ /F2 1 Tf 0.94579 0 Td [(est)]TJ -37.934 -1.1988 Td [(d)30(\266)469(eriv)54(able)-307(de)-307(d)30(\266)469(eriv)54(\266)469(ee)-307(n)24(ulle)-301(donc)-307(constan)24(te.)-439(La)-307(con)24(tin)24(uit)24(\266)469(e)-307(de)]TJ /F8 1 Tf 25.873 0 Td [(\301)]TJ /F2 1 Tf 0.89759 0 Td [(assure)-307(que)-307(la)-301(v)48(aleur)-301(de)]TJ /F8 1 Tf 10.151 0 Td [(\301)]TJ /F2 1 Tf 0.89759 0 Td [(sur)]TJ /F8 1 Tf -37.819 -1.2048 Td [(J)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4224 46.53 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8916 32.665 Td [(ainsi)-331(que)-337(celle)-337(sur)]TJ /F8 1 Tf 8.0964 0 Td [(J)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 22.241 46.53 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.542 32.665 Td [(est)-307(\266)469(egale)-337(\265)500(a)]TJ /F8 1 Tf 4.8916 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(\),)-337(par)-337(suite)]TJ /F8 1 Tf 5.1747 0 Td [(\301)]TJ /F2 1 Tf 0.92771 0 Td [(est)-337(constan)24(te)-337(sur)]TJ /F8 1 Tf 7.7169 0 Td [(J)]TJ /F2 1 Tf 0.98193 0 Td [(donc)-307(\266)469(egalemen)24(t)]TJ -32.41 -1.1988 Td [(d)30(\266)469(eriv)54(able)-337(de)-331(d)24(\266)469(eriv)54(\266)469(ee)-337(n)30(ulle)-337(au)-337(p)-30(oin)24(t)]TJ /F8 1 Tf 15.398 0 Td [(d)]TJ /F2 1 Tf 0.5241 0 Td [(.)]TJ -15.922 -1.6325 Td [(En)-445(conclusion)]TJ /F8 1 Tf 6.6325 0 Td [(\301)]TJ /F2 1 Tf 1.0361 0 Td [(a)-445(une)-445(d)24(\266)475(eriv)48(\266)475(ee)-451(n)30(ulle)-445(en)-445(tout)-445(p)-30(oin)24(t)-445(de)]TJ /F8 1 Tf 16.867 0 Td [(I)]TJ /F2 1 Tf 0.96385 0 Td [(donc)-445(est)-445(constan)24(te.)-777(Or)]TJ /F8 1 Tf 10.982 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-445(a)]TJ -37.831 -1.7711 Td [(une)-373(limite)-379(lorsque)]TJ /F8 1 Tf 8.2771 0 Td [(t)]TJ /F2 1 Tf 0.73494 0 Td [(tend)-379(v)24(ers)]TJ /F8 1 Tf 4.4277 0 Td [(a)]TJ /F2 1 Tf 0.90361 0 Td [(\(resp.)]TJ /F8 1 Tf 3.0241 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))-349(\266)469(egale)-379(\265)499(a)-524(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.078 39.306 Td [(x)]TJ /F10 1 Tf 0.65517 0 Td [(!)]TJ /F7 1 Tf 1.1379 0 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.163 28.063 Td [(f)-108(g)]TJ /F2 1 Tf 1.1084 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.63855 0 Td [(\241)]TJ /F9 1 Tf 1.0241 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.509 41.754 Td [(a)]TJ -0.62931 -2.8965 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.217 28.063 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.112 40.754 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.247 28.063 Td [(+)]TJ /F8 1 Tf 1.0301 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.198 40.754 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.157 28.063 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\)\()]TJ /F8 1 Tf 0.77711 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4699 0 Td [(\(resp.)]TJ -36.608 -2.6084 Td [(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.6293 35.487 Td [(x)]TJ /F10 1 Tf 0.64655 0 Td [(!)]TJ /F7 1 Tf 1.1379 0 Td [(b)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1024 25.454 Td [(f)-108(g)]TJ /F2 1 Tf 1.1084 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.53614 0 Td [(\241)]TJ /F9 1 Tf 0.91566 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.638 38.021 Td [(b)]TJ -0.63793 -2.9052 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.861 25.454 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.121 37.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.783 25.454 Td [(+)]TJ /F8 1 Tf 0.92169 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.897 37.012 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.578 25.454 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\)\()]TJ /F8 1 Tf 0.77711 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(\).)-439(L')24(\266)475(egalit)18(\266)475(e)-301(de)-295(ces)-289(limites)-301(\(puisque)]TJ /F8 1 Tf 15.187 0 Td [(\301)]TJ /F2 1 Tf 0.89157 0 Td [(est)-295(constan)24(te\))-301(fournit)]TJ -29.837 -1.7108 Td [(la)-331(relation)-337(demand)30(\266)469(ee)-337(gr)0(^)500(ace)-337(\265)500(a)-331(la)-337(relation)-331(de)-337(Chasles.)]TJ ET 512.64 282.63 m 519.84 282.63 l 519.84 291.06 l 512.64 291.06 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 21.479 Td [(Remarque.)]TJ /F2 1 Tf 6.8373 0 Td [(L')18(\266)475(enonc)24(\266)475(e)-487(pr)24(\266)475(ec)24(\266)469(eden)30(t)-487(supp)-30(ose)]TJ /F14 1 Tf 12.922 0 Td [(a)-493(priori)]TJ /F2 1 Tf 4.1265 0 Td [(l'in)24(t)24(\266)469(egrabilit)24(\266)475(e)-487(des)-481(fonctions)]TJ /F8 1 Tf 12.572 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.388 31.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 21.479 Td [(et)]TJ /F8 1 Tf -38.331 -1.2048 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4828 29.607 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.2108 20.274 Td [(g)]TJ /F2 1 Tf 0.96386 0 Td [(sur)]TJ /F8 1 Tf 1.8012 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-813(Il)-457(ne)-457(p)-30(eut)-463(pas)-433(^)475(etre)-457(utilis)24(\266)469(e)-463(\(sans)-457(autre)-463(pr)24(\266)475(ecaution\))-463(p)-24(our)-463(d)30(\266)469(eduire)-457(une)-457(de)-457(ces)]TJ -4.1627 -1.1988 Td [(in)24(t)24(\266)475(egrabilit)24(\266)475(es)-493(\265)500(a)-487(partir)-487(de)-493(l'autre.)-909(En)-487(fait)-493(l'in)24(t)24(\266)469(egrabilit)30(\266)469(e)-493(de)]TJ /F8 1 Tf 27.114 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.284 27.892 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.325 19.075 Td [(g)]TJ /F2 1 Tf 1 0 Td [(\(par)-493(exemple\))-493(join)24(te)-487(\265)500(a)]TJ -28.994 -1.1988 Td [(l'h)24(yp)-30(oth)24(\265)469(ese)-439(d'existence)-439(des)-439(limites)-445(de)]TJ /F8 1 Tf 17.157 0 Td [(f)-108(g)]TJ /F2 1 Tf 1.1145 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-439(lorsque)]TJ /F8 1 Tf 4.3554 0 Td [(x)]TJ /F2 1 Tf 1.012 0 Td [(tend)-439(v)24(ers)]TJ /F8 1 Tf 4.5482 0 Td [(a)]TJ /F2 1 Tf 0.96386 0 Td [(ou)-439(v)24(ers)]TJ /F8 1 Tf 3.6627 0 Td [(b)]TJ /F2 1 Tf 0.86747 0 Td [(p)-30(ermet)-439(de)]TJ -34.639 -1.8976 Td [(prouv)30(er)-409(que)]TJ /F9 1 Tf 5.6747 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.181 24.461 Td [(t)]TJ -0.63793 -2.8966 Td [(c)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.361 15.979 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.276 23.461 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.753 15.979 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.5 0 Td [(a)-403(une)-403(limite)-403(lorsque)]TJ /F8 1 Tf 9.2651 0 Td [(t)]TJ /F2 1 Tf 0.76506 0 Td [(tend)-409(v)24(ers)]TJ /F8 1 Tf 4.4819 0 Td [(a)]TJ /F2 1 Tf 0.93374 0 Td [(ou)-403(v)24(ers)]TJ /F8 1 Tf 3.5964 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(.)-656(Mais)-409(nous)-403(a)30(v)24(ons)]TJ -30.904 -1.8133 Td [(remarqu)30(\266)469(e)-337(que)-343(cela)-337(n'est)-337(pas)-337(su\261san)18(t)-337(p)-30(our)-337(assurer)-337(l'in)24(t)24(\266)475(egrabilit)24(\266)475(e)-343(de)]TJ /F8 1 Tf 29.663 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.664 20.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.717 14.165 Td [(\(sauf)-337(si)]TJ /F8 1 Tf 3.488 0 Td [(f)-108(g)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.121 20.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.934 14.165 Td [(est)-337(de)]TJ -36.602 -1.1988 Td [(signe)-337(constan)24(t\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 8.2485 Td [(VI.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 9.8882 Td [(6.)]TJ /F15 1 Tf 1.753 0 Td [(Th)]TJ 1.3976 0.084337 Td [(\266)]TJ 0 -0.084337 Td [(eor)]TJ 1.8012 0.084337 Td [(\265)]TJ -0.0060244 -0.084337 Td [(eme.)]TJ /F1 1 Tf 2.6687 0 Td [(\(de)-355(c)24(hangemen)24(t)-349(de)-361(v)60(ariable\))]TJ /F12 1 Tf 14.494 0 Td [(Soit)]TJ /F8 1 Tf 2.0301 0 Td [(')]TJ /F12 1 Tf 0.96385 0 Td [(une)-307(fonction)-313(con)30(tin)-6(^)530(umen)18(t)]TJ -28.048 -1.1988 Td [(d)30(\266)469(eriv)54(able)-307(et)-301(monotone)-301(sur)-307(un)-301(in)24(terv)54(alle)]TJ /F8 1 Tf 17.295 0 Td [(I)]TJ /F12 1 Tf 0.81928 0 Td [(\(de)-301(longueur)-307(non)-307(n)30(ulle\))-307(de)]TJ /F3 1 Tf 11.759 0 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)-433(Alors,)-307(si)]TJ /F8 1 Tf 4.5843 0 Td [(f)]TJ /F12 1 Tf 0.89759 0 Td [(est)-307(une)]TJ -36.078 -1.1988 Td [(fonction)-391(mesurable)-391(p)-30(ositiv)30(e)-397(\(resp.)-614(in)30(t)24(\266)469(egrable\))-391(sur)]TJ /F8 1 Tf 22.349 0 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(,)]TJ /F2 1 Tf 0.68072 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.328 11.306 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.054 7.4906 Td [(j)]TJ /F12 1 Tf 0.66868 0 Td [(est)-391(mesurable)-391(p)-30(ositiv)30(e)]TJ -29.392 -1.2048 Td [(\(resp.)-445(in)24(t)24(\266)475(egrable\))-337(sur)]TJ /F8 1 Tf 9.6084 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(et)-337(l'on)-337(a)-331(:)]TJ /F2 1 Tf 8.4578 -2.3976 Td [(83)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 285 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 284 0 R /ProcSet 2 0 R >> /Contents 283 0 R >> endobj 284 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F14 43 0 R /F15 68 0 R >> endobj 286 0 obj << /Length 22887 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F9 1 Tf 11.837 -1.4036 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.371 87.219 Td [(')]TJ /F4 1 Tf 0.75 0 Td [(\()]TJ /F7 1 Tf 0.44827 0 Td [(I)]TJ /F4 1 Tf 0.58621 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.337 61.858 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.966 87.219 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.892 61.858 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\241)]TJ /F8 1 Tf 0.46386 -0.81325 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.81325 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.931 89.116 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.578 61.858 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.52409 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dy)]TJ /F12 1 Tf 1.3795 0 Td [(,)]TJ -27.38 -2.4458 Td [(en)-307(outre,)-319(si)]TJ /F8 1 Tf 5.1627 0 Td [(I)]TJ /F12 1 Tf 0.83132 0 Td [(est)-313(l'in)24(terv)54(alle)]TJ /F2 1 Tf 6.4096 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(,)-313(en)-307(prolongean)18(t)-283(\266)469(ev)24(en)30(tuellemen)24(t)]TJ /F8 1 Tf 14.169 0 Td [(')]TJ /F12 1 Tf 0.96385 0 Td [(par)-307(con)24(tin)24(uit)18(\266)475(e)-313(\265)500(a)]TJ /F2 1 Tf 7.2108 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])]TJ /F12 1 Tf 0.2771 0 Td [(,)]TJ -38.886 -1.2048 Td [(p)-30(our)]TJ /F8 1 Tf 2.3614 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(in)24(t)24(\266)475(egrable)-337(sur)]TJ /F8 1 Tf 6.3193 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.81326 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\);)]TJ /F8 1 Tf 0.83735 0 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81326 Td [(\242)]TJ /F12 1 Tf 0.45783 -0.81326 Td [(,)-337(on)-331(a)]TJ /F9 1 Tf -6.3554 -0.92771 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.845 81.616 Td [(')]TJ /F4 1 Tf 0.75862 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(b)]TJ /F4 1 Tf 0.5 0 Td [(\))]TJ /F7 1 Tf -2.3448 -2.8966 Td [(')]TJ /F4 1 Tf 0.75862 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(a)]TJ /F4 1 Tf 0.62069 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.614 55.918 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36 81.616 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.56 55.918 Td [(f)]TJ /F9 1 Tf 0.59638 0.81326 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.81326 Td [(\242)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.483 80.616 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.97 55.918 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F12 1 Tf 1.3795 0 Td [(.)]TJ /F1 1 Tf -27.488 -2.9036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Notons)]TJ /F6 1 Tf 3.5602 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 26.284 75.65 Td [(+)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 19.03 53.822 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F2 1 Tf 0.92771 -0.80723 Td [(l'ensem)24(ble)-469(des)-475(fonctions)-469(b)-30(or)30(\266)469(eliennes)-475(p)-30(ositiv)30(es)-469(sur)]TJ /F8 1 Tf -17.036 -1.3012 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-337(et)-337(consid)30(\266)469(erons)-337(dans)-331(une)-337(premi)24(\265)469(ere)-307(\266)469(etap)-30(e)-337(le)-337(cas)]TJ /F8 1 Tf 21.012 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-277(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.474 73.788 Td [(+)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 31.048 52.527 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F2 1 Tf 0.78916 -0.81325 Td [(.)]TJ -28.916 -1.7108 Td [(Comme)]TJ /F8 1 Tf 3.7229 0 Td [(')]TJ /F2 1 Tf 1.0422 0 Td [(est)-391(con)30(tin)18(ue)-385(\(puisque)-391(d)24(\266)475(eriv)48(able\))-391(ainsi)-385(que)]TJ /F8 1 Tf 18.771 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.241 72.142 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.187 50.003 Td [(\(par)-391(h)24(yp)-36(oth)24(\265)475(ese\),)-409(il)-391(est)-391(clair)-385(que,)]TJ -24.855 -1.2048 Td [(p)-30(our)]TJ /F8 1 Tf 2.3434 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(fonction)-319(b)-30(or)24(\266)475(elienne)-319(et)-319(p)-30(ositiv)30(e)-325(sur)]TJ /F8 1 Tf 15.133 0 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\),)-325(\()]TJ /F8 1 Tf 1.3735 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.552 70.426 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.012 48.798 Td [(j)]TJ /F2 1 Tf 0.59639 0 Td [(est)-319(b)-30(or)24(\266)475(elienne)-319(\(donc)-319(mesurable\))]TJ -25.277 -1.1988 Td [(et)-337(p)-24(ositiv)24(e)-337(sur)]TJ /F8 1 Tf 6.5422 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)]TJ -7.0602 -2.1687 Td [(Prouv)30(ons)-337(main)18(tenan)24(t)-337(l')24(\266)475(egalit)24(\266)469(e)]TJ /F9 1 Tf 13.416 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.621 63.711 Td [(')]TJ /F4 1 Tf 0.75862 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(I)]TJ /F4 1 Tf 0.57759 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.916 45.43 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.224 63.711 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.47 45.43 Td [(f)]TJ /F9 1 Tf 0.59638 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.81325 Td [(\242)]TJ /F6 1 Tf 0.46385 -0.81325 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.181 65.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.157 45.43 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dy)]TJ /F2 1 Tf 1.3855 0 Td [(p)-24(our)]TJ /F8 1 Tf 2.3614 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 55.991 64.797 Td [(+)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 39.795 46.244 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.39156 0.81325 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(.)]TJ -37.331 -2.5482 Td [(Remarquons)-253(que,)-277(lorsque)]TJ /F8 1 Tf 11.12 0 Td [(f)]TJ /F2 1 Tf 0.8494 0 Td [(v)48(arie)-253(dans)]TJ /F6 1 Tf 4.6024 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32.284 61.15 Td [(+)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 23.229 43.695 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(,)-271(le)-259(second)-259(mem)24(bre)-253(de)-259(cette)-259(relation)-253(d)30(\266)469(e\257nit)]TJ -20.765 -1.3012 Td [(une)-277(fonctionnelle)-277(sur)]TJ /F6 1 Tf 9.259 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 21.819 59.288 Td [(+)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 15.91 42.394 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.81325 Td [(\242)]TJ /F2 1 Tf 0.73494 -0.81325 Td [(qui)-277(est)-283(clairemen)24(t)-283(p)-24(ositiv)24(emen)24(t)-277(homog)24(\265)469(ene)-283(et)-277(d)24(\266)475(enom)24(brable-)]TJ -13.723 -1.1988 Td [(men)24(t)-445(additiv)24(e.)-783(La)-451(prop)-36(osition)]TJ /F1 1 Tf 14.145 0 Td [(IV.2.2.)]TJ /F2 1 Tf 3.8614 0 Td [(appliqu)24(\266)469(ee)-451(\265)500(a)-445(l'espace)]TJ /F8 1 Tf 9.3193 0 Td [(')]TJ /F2 1 Tf 0.65662 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.5241 0 Td [(\))-445(m)24(uni)-451(de)-445(sa)-445(tribu)-451(des)]TJ -28.892 -1.1988 Td [(b)-30(or)30(\266)469(eliens)-445(assure)-445(alors)-445(l'existence)-445(d'une)-445(mesure)]TJ /F8 1 Tf 21.211 0 Td [(m)]TJ /F2 1 Tf 1.3193 0 Td [(sur)-445(les)-445(b)-30(or)24(\266)475(eliens)-445(de)]TJ /F8 1 Tf 9.1024 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-445(telle)-445(que)-451(ce)]TJ -33.193 -1.7169 Td [(second)-361(mem)24(bre)-361(s')18(\266)475(ecriv)24(e)]TJ /F9 1 Tf 10.633 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.638 52.314 Td [(')]TJ /F4 1 Tf 0.75 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(I)]TJ /F4 1 Tf 0.58621 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.127 37.467 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5241 0 Td [(p)-30(our)-367(toute)]TJ /F8 1 Tf 5.0361 0 Td [(f)]TJ /F6 1 Tf 0.92169 0 Td [(2)-325(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.431 53.409 Td [(+)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 28.922 38.28 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(.)-524(T)72(out)-361(revien)24(t)-367(\265)500(a)-361(prouv)24(er)-361(que)]TJ /F8 1 Tf -26.458 -1.8795 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(co)108(\304)385(\263ncide)-337(a)30(v)24(ec)-337(la)-337(mesure)-337(de)-331(Leb)-36(esgue.)]TJ -1.2108 -1.7108 Td [(En)-361(utilisan)24(t)-361(la)-361(prop)-36(osition)-361(d'unicit)24(\266)469(e)-367(\(cf.)]TJ /F1 1 Tf 18.066 0 Td [(I)-36(I)-36(I.4.5.)]TJ /F2 1 Tf 3.4819 0 Td [(\),)-373(on)-367(v)24(oit)-361(qu'il)-367(su\261t)-367(de)-361(prouv)30(er)-367(la)-367(co)108(\304)391(\263n-)]TJ -21.548 -1.1988 Td [(cidence)-397(sur)-403(les)-397(in)24(terv)54(alles)-397(de)]TJ /F8 1 Tf 13.006 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\),)-415(c'est-)0(\265)499(a-dire)-403(prouv)24(er)-397(l')24(\266)475(egalit)24(\266)469(e)-403(lorsque)]TJ /F8 1 Tf 17.277 0 Td [(f)]TJ /F2 1 Tf 0.99397 0 Td [(est)-397(la)-397(fonction)]TJ -32.837 -1.2048 Td [(indicatrice)-331(d'un)-337(in)24(terv)54(alle)]TJ /F8 1 Tf 11.512 0 Td [(J)]TJ /F2 1 Tf 0.98193 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\).)]TJ -15.392 -2.2229 Td [(Il)-331(s'agit)-337(donc)-337(de)-331(mon)24(trer)]TJ /F9 1 Tf 11.181 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.422 40.556 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.53 29.25 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.888 41.642 Td [(J)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 18.651 30.063 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.81325 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.138 42.452 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.735 29.25 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.241 40.556 Td [(')]TJ /F4 1 Tf 0.75862 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(I)]TJ /F4 1 Tf 0.57759 0 Td [(\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.036 29.25 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.353 41.642 Td [(J)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.157 29.25 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.0964 0 Td [(.)]TJ -27.434 -2.5482 Td [(Notons)]TJ /F8 1 Tf 3.3916 0 Td [(K)]TJ /F2 1 Tf 1.2229 0 Td [(l'image)-301(r)24(\266)475(ecipro)-30(que)-307(de)]TJ /F8 1 Tf 9.5783 0 Td [(J)]TJ /F2 1 Tf 0.95181 0 Td [(par)]TJ /F8 1 Tf 1.747 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(,)-307(c'est)-307(un)-301(in)24(terv)54(alle)-301(de)]TJ /F8 1 Tf 9.8554 0 Td [(I)]TJ /F2 1 Tf 0.81928 0 Td [(et)-307(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.448 37.995 Td [(J)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 35.813 27.509 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F2 1 Tf 0.73494 -0.80723 Td [(=)-277(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.078 37.995 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.355 26.701 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\).)-433(La)]TJ -36.94 -1.8133 Td [(relation)-331(\265)500(a)-337(prouv)30(er)-337(est)-343(alors)]TJ /F9 1 Tf 12.139 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.793 34.314 Td [(K)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.795 24.888 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.233 36.211 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.012 24.888 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.336 34.314 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.422 24.888 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)-283(longueur)-337(de)]TJ /F8 1 Tf 6.506 0 Td [(J)]TJ /F2 1 Tf 0.6506 0 Td [(.)]TJ -28.614 -2.2169 Td [(Lorsque)]TJ /F8 1 Tf 3.7892 0 Td [(')]TJ /F2 1 Tf 1.006 0 Td [(est)-361(croissan)24(te)-349(cela)-361(d)30(\266)469(ecoule)-355(du)-355(corollaire)]TJ /F1 1 Tf 17.62 0 Td [(VI.2.3.)]TJ /F2 1 Tf 3.4157 0 Td [(.)-499(Lorsque)]TJ /F8 1 Tf 4.5723 0 Td [(')]TJ /F2 1 Tf 1.006 0 Td [(est)-355(d)24(\266)475(ecroissan)24(te)-355(il)]TJ -31.41 -1.1988 Td [(su\261t)-337(de)-331(remplacer)]TJ /F8 1 Tf 8.4578 0 Td [(')]TJ /F2 1 Tf 0.98795 0 Td [(par)-337(son)-337(opp)-30(os)24(\266)469(e)-337(p)-30(our)-331(obtenir)-337(le)-337(r)24(\266)475(esultat.)]TJ -9.4458 -2.0723 Td [(P)30(our)]TJ /F8 1 Tf 2.4096 0 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(b)-30(or)24(\266)475(elienne)-313(de)-313(signe)-313(quelconque,)-325(en)-313(appliquan)30(t)-313(le)-319(r)30(\266)469(esultat)-319(pr)30(\266)469(ec)24(\266)469(eden)30(t)-319(\265)500(a)]TJ /F6 1 Tf 30.319 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(,)]TJ /F8 1 Tf 0.59036 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 58.974 27.547 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.187 19.4 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.707 27.547 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 18.201 Td [(on)-313(obtien)30(t)-313(que,)-319(si)]TJ /F8 1 Tf 7.8614 0 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(est)-313(in)24(t)24(\266)469(egrable,)-313(\()]TJ /F8 1 Tf 6.8313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.353 26.642 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.289 18.201 Td [(j)]TJ /F2 1 Tf 0.58434 0 Td [(est)-319(aussi)-313(in)30(t)24(\266)469(egrable)-313(et)-313(l')30(\266)469(egalit)24(\266)469(e)-313(pr)24(\266)475(ec)24(\266)469(eden)30(te)-313(est)]TJ -19.542 -1.1988 Td [(encore)-337(v)54(alable.)]TJ 0 -1.7169 Td [(P)30(ar)-373(ailleurs,)-379(si)]TJ /F8 1 Tf 6.7349 0 Td [(I)]TJ /F2 1 Tf 0.89157 0 Td [(est)-373(un)-373(in)24(terv)54(alle)-367(d'extr)24(\266)469(emit)24(\266)469(es)]TJ /F8 1 Tf 13.199 0 Td [(a)]TJ /F2 1 Tf 0.89759 0 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(,)]TJ /F8 1 Tf 0.65663 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-373(est)-373(un)-373(in)24(terv)54(alle)-367(d'extr)24(\266)469(emit)24(\266)469(es)]TJ /F8 1 Tf -25.578 -1.1988 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-246(et)]TJ /F8 1 Tf 1.7108 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))-246(\(en)-253(prolongean)24(t)-222(\266)475(ev)24(en)24(tuellemen)30(t)]TJ /F8 1 Tf 14.41 0 Td [(')]TJ /F2 1 Tf 0.89759 0 Td [(par)-246(con)24(tin)24(uit)24(\266)469(e)-253(aux)-246(p)-30(oin)24(ts)]TJ /F8 1 Tf 11.024 0 Td [(a)]TJ /F2 1 Tf 0.77108 0 Td [(et)]TJ /F8 1 Tf 1.0843 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\).)-421(La)-246(relation)]TJ -33.373 -1.988 Td [(obten)24(ue)-361(s')24(\266)469(ecrit)-367(alors)]TJ /F6 1 Tf 9.1928 0 Td [(\247)]TJ /F9 1 Tf 0.94578 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.569 18.909 Td [(')]TJ /F4 1 Tf 0.75 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(b)]TJ /F4 1 Tf 0.5 0 Td [(\))]TJ /F7 1 Tf -2.3362 -2.9052 Td [(')]TJ /F4 1 Tf 0.75 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(a)]TJ /F4 1 Tf 0.62069 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.024 12.099 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.4157 0 Td [(=)]TJ /F6 1 Tf 1.1084 0 Td [(\247)]TJ /F9 1 Tf 0.94578 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.216 18.909 Td [(b)]TJ -0.63793 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.006 12.099 Td [(f)]TJ /F9 1 Tf 0.60241 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)0(\241)]TJ /F6 1 Tf 1.1566 -0.80723 Td [(\247)]TJ /F8 1 Tf 1.0241 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.155 17.9 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.139 12.099 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62651 -0.80723 Td [(dy)]TJ /F2 1 Tf 1.4096 0 Td [(o)-30(\265)530(u)-367(les)]TJ /F6 1 Tf 2.9036 0 Td [(\247)]TJ /F2 1 Tf 1.1386 0 Td [(d)24(\266)475(ep)-30(enden)24(t)-367(du)]TJ -33.187 -1.988 Td [(fait)-427(que)]TJ /F8 1 Tf 3.8554 0 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-427(est)-433(inf)30(\266)469(erieur)-427(ou)-427(non)-427(\265)500(a)]TJ /F8 1 Tf 10.988 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\),)]TJ /F8 1 Tf 1.1145 0 Td [(a)]TJ /F2 1 Tf 0.95783 0 Td [(est)-433(inf)30(\266)469(erieur)-427(ou)-427(non)-427(\265)500(a)]TJ /F8 1 Tf 10.175 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(,)]TJ /F8 1 Tf 0.72892 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.353 15.056 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.988 10.111 Td [(est)-433(p)-30(ositiv)30(e)-433(ou)]TJ -32.657 -1.2048 Td [(n)30(\266)469(egativ)24(e.)-445(En)-337(examinan)24(t)-337(les)-337(di\256)24(\266)475(eren)24(ts)-337(cas)-337(on)-331(constate)-337(ais)24(\266)469(emen)24(t)-337(que)-337(l'on)-337(a)-331(toujours)-337(:)]TJ /F9 1 Tf 11.729 -0.57229 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.845 11.573 Td [(')]TJ /F4 1 Tf 0.75862 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(b)]TJ /F4 1 Tf 0.5 0 Td [(\))]TJ /F7 1 Tf -2.3448 -2.8966 Td [(')]TJ /F4 1 Tf 0.75862 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(a)]TJ /F4 1 Tf 0.62069 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.614 6.9726 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36 11.573 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.56 6.9726 Td [(f)]TJ /F9 1 Tf 0.59638 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.81325 Td [(\242)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.483 10.573 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.97 6.9726 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 1.0482 0 Td [(.)]TJ /F5 1 Tf -27.157 -3.0843 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(84)]TJ ET endstream endobj 288 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 287 0 R /ProcSet 2 0 R >> /Contents 286 0 R >> endobj 287 0 obj << /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R >> endobj 289 0 obj << /Length 16300 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F2 1 Tf 0 -2.4036 Td [(P)30(our)-337(passer)-343(au)-337(cas)-337(g)24(\266)475(en)24(\266)475(eral,)-343(i.e.)]TJ /F8 1 Tf 13.988 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(mesurable)-337(\(au)-343(sens)-337(de)-337(Leb)-30(esgue\),)-343(nous)-337(allons)-337(utiliser)-337(le)]TJ -14.928 -1.1988 Td [(fait)-289(que)-295(la)-289(mesure)-295(de)-289(Leb)-36(esgue)-289(s'obtien)24(t)-295(en)-289(compl)24(\266)469(etan)24(t)-295(la)-289(mesure)-295(de)-289(Borel)-295(\(cf.)-427(prop)-30(osition)]TJ /F1 1 Tf 0 -1.1988 Td [(VI)-36(I.2.2.)]TJ /F2 1 Tf 3.8795 0 Td [(\).)-451(Plus)-337(pr)24(\266)469(ecis)30(\266)469(emen)24(t)-337(\(cf.)]TJ /F1 1 Tf 10.705 0 Td [(IV.6.2.)]TJ /F2 1 Tf 3.4157 0 Td [(\),)-337(cela)-337(en)30(tra)102(^)391(\263ne)-337(:)]TJ /F1 1 Tf -17.187 -1.6446 Td [(1.)]TJ /F2 1 Tf 1.6747 0 Td [(toute)-391(fonction)-397(mesurable)-391(est)-397(la)-391(somme)-397(d'une)-391(fonction)-397(b)-30(or)30(\266)469(elienne)-391(et)-397(d'une)-391(fonction)]TJ -2.488 -1.2048 Td [(n)30(\266)469(egligeable)-337(;)]TJ /F1 1 Tf 0.81325 -1.6446 Td [(2.)]TJ /F2 1 Tf 1.3916 0 Td [(la)-253(v)54(aleur)-253(absolue)-253(de)-246(toute)-253(fonction)-259(n)30(\266)469(egligeable)-253(est)-253(ma)-60(jor)24(\266)469(ee)-253(par)-253(une)-253(fonction)-253(b)-30(or)30(\266)469(elienne)]TJ -2.2048 -1.1988 Td [(n)30(\266)469(egligeable)-337(;)]TJ /F1 1 Tf 0.81325 -1.6446 Td [(3.)]TJ /F2 1 Tf 1.5602 0 Td [(toute)-337(fonction)-337(n)24(\266)475(egligeable)-337(est)-337(mesurable)-337(et)-331(m)24(^)469(eme)-331(in)24(t)24(\266)469(egrable)-331(d'in)24(t)24(\266)469(egrale)-337(n)24(ulle.)]TJ -2.3735 -1.9639 Td [(Compte)-373(ten)24(u)-379(du)-379(r)30(\266)469(esultat)-379(lorsque)]TJ /F8 1 Tf 14.789 0 Td [(f)]TJ /F2 1 Tf 0.9759 0 Td [(est)-379(b)-30(or)30(\266)469(elienne,)-385(le)-385(p)-30(oin)30(t)]TJ /F1 1 Tf 10.494 0 Td [(1.)]TJ /F2 1 Tf 1.4699 0 Td [(mon)24(tre)-379(qu'il)-373(ne)-379(reste)-379(qu')0(\265)500(a)]TJ -27.729 -1.1988 Td [(prouv)30(er)-343(le)-337(th)24(\266)469(eor)24(\265)469(eme)-331(p)-30(our)]TJ /F8 1 Tf 11.434 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(n)24(\266)469(egligeable.)]TJ -12.361 -1.6446 Td [(Si)]TJ /F8 1 Tf 1.2229 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(est)-391(n)24(\266)469(egligeable,)-403(le)-397(p)-24(oin)24(t)]TJ /F1 1 Tf 10.861 0 Td [(2.)]TJ /F2 1 Tf 1.506 0 Td [(assure)-391(l'existence)-391(de)]TJ /F8 1 Tf 9.3313 0 Td [(g)]TJ /F2 1 Tf 0.90361 0 Td [(b)-30(or)30(\266)469(elienne)-391(sur)]TJ /F8 1 Tf 6.5964 0 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-391(telle)]TJ /F6 1 Tf 3 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.6506 0 Td [(6)]TJ /F8 1 Tf 1.1506 0 Td [(g)]TJ /F2 1 Tf -38.651 -1.7108 Td [(et)]TJ /F9 1 Tf 1.2229 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.181 64.469 Td [(')]TJ /F4 1 Tf 0.75 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(I)]TJ /F4 1 Tf 0.58621 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.7229 45.967 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4699 0 Td [(=)-379(0.)-626(Le)-397(r)24(\266)475(esultat)-397(pr)24(\266)469(ec)24(\266)475(eden)24(t)-397(implique)]TJ /F6 1 Tf 16.5 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F9 1 Tf 0.59638 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F9 1 Tf 0.39156 0.80723 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.80723 Td [(jj)]TJ /F8 1 Tf 0.55422 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.207 66.366 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.964 45.967 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.52409 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.65663 0 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(g)]TJ /F9 1 Tf 0.51205 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(')]TJ /F2 1 Tf 0.65662 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.80723 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.25 66.366 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.687 45.967 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F2 1 Tf 0.6747 0 Td [(et)]TJ /F9 1 Tf -38.331 -1.1867 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4224 60.823 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.3434 43.412 Td [(g)]TJ /F9 1 Tf 0.51205 0.81325 Td [(\241)]TJ /F8 1 Tf 0.46386 -0.81325 Td [(')]TJ /F2 1 Tf 0.6506 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.81325 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.267 62.719 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.946 43.412 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dy)]TJ /F2 1 Tf 1.4337 0 Td [(=)-385(0.)-644(Il)-397(s'ensuit)-403(que)]TJ /F8 1 Tf 9.241 0 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.81325 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.81325 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.75 62.719 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.06 43.412 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F2 1 Tf 0.68072 0 Td [(est)-403(n)30(\266)469(egligeable)-397(donc)-403(\(d'apr)24(\265)475(es)]TJ /F1 1 Tf 13.169 0 Td [(3.)]TJ /F2 1 Tf 0.89759 0 Td [(\))]TJ -38.777 -2.3675 Td [(mesurable)-331(et)-337(in)24(t)24(\266)469(egrable)-331(a)30(v)18(ec)]TJ /F9 1 Tf 14.741 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.517 57.426 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.084 41.045 Td [(f)]TJ /F9 1 Tf 0.59638 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(')]TJ /F2 1 Tf 0.65663 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.80723 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.483 59.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.771 41.045 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dy)]TJ /F2 1 Tf 1.3193 0 Td [(=)-283(0)-277(=)]TJ /F9 1 Tf 2.8916 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.216 57.426 Td [(')]TJ /F4 1 Tf 0.75 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(I)]TJ /F4 1 Tf 0.5862 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.904 41.045 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4277 0 Td [(.)]TJ ET 512.64 489.34 m 519.84 489.34 l 519.84 497.77 l 512.64 497.77 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 29.786 Td [(VI.3.)-236(Crit)15(\265)542(eres)-381(d'in)30(t)20(\266)542(egrabilit)20(\266)537(e.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 34.063 Td [(Soit)]TJ /F8 1 Tf 2.1627 0 Td [(X)]TJ /F2 1 Tf 1.3554 0 Td [(une)-445(partie)-445(mesurable)-445(de)]TJ /F3 1 Tf 11.289 0 Td [(R)]TJ /F2 1 Tf 1.1687 0 Td [(\(usuellemen)30(t)]TJ /F8 1 Tf 5.9759 0 Td [(X)]TJ /F2 1 Tf 1.3554 0 Td [(est)-445(un)-445(in)24(terv)54(alle\))-445(;)-500(consid)24(\266)475(erons)-451(une)]TJ -23.307 -1.1988 Td [(fonction)]TJ /F8 1 Tf 3.8373 0 Td [(f)]TJ /F2 1 Tf 0.90361 0 Td [(qui)-307(est)-313(mesurable)-307(sur)]TJ /F8 1 Tf 9.5602 0 Td [(X)]TJ /F2 1 Tf 1.2169 0 Td [(\(le)-307(plus)-307(fr)18(\266)475(equemmen)24(t)]TJ /F8 1 Tf 9.5181 0 Td [(f)]TJ /F2 1 Tf 0.90362 0 Td [(est)-313(con)30(tin)18(ue\).)-439(Le)-307(probl)18(\265)475(eme)-307(est)]TJ -25.94 -1.1988 Td [(alors)-337(de)-331(d)24(\266)469(eterminer)-337(si)]TJ /F8 1 Tf 9.8012 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-343(in)30(t)24(\266)469(egrable.)]TJ -10.729 -2.1024 Td [(Remarquons)-331(que)-337(l'in)18(t)24(\266)475(egrabilit)24(\266)475(e)-343(de)]TJ /F8 1 Tf 15.114 0 Td [(f)]TJ /F2 1 Tf 0.90361 0 Td [(\266)475(equiv)48(aut)-337(\265)500(a)]TJ /F9 1 Tf 4.9157 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.388 41.004 Td [(X)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.584 29.563 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)-283(<)]TJ /F2 1 Tf 2.4277 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)]TJ -29.127 -2.1506 Td [(Lorsque)]TJ /F8 1 Tf 3.8494 0 Td [(f)]TJ /F2 1 Tf 1.012 0 Td [(est)-415(in)24(t)24(\266)469(egrable)-409(on)-415(dit)-421(parfois)-415(que)-415(l'in)24(t)24(\266)469(egrale)-415(de)]TJ /F8 1 Tf 20.886 0 Td [(f)]TJ /F2 1 Tf 1.006 0 Td [(est)]TJ /F1 1 Tf 1.6446 0 Td [(con)30(v)30(ergen)30(te)]TJ /F2 1 Tf 5.9518 0 Td [(,)-433(lorsque)]TJ /F8 1 Tf 4.2169 0 Td [(f)]TJ /F2 1 Tf -38.566 -1.1988 Td [(n'est)-469(pas)-469(in)24(t)24(\266)475(egrable)-469(on)-469(dit)-469(parfois)-475(que)-469(l'in)24(t)24(\266)475(egrale)-475(de)]TJ /F8 1 Tf 24.066 0 Td [(f)]TJ /F2 1 Tf 1.0663 0 Td [(est)]TJ /F1 1 Tf 1.6988 0 Td [(div)30(ergen)30(te)]TJ /F2 1 Tf 5.2169 0 Td [(,)-500(toutefois)-475(cette)]TJ -32.048 -1.1988 Td [(terminologie)-337(p)-24(eut)-337(pr)24(^)469(eter)-331(\265)500(a)-337(confusion)-331(comme)-337(nous)-337(le)-337(v)24(errons.)]TJ 0 -1.6446 Td [(P)30(our)-337(r)30(\266)469(esoudre)-337(ce)-337(probl)24(\265)469(eme)-331(on)-337(utilise)-337(essen)24(tiellemen)24(t)-337(deux)-331(grandes)-337(id)24(\266)469(ees.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 16.917 Td [(VI.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 20.28 Td [(1.)-289(P)30(assage)-385(au)-391(lo)-36(cal.)]TJ /F2 1 Tf -2.9458 -1.6446 Td [(Av)30(ec)-361(les)-361(notations)-361(pr)30(\266)469(ec)24(\266)469(eden)30(tes)-361(rapp)-36(elons)-361(que,)-367(p)-30(our)]TJ /F8 1 Tf 22.904 0 Td [(Y)]TJ /F2 1 Tf 1.1626 0 Td [(partie)-361(mesurable)-355(de)]TJ /F8 1 Tf 9.0301 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-367(on)-355(dit)-361(que)]TJ /F8 1 Tf -34 -1.1988 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-343(in)30(t)24(\266)469(egrable)-331(sur)]TJ /F8 1 Tf 7.8855 0 Td [(Y)]TJ /F2 1 Tf 1.1325 0 Td [(lorsque)-337(le)-337(pro)-30(duit)-337(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.216 24.737 Td [(Y)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.602 17.436 Td [(\242)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)469(egrable.)-445(On)-331(a)-337(alors)]TJ /F15 1 Tf -20.482 -1.9639 Td [(Lemme.)]TJ /F12 1 Tf 4.1084 0 Td [(Si)]TJ /F8 1 Tf 1.1627 0 Td [(f)]TJ /F12 1 Tf 0.93374 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(sur)]TJ /F8 1 Tf 7.8795 0 Td [(Y)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.621 21.926 Td [(1)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.783 15.473 Td [(et)]TJ /F8 1 Tf 1.1626 0 Td [(Y)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32.241 21.926 Td [(2)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 22.976 15.473 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-313(\266)469(egalemen)30(t)-337(in)24(t)24(\266)469(egrable)-331(sur)]TJ /F8 1 Tf 12.578 0 Td [(Y)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 53.922 21.926 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.349 15.473 Td [([)]TJ /F8 1 Tf 0.89157 0 Td [(Y)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.983 21.926 Td [(2)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 40.271 15.473 Td [(.)]TJ /F1 1 Tf -34.94 -2.4217 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Il)-259(su\261t)-259(de)-259(v)24(oir)-259(que)-265(l'on)-259(a)-259(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.69 18.461 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 52.217 25.596 Td [(1)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.922 18.461 Td [([)]TJ /F7 1 Tf 0.76724 0 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 55.012 25.596 Td [(2)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 28.229 13.051 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.707 18.461 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 60.627 25.596 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.825 13.051 Td [(+)-72(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.129 18.461 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 65.41 25.596 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.404 13.051 Td [(et)-259(donc)]TJ /F9 1 Tf 3.4036 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.474 17.366 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 75.675 24.078 Td [(1)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.707 17.366 Td [([)]TJ /F7 1 Tf 0.76725 0 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 78.47 24.078 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.675 13.051 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F8 1 Tf 0.44579 0 Td [(dx)]TJ /F11 1 Tf 1.3675 0 Td [(6)]TJ /F9 1 Tf -38.386 -1.1145 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4224 13.831 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 12.711 19.126 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7952 10.575 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dx)]TJ /F2 1 Tf 1.3193 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.655 13.831 Td [(Y)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 25.614 19.126 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.247 10.575 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dx)]TJ /F2 1 Tf 1.4277 0 Td [(.)]TJ ET 512.64 125.24 m 519.84 125.24 l 519.84 133.66 l 512.64 133.66 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 7.4906 Td [(Alors,)-403(si)]TJ /F8 1 Tf 4.0602 0 Td [(a)]TJ /F2 1 Tf 0.91566 0 Td [(est)-397(un)-391(p)-30(oin)24(t)-391(de)-391(l'adh)24(\266)469(erence)]TJ ET 0.504 w 272.16 99.356 m 283.03 99.356 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.771 7.4906 Td [(X)]TJ /F2 1 Tf 1.3012 0 Td [(de)]TJ /F8 1 Tf 1.3916 0 Td [(X)]TJ /F2 1 Tf 1.2952 0 Td [(dans)]TJ ET 348.48 99.428 m 357.12 99.428 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 29.157 7.4906 Td [(R)]TJ /F2 1 Tf 1.1084 0 Td [(on)-397(dit)-391(que)]TJ /F8 1 Tf 4.9819 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(est)]TJ /F1 1 Tf 1.6145 0 Td [(in)30(t)18(\266)548(egrable)-445(au)]TJ -32.518 -1.2048 Td [(v)30(oisinage)-373(de)]TJ /F8 1 Tf 6.4458 0 Td [(a)]TJ /F2 1 Tf 0.8494 0 Td [(ou)-319(que)-325(l'in)24(t)24(\266)475(egrale)-325(de)]TJ /F8 1 Tf 9.1747 0 Td [(f)]TJ /F1 1 Tf 0.91566 0 Td [(con)30(v)30(erge)]TJ /F2 1 Tf 4.6928 0 Td [(au)-319(p)-30(oin)24(t)]TJ /F8 1 Tf 3.9699 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)-325(lorsqu'il)-319(existe)-325(un)-319(v)24(oisinage)]TJ -7.6627 -2.3976 Td [(85)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 291 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 290 0 R /ProcSet 2 0 R >> /Contents 289 0 R >> endobj 290 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F19 276 0 R >> endobj 292 0 obj << /Length 17101 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F8 1 Tf 0 -2.4036 Td [(V)]TJ /F2 1 Tf 1.1325 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(a)]TJ /F2 1 Tf 0.85542 0 Td [(tel)-331(que)]TJ /F8 1 Tf 3.2952 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(soit)-331(in)24(t)24(\266)475(egrable)-331(sur)]TJ /F8 1 Tf 8.1988 0 Td [(V)]TJ /F6 1 Tf 1.0241 0 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(.)-445(Dans)-331(le)-337(cas)-331(con)30(traire)-331(on)-331(dit)-337(que)-331(l'in)24(t)24(\266)469(egrale)-331(de)]TJ /F8 1 Tf -18.548 -1.1988 Td [(f)]TJ /F1 1 Tf 0.92771 0 Td [(div)24(erge)]TJ /F2 1 Tf 3.9759 0 Td [(au)-331(p)-30(oin)24(t)]TJ /F8 1 Tf 4 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-445(Av)30(ec)-337(cette)-337(d)24(\266)475(e\257nition)-337(on)-337(a)-331(:)]TJ /F15 1 Tf -9.4337 -1.747 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8614 0 Td [(P)30(our)-319(que)]TJ /F8 1 Tf 4.259 0 Td [(f)]TJ /F12 1 Tf 0.90964 0 Td [(soit)-319(in)30(t)24(\266)469(egrable)-313(sur)]TJ /F8 1 Tf 8.1566 0 Td [(X)]TJ /F12 1 Tf 1.2229 0 Td [(il)-319(faut)-313(et)-319(il)-313(su\261t)-319(que)]TJ /F8 1 Tf 9.2831 0 Td [(f)]TJ /F12 1 Tf 0.91566 0 Td [(soit)-313(in)24(t)24(\266)469(egrable)-313(au)]TJ -31.608 -1.1988 Td [(v)24(oisinage)-337(de)-337(tous)-331(les)-337(p)-30(oin)24(ts)-331(de)]TJ ET 0.504 w 226.08 703.94 m 236.95 703.94 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.916 58.075 Td [(X)]TJ /F12 1 Tf 0.90361 0 Td [(.)]TJ /F1 1 Tf -14.488 -1.747 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-391(condition)-391(est)-367(\266)469(evidemmen)24(t)-391(n)24(\266)469(ecessaire.)-614(R)24(\266)475(ecipro)-30(quemen)24(t,)-403(si)-397(elle)-391(est)]TJ -8.8193 -1.2048 Td [(r)30(\266)469(ealis)24(\266)475(ee,)-331(p)-30(our)-319(tout)]TJ /F8 1 Tf 8.2831 0 Td [(a)]TJ /F2 1 Tf 0.8494 0 Td [(de)]TJ ET 188.64 668.73 m 199.51 668.73 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.783 55.123 Td [(X)]TJ /F2 1 Tf 1.2229 0 Td [(il)-325(y)-325(a)-319(un)-319(v)24(oisinage)]TJ /F8 1 Tf 8.253 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.974 78.668 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.645 55.123 Td [(de)]TJ /F8 1 Tf 1.3193 0 Td [(a)]TJ /F2 1 Tf 0.8494 0 Td [(tel)-325(que)]TJ /F8 1 Tf 3.2771 0 Td [(f)]TJ /F2 1 Tf 0.92168 0 Td [(soit)-319(in)24(t)24(\266)475(egrable)-325(sur)]TJ /F8 1 Tf 8.1747 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.776 78.668 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.446 55.123 Td [(\\)]TJ /F8 1 Tf 0.86144 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -38.886 -1.1988 Td [(Comme)]TJ ET 107.57 654.33 m 118.44 654.33 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9 53.924 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-337(ferm)30(\266)469(e)-343(dans)]TJ ET 186.77 654.4 m 195.41 654.4 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 15.627 53.924 Td [(R)]TJ /F2 1 Tf 0.71687 0 Td [(,)-343(est)-337(compact)-343(on)-337(p)-30(eut)-343(recouvrir)]TJ /F8 1 Tf 14.151 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(par)-337(un)-337(nom)18(bre)-337(\257ni)-337(de)]TJ /F8 1 Tf 9.8855 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.405 76.952 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.693 53.924 Td [(.)-457(Le)]TJ -37.361 -1.1988 Td [(lemme)-379(pr)30(\266)469(ec)24(\266)469(eden)30(t)-379(assure)-373(alors)-379(que)]TJ /F8 1 Tf 15.181 0 Td [(f)]TJ /F2 1 Tf 0.9759 0 Td [(est)-379(in)24(t)24(\266)475(egrable)-379(sur)-379(la)-373(r)24(\266)475(eunion)-379(\(\257nie\))-379(des)-379(in)24(tersections)]TJ -16.157 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(a)24(v)24(ec)-337(ces)]TJ /F8 1 Tf 3.8133 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.603 73.512 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.783 51.521 Td [(,)-337(c'est-)0(\265)500(a-dire)-337(sur)]TJ /F8 1 Tf 7.4096 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ ET 512.64 614.62 m 519.84 614.62 l 519.84 623.05 l 512.64 623.05 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 49.551 Td [(Remarquons)-421(main)18(tenan)24(t)-427(que,)-451(p)-24(our)]TJ /F8 1 Tf 15.94 0 Td [(a)]TJ /F2 1 Tf 0.95181 0 Td [(\257ni)]TJ ET 265.61 590.61 m 282.24 590.61 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.036 49.551 Td [(dans)]TJ ET 316.3 602.13 m 327.17 602.13 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.464 49.551 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-445(si)]TJ /F8 1 Tf 1.8193 0 Td [(f)]TJ /F2 1 Tf 1.0181 0 Td [(est)-433(b)-24(orn)24(\266)469(ee)-427(au)-421(v)24(oisinage)-433(de)]TJ /F8 1 Tf 12.277 0 Td [(a)]TJ /F2 1 Tf 0.95181 0 Td [(on)]TJ -38.108 -1.1988 Td [(p)-30(eut)-319(trouv)24(er)-319(un)-319(v)24(oisinage)]TJ /F8 1 Tf 11.47 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.879 68.978 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.181 48.352 Td [(de)]TJ /F8 1 Tf 1.3193 0 Td [(a)]TJ /F2 1 Tf 0.84337 0 Td [(et)-313(une)-319(ma)-60(joration)]TJ /F6 1 Tf 8.1446 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.56024 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F2 1 Tf 1.3916 0 Td [(v)48(alable)-319(p)-24(our)]TJ /F8 1 Tf 5.6928 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.802 68.978 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.458 48.352 Td [(\\)]TJ /F8 1 Tf 0.8494 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -38.886 -1.1988 Td [(Quitte)-289(\265)500(a)-289(restreindre)]TJ /F8 1 Tf 8.8795 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.172 67.262 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.566 47.153 Td [(on)-289(p)-30(eut)-289(supp)-30(oser)-295(qu'il)-289(s'agit)-289(d'un)-295(in)30(terv)54(alle)-289(b)-30(orn)24(\266)475(e)-295(de)-289(cen)24(tre)]TJ /F8 1 Tf 26.054 0 Td [(a)]TJ /F2 1 Tf 0.81927 0 Td [(donc)]TJ -37.108 -1.6566 Td [(que)]TJ /F8 1 Tf 1.8012 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.043 64.892 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3012 45.497 Td [(\\)]TJ /F8 1 Tf 0.76506 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(est)-277(de)-271(mesure)-277(\257nie.)-427(On)-277(a)-271(alors)]TJ /F9 1 Tf 13.361 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.578 63.797 Td [(V)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 49.277 88.969 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.879 63.797 Td [(\\)]TJ /F7 1 Tf 0.77586 0 Td [(X)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.373 45.497 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dx)]TJ /F11 1 Tf 1.3735 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F6 1 Tf 1.1807 0 Td [(\242)]TJ /F2 1 Tf 0.37952 0 Td [(\()0(mesure)-343(de)]TJ /F8 1 Tf 5.1205 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.431 64.892 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.319 45.497 Td [(\\)]TJ /F8 1 Tf 0.76506 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(<)]TJ /F2 1 Tf 1.0542 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf -38.163 -1.8072 Td [(d'o)-30(\265)530(u)-337(l'in)24(t)24(\266)475(egrabilit)24(\266)469(e)-337(de)]TJ /F8 1 Tf 9.6988 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(au)-331(v)18(oisinage)-337(de)]TJ /F8 1 Tf 7.006 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)]TJ -18.169 -1.5241 Td [(Ainsi,)-506(la)-463(fonction)]TJ /F8 1 Tf 8.2771 0 Td [(f)]TJ /F2 1 Tf 1.0602 0 Td [(est)-475(in)24(t)24(\266)475(egrable)-469(au)-469(v)24(oisinage)-469(de)-469(tous)-469(les)-469(p)-30(oin)24(ts)-463(\257nis)-469(de)]TJ ET 469.8 513.79 m 480.67 513.79 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.307 42.165 Td [(X)]TJ /F2 1 Tf 1.3735 0 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.5241 0 Td [(f)]TJ /F2 1 Tf 1.0602 0 Td [(est)]TJ -37.934 -1.1988 Td [(lo)-30(calemen)30(t)-343(b)-30(orn)30(\266)469(ee.)-463(En)-337(particulier,)-343(si)]TJ /F8 1 Tf 16.367 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-343(con)24(tin)24(ue,)]TJ /F8 1 Tf 5.8554 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-343(in)24(t)24(\266)475(egrable)-337(au)-343(v)24(oisinage)-343(de)-337(tous)]TJ -24.09 -1.2048 Td [(les)-337(p)-24(oin)24(ts)-337(de)]TJ /F8 1 Tf 5.7892 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -6.6988 -1.5241 Td [(Autremen)24(t)-451(dit)-457(les)-451(probl)24(\265)469(emes)-451(d'in)24(t)24(\266)469(egrabilit)30(\266)469(e)-451(se)-451(p)-30(osen)24(t)-451(aux)-457(p)-30(oin)24(ts)]TJ /F6 1 Tf 29.536 0 Td [(\2411)]TJ /F2 1 Tf 2.2289 0 Td [(et)-451(+)]TJ /F6 1 Tf 2.0663 0 Td [(1)]TJ /F2 1 Tf 1.4458 0 Td [(s'ils)-451(son)18(t)]TJ -35.277 -1.1988 Td [(adh)24(\266)475(eren)18(ts)-331(\265)500(a)]TJ /F8 1 Tf 5.3735 0 Td [(X)]TJ /F2 1 Tf 1.2349 0 Td [(ainsi)-337(qu'aux)-331(p)-30(oin)24(ts)-331(\257nis)-337(de)]TJ ET 287.06 452.59 m 297.94 452.59 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.018 37.039 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.3916 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(n'est)-331(pas)-343(lo)-24(calemen)24(t)-337(b)-30(orn)24(\266)475(ee.)]TJ -22.253 -1.5241 Td [(Lorsque)]TJ /F8 1 Tf 3.8133 0 Td [(f)]TJ /F2 1 Tf 0.9759 0 Td [(est)-379(une)-379(fonction)-379(con)24(tin)24(ue)-373(sur)-385(un)-379(in)30(terv)54(alle,)-391(le)-379(probl)24(\265)469(eme)-373(de)-379(l'in)24(t)24(\266)475(egrabilit)24(\266)475(e)-385(de)]TJ /F8 1 Tf 33.777 0 Td [(f)]TJ /F2 1 Tf -38.566 -1.1988 Td [(ne)-331(se)-337(p)-30(ose)-331(donc)-337(qu'aux)-337(extr)30(\266)469(emit)24(\266)469(es)-331(de)-337(l'in)24(terv)54(alle)-331(o)-30(\265)530(u)]TJ /F8 1 Tf 22.922 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(n'est)-337(pas)-337(d)24(\266)469(e\257nie.)]TJ -23.849 -1.5241 Td [(Plus)-355(g)30(\266)469(en)30(\266)469(eralemen)30(t)-355(on)-349(dira)-349(qu'une)-355(fonction)-349(d)24(\266)469(e\257nie)-349(sur)]TJ /F8 1 Tf 23.855 0 Td [(X)]TJ /F2 1 Tf 1.259 0 Td [(est)]TJ /F1 1 Tf 1.5783 0 Td [(lo)-36(calemen)36(t)-403(in)30(t)24(\266)548(egrable)]TJ /F2 1 Tf 11.127 0 Td [(sur)]TJ /F8 1 Tf -37.819 -1.1988 Td [(X)]TJ /F2 1 Tf 1.2771 0 Td [(lorsque)-367(elle)-373(est)-373(in)24(t)24(\266)469(egrable)-367(au)-367(v)24(oisinage)-379(de)-367(tout)-367(p)-30(oin)24(t)-373(de)]TJ /F8 1 Tf 24.88 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)-548(C'est)-373(en)-367(particulier)-373(le)-373(cas)]TJ -27.066 -1.2048 Td [(p)-30(our)-271(les)-271(fonctions)-271(con)24(tin)24(ues.)-427(La)-277(prop)-30(osition)-271(pr)24(\266)469(ec)24(\266)475(eden)24(te)-271(mon)24(tre)-271(qu'une)-271(fonction)-277(lo)-30(calemen)24(t)]TJ 0 -1.1988 Td [(in)24(t)24(\266)475(egrable)-228(sur)]TJ /F8 1 Tf 6.1024 0 Td [(X)]TJ /F2 1 Tf 1.1265 0 Td [(est)-234(in)30(t)24(\266)469(egrable)-222(sur)-228(toute)-228(partie)-228(compacte,)-253(et)-228(en)-222(particulier)-222(sur)-228(tout)-228(segmen)24(t,)]TJ -7.2289 -1.1988 Td [(con)30(ten)24(ue)-337(dans)]TJ /F8 1 Tf 6.506 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 20.952 Td [(VI.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 25.117 Td [(2.)-289(Comparaison)-373(\265)560(a)-385(des)-391(fonctions)-385(tests.)]TJ /F2 1 Tf -2.9458 -1.5241 Td [(L'id)30(\266)469(ee)-337(de)-337(base)-337(de)-331(cette)-337(comparaison)-337(est)-337(:)]TJ /F12 1 Tf 0 -1.5241 Td [(si)]TJ /F8 1 Tf 1.1205 0 Td [(f)]TJ /F12 1 Tf 1.0482 0 Td [(et)]TJ /F8 1 Tf 1.2831 0 Td [(g)]TJ /F12 1 Tf 0.95783 0 Td [(son)18(t)-451(deux)-445(fonctions)-451(mesurables)-457(sur)-451(une)-451(partie)]TJ /F8 1 Tf 21.217 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-481(telles)-451(que)]TJ /F6 1 Tf 5.4096 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.74699 0 Td [(6)]TJ /F6 1 Tf 1.247 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F6 1 Tf 0.51807 0 Td [(j)]TJ /F12 1 Tf 0.72289 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.3133 0 Td [(g)]TJ /F12 1 Tf -38.651 -1.1988 Td [(in)24(t)24(\266)475(egrable)-337(alors)]TJ /F8 1 Tf 7.0422 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-343(in)30(t)24(\266)469(egrable.)]TJ /F2 1 Tf -7.9699 -1.9819 Td [(En)-331(e\256et)-337(on)-337(a)-331(alors)]TJ /F9 1 Tf 8.3855 1.3675 Td [(Z)]TJ /F6 1 Tf 1 -1.3675 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dx)]TJ /F11 1 Tf 1.3735 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3675 Td [(Z)]TJ /F6 1 Tf 1 -1.3675 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dx)-283(<)]TJ /F2 1 Tf 2.4277 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(.)]TJ -22.277 -1.9759 Td [(Ainsi,)-301(p)-30(our)-289(mon)24(trer)-289(qu'une)-295(fonction)-289(mesurable)-289(est)-295(in)24(t)24(\266)469(egrable,)-295(il)-289(su\261t)-295(de)-289(ma)-60(jorer)-289(sa)-289(v)48(aleur)]TJ 0 -1.1988 Td [(absolue)-313(par)-313(celle)-313(d'une)-319(fonction)-313(que)-319(l'on)-313(sait)-283(^)469(etre)-313(in)24(t)24(\266)475(egrable.)-439(P)24(our)-313(mon)24(trer)-313(qu'elle)-319(est)-319(non)]TJ 0 -1.2048 Td [(in)24(t)24(\266)475(egrable)-403(il)-397(su\261t)-409(de)-397(minorer)-403(sa)-403(v)48(aleur)-397(absolue)-403(par)-403(celle)-403(d'une)-403(fonction)-403(que)-403(l'on)-403(sait)-403(ne)]TJ 0 -1.1988 Td [(pas)-307(^)469(etre)-337(in)30(t)24(\266)469(egrable.)-445(Mieux,)-337(on)-331(p)-30(eut)-313(\266)475(enoncer)-337(:)]TJ /F15 1 Tf 0 -1.747 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.0422 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0843 0 Td [(f)]TJ /F12 1 Tf 0.98795 0 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(g)]TJ /F12 1 Tf 0.90361 0 Td [(des)-391(fonctions)-391(mesurables)-391(d)24(\266)469(e\257nies)-391(sur)-391(une)-391(partie)-397(mesurable)]TJ /F8 1 Tf -13.241 -1.1988 Td [(X)]TJ /F12 1 Tf 1.2952 0 Td [(de)]TJ /F3 1 Tf 1.3855 0 Td [(R)]TJ /F12 1 Tf 1.1084 0 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(a)]TJ /F12 1 Tf 0.91566 0 Td [(un)-385(p)-30(oin)24(t)-391(de)]TJ ET 200.88 133.77 m 209.52 133.77 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 16.807 10.364 Td [(R)]TJ /F12 1 Tf 1.1084 0 Td [(adh)24(\266)469(eren)24(t)-391(\265)500(a)]TJ /F8 1 Tf 5.0843 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)-602(Si,)-403(lorsque)]TJ /F8 1 Tf 5.8735 0 Td [(x)]TJ /F12 1 Tf 0.95783 0 Td [(tend)-391(v)24(ers)]TJ /F8 1 Tf 4.4458 0 Td [(a)]TJ /F12 1 Tf 0.91566 0 Td [(en)-391(restan)30(t)-391(dans)]TJ /F8 1 Tf 7.2048 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)]TJ /F6 1 Tf -38.886 -1.2048 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F12 1 Tf 0.63855 0 Td [(est)-361(du)-355(m)24(^)469(eme)-355(ordre)-361(que)]TJ /F6 1 Tf 10.494 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F12 1 Tf 0.63253 0 Td [(\(i.e.)]TJ /F6 1 Tf 2.1867 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F12 1 Tf 0.60843 0 Td [(\266)475(equiv)48(alen)30(t)-361(\265)499(a)]TJ /F8 1 Tf 5.6626 0 Td [(\270)]TJ /F6 1 Tf 0.57832 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F12 1 Tf 0.63253 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.2229 0 Td [(\270)-319(>)]TJ /F2 1 Tf 1.994 0 Td [(0)]TJ /F12 1 Tf 0.5 0 Td [(\))-361(alors)-361(les)]TJ -34.88 -1.1988 Td [(in)24(t)24(\266)475(egrales)-313(de)]TJ /F8 1 Tf 5.7651 0 Td [(f)]TJ /F12 1 Tf 0.90964 0 Td [(et)-313(de)]TJ /F8 1 Tf 2.4518 0 Td [(g)]TJ /F12 1 Tf 0.8253 0 Td [(son)24(t)-313(du)-313(m)24(^)475(eme)-313(t)24(yp)-30(e)-313(au)-313(p)-30(oin)24(t)]TJ /F8 1 Tf 12.59 0 Td [(a)]TJ /F12 1 Tf 0.83735 0 Td [(\(i.e.)-439(toutes)-313(les)-313(deux)-313(con)30(v)24(ergen)18(tes)-313(ou)]TJ -23.38 -1.1988 Td [(toutes)-337(les)-337(deux)-331(div)24(ergen)24(tes\).)]TJ /F5 1 Tf 0 -2.8735 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(86)]TJ ET endstream endobj 294 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 293 0 R /ProcSet 2 0 R >> /Contents 292 0 R >> endobj 293 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 295 0 obj << /Length 17032 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ 0 -2.4036 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(En)-433(prenan)24(t)]TJ /F8 1 Tf 5.4639 0 Td [(\256)]TJ /F2 1 Tf 1.0783 0 Td [(et)]TJ /F8 1 Tf 1.2651 0 Td [(\257)]TJ /F2 1 Tf 1.0542 0 Td [(tels)-433(que)-433(0)]TJ /F8 1 Tf 4.8434 0 Td [(<)-439(\256)-451(<)-445(\270)-445(<)-445(\257)]TJ /F2 1 Tf 6.8253 0 Td [(on)-433(sait)-433(qu'il)-439(existe)]TJ /F8 1 Tf 8.747 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.983 88.823 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 61.021 Td [(v)24(oisinage)-349(de)]TJ /F8 1 Tf 5.6325 0 Td [(a)]TJ /F2 1 Tf 0.87349 0 Td [(tel)-343(que,)-349(sur)]TJ /F8 1 Tf 5.2892 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.345 87.107 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.422 61.021 Td [(\\)]TJ /F8 1 Tf 0.89759 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-343(on)-349(ait)]TJ /F8 1 Tf 3.5301 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(j)]TJ /F11 1 Tf 0.57229 0 Td [(6)]TJ /F6 1 Tf 1.0663 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(j)]TJ /F11 1 Tf 0.56627 0 Td [(6)]TJ /F8 1 Tf 1.0723 0 Td [(\257)]TJ /F6 1 Tf 0.62048 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-469(Alors)-349(la)-337(premi)18(\265)475(ere)-349(ma)-54(joration)]TJ -25.699 -1.1988 Td [(mon)24(tre)-391(que)-385(si)-391(l'in)24(t)24(\266)469(egrale)-391(de)]TJ /F8 1 Tf 12.53 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(con)30(v)24(erge)-391(au)-391(p)-30(oin)24(t)]TJ /F8 1 Tf 8.247 0 Td [(a)]TJ /F2 1 Tf 0.91566 0 Td [(il)-391(en)-385(v)48(a)-385(de)-385(m)24(^)469(eme)-385(p)-30(our)-385(celle)-391(de)]TJ /F8 1 Tf 14.108 0 Td [(\256)0(g)]TJ /F2 1 Tf 1.5422 0 Td [(et)]TJ -38.331 -1.2048 Td [(donc)-319(de)]TJ /F8 1 Tf 3.6988 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(.)-439(La)-325(deuxi)30(\265)469(eme)-319(ma)-60(joration)-325(mon)24(tre)-325(que)-325(si)-319(l'in)24(t)24(\266)469(egrale)-325(de)]TJ /F8 1 Tf 23.898 0 Td [(f)]TJ /F2 1 Tf 0.92169 0 Td [(div)30(erge)-325(au)-319(p)-30(oin)24(t)]TJ /F8 1 Tf 7.4096 0 Td [(a)]TJ /F2 1 Tf 0.8494 0 Td [(il)-319(en)]TJ -37.289 -1.1988 Td [(v)54(a)-337(de)-331(m)24(^)475(eme)-337(p)-30(our)-331(celle)-337(de)]TJ /F8 1 Tf 11.452 0 Td [(\257)-60(g)]TJ /F2 1 Tf 1.4639 0 Td [(et)-337(donc)-337(de)]TJ /F8 1 Tf 4.8916 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(.)]TJ ET 512.64 685.04 m 519.84 685.04 l 519.84 693.46 l 512.64 693.46 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 55.039 Td [(P)30(our)-445(appliquer)-451(cette)-445(m)24(\266)469(etho)-30(de)-445(de)-445(comparaison)-451(on)-445(doit)-445(donc)-445(disp)-30(oser)-445(de)-445(fonctions)-445(tests)]TJ 0 -1.1988 Td [(p)-30(our)-325(lesquelles)-325(on)-325(conna)108(^)385(\263t)-325(la)-325(nature)-325(de)-325(l'in)24(t)24(\266)475(egrale.)-445(On)-325(utilise)-325(essen)24(tiellemen)24(t)-325(p)-30(our)-325(cela)-331(les)]TJ 0 -1.6205 Td [(fonctions)]TJ /F8 1 Tf 4.259 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(!)]TJ /F2 1 Tf 2.8675 0.68072 Td [(1)]TJ ET 0.504 w 141.48 627.12 m 182.74 627.12 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.837 51.533 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.06 74.159 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.741 52.22 Td [(p)-24(our)-337(lesquelles)-337(on)-337(a)-331(:)]TJ /F15 1 Tf -10.41 -3.1928 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(L'in)24(t)24(\266)475(egrale)-343(de)]TJ /F2 1 Tf 7.9217 0.68072 Td [(1)]TJ ET 222.84 589.03 m 264.1 589.03 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.645 48.34 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.802 69.59 Td [(\256)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 22.217 49.027 Td [(con)30(v)24(erge)-337(en)]TJ /F6 1 Tf 5.4217 0 Td [(\2471)]TJ /F12 1 Tf 2.1084 0 Td [(si)-337(et)-337(seulemen)30(t)-337(si)]TJ /F8 1 Tf 7.8253 0 Td [(\256)-283(>)]TJ /F2 1 Tf 1.9819 0 Td [(1)]TJ /F12 1 Tf 0.5 0 Td [(.)]TJ -34.723 -2.2169 Td [(Elle)-331(con)24(v)24(erge)-337(au)-337(p)-24(oin)24(t)]TJ /F8 1 Tf 10.102 0 Td [(a)]TJ /F12 1 Tf 0.86144 0 Td [(si)-337(et)-337(seulemen)30(t)-337(si)]TJ /F8 1 Tf 7.8253 0 Td [(\256)-283(<)]TJ /F2 1 Tf 1.9819 0 Td [(1)]TJ /F12 1 Tf 0.5 0 Td [(.)]TJ /F1 1 Tf -21.271 -2.5602 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(On)-259(a)-259(une)-259(primitiv)30(e)]TJ /F8 1 Tf 8.3675 0 Td [(F)]TJ /F2 1 Tf 1.0361 0 Td [(de)]TJ 2.8554 0.68073 Td [(1)]TJ ET 298.01 531.86 m 339.26 531.86 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.934 43.563 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.802 62.754 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.759 44.25 Td [(en)-259(p)-30(osan)24(t)]TJ /F8 1 Tf 4.4096 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)]TJ /F8 1 Tf 3.8675 0.68073 Td [(")]TJ /F2 1 Tf 0.46385 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ ET 439.34 531.86 m 530.42 531.86 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.759 43.563 Td [(\(1)]TJ /F6 1 Tf 1.1084 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F6 1 Tf 0.52409 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.224 62.754 Td [(\256)]TJ /F10 1 Tf 0.74138 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 42.467 Td [(si)]TJ /F8 1 Tf 0.99398 0 Td [(\256)]TJ /F6 1 Tf 0.92169 0 Td [(6)]TJ /F2 1 Tf [(=)-277(1)-325(et)]TJ /F8 1 Tf 3.0361 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.77108 0 Td [(\241)]TJ /F8 1 Tf 0.98193 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))-174(ln)]TJ /F6 1 Tf 1.5602 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.77108 0 Td [(\241)]TJ /F8 1 Tf 0.9759 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F2 1 Tf 0.60241 0 Td [(si)]TJ /F8 1 Tf 0.99397 0 Td [(\256)]TJ /F2 1 Tf 0.92169 0 Td [(=)-277(1,)-325(o)-30(\265)530(u)]TJ /F8 1 Tf 3.5361 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-325(v)48(aut)-325(1)-325(p)-24(our)]TJ /F8 1 Tf 6.1265 0 Td [(t)-283(>)]TJ /F2 1 Tf 1.6988 0 Td [(0)-319(et)]TJ /F6 1 Tf 1.9759 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(1)-325(p)-30(our)]TJ /F8 1 Tf 3.1807 0 Td [(t)-283(<)]TJ /F2 1 Tf 1.6928 0 Td [(0.)]TJ -38.386 -1.6928 Td [(S'agissan)24(t)-481(de)-481(l'in)24(t)24(\266)475(egrale)-487(d'une)-481(fonction)-487(p)-24(ositiv)24(e,)-518(sa)-487(con)30(v)24(ergence)-481(en)]TJ /F6 1 Tf 30.639 0 Td [(\2471)]TJ /F2 1 Tf 2.2289 0 Td [(\266)469(equiv)54(aut)-481(alors)]TJ -32.867 -1.1988 Td [(clairemen)24(t)-427(\265)499(a)-433(l'existence)-433(d'une)-433(limite)-427(\257nie)-433(en)]TJ /F6 1 Tf 20.241 0 Td [(\2471)]TJ /F2 1 Tf 2.2108 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.4578 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(.)-734(De)-433(m)30(^)469(eme)-427(la)-433(con)30(v)24(ergence)-433(de)]TJ -25.693 -1.2048 Td [(l'in)24(t)24(\266)475(egrale)-313(au)-301(p)-30(oin)24(t)]TJ /F8 1 Tf 8.5542 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(\266)475(equiv)48(aut)-307(\265)500(a)-301(l'existence)-307(d'une)-313(limite)-301(\257nie)-307(\265)500(a)-307(droite)-307(et)-307(\265)500(a)-301(gauc)24(he)-301(au)-307(p)-30(oin)24(t)]TJ /F8 1 Tf -9.3614 -1.1988 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)-439(Le)-337(r)24(\266)469(esultat)-337(annonc)24(\266)469(e)-337(d)30(\266)469(ecoule)-337(donc)-337(de)-331(l')24(\266)469(etude)-337(classique)-337(de)-331(la)-337(fonction)]TJ /F8 1 Tf 30.711 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(.)]TJ ET 512.64 443.12 m 519.84 443.12 l 519.84 451.54 l 512.64 451.54 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 34.376 Td [(On)-337(p)-30(eut)-343(donc)-343(dire)-337(que)-343(si)]TJ /F8 1 Tf 11.265 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-343(tend)-343(v)24(ers)-337(0)-343(\265)500(a)-337(l'in\257ni)-343(plus)-337(vite)-337(que)]TJ 16.566 0.6747 Td [(1)]TJ ET 407.81 413.85 m 428.11 413.85 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.12 33.689 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.57229 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.44 48.625 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.277 34.376 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.2048 0 Td [(\256)-289(>)]TJ /F2 1 Tf 1.994 0 Td [(1)-337(alors)-343(la)]TJ -35.145 -1.7289 Td [(fonction)]TJ /F8 1 Tf 3.8614 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)475(egrable)-337(\265)500(a)-331(l'in\257ni.)]TJ -4.7952 -2.1084 Td [(De)-415(m)30(^)469(eme)-409(si,)-439(lorsque)]TJ /F8 1 Tf 9.4699 0 Td [(x)]TJ /F2 1 Tf 0.98193 0 Td [(tend)-421(v)24(ers)]TJ /F8 1 Tf 4.5 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)]TJ /F8 1 Tf 0.71084 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-415(tend)-415(v)24(ers)-415(l'in\257ni)-409(moins)-415(vite)-409(que)]TJ 17.048 0.6747 Td [(1)]TJ ET 461.95 367.99 m 503.21 367.99 l S BT /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.651 29.852 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.431 43.133 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.633 30.539 Td [(a)24(v)24(ec)]TJ /F8 1 Tf -37.301 -1.7289 Td [(\256)-283(<)]TJ /F2 1 Tf 1.9759 0 Td [(1)-331(alors)-343(la)-331(fonction)]TJ /F8 1 Tf 8.2048 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(au)-337(p)-30(oin)30(t)]TJ /F8 1 Tf 10.205 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)]TJ /F1 1 Tf -21.849 -2.3795 Td [(A)30(tten)30(tion)]TJ /F2 1 Tf 4.8494 0 Td [(,)-307(les)-301(conditions)-301(pr)24(\266)469(ec)24(\266)475(eden)24(ts)-301(son)24(t)-307(seulemen)30(t)-307(su\261san)24(tes.)-433(Elles)-307(ne)-301(son)24(t)-301(pas)-307(du)-307(tout)]TJ -4.8494 -1.2048 Td [(n)30(\266)469(ecessaires.)-825(En)-463(particulier)-463(une)-463(fonction)-463(p)-30(eut)-439(^)475(etre)-463(in)24(t)24(\266)475(egrable)-463(en)-463(+)]TJ /F6 1 Tf 30.042 0 Td [(1)]TJ /F2 1 Tf 1.4578 0 Td [(sans)-463(p)-30(our)-463(autan)24(t)]TJ -31.5 -1.1988 Td [(tendre)-457(v)24(ers)-451(0)-457(en)-451(ce)-457(p)-30(oin)24(t)-457(comme)-451(le)-457(mon)18(tre)-451(l'exemple)-457(de)-451(la)-451(fonction)-463(caract)30(\266)469(eristique)-457(de)]TJ 0 -1.1988 Td [(l'ensem)30(ble)]TJ /F8 1 Tf 4.8494 0 Td [(A)]TJ /F2 1 Tf 1.1205 0 Td [(r)30(\266)469(eunion)-373(des)-373(segmen)24(ts)-373([)]TJ /F8 1 Tf 9.994 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(n)]TJ /F2 1 Tf 0.84337 0 Td [(+)-246(2)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.362 33.254 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.886 22.828 Td [(])-367(p)-30(our)]TJ /F8 1 Tf 3.0482 0 Td [(n)]TJ /F6 1 Tf 0.93976 0 Td [(2)]TJ /F3 1 Tf 1.006 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)-554(En)-367(e\256et)-373(on)-373(v)54(\266)469(eri\257e)-367(ais)24(\266)469(emen)24(t)]TJ /F9 1 Tf -26.271 -0.3494 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.8621 30.004 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.5422 21.117 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)-283(mesure)-337(de)]TJ /F8 1 Tf 5.7892 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F2 1 Tf 1.6145 -0.94578 Td [(2)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.216 30.806 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.861 21.117 Td [(=)-283(2.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 13.294 Td [(VI.4.)-236(Lien)-381(a)25(v)30(ec)-381(l'in)30(t)25(\266)537(egrale)-376(de)-381(Riemann.)]TJ 0 -2.6332 Td [(VI.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 12.78 Td [(1.)-289(Rapp)-36(el)-385(de)-385(la)-391(th)24(\266)548(eorie)-385(de)-385(Riemann.)]TJ /F2 1 Tf -2.9458 -1.6928 Td [(Rapp)-30(elons)-415(que)-415(la)-409(th)24(\266)469(eorie)-415(de)-409(l'in)24(t)24(\266)475(egrale)-415(de)-409(Riemann)-415(concerne)-409(des)-415(fonctions)]TJ /F8 1 Tf 33.681 0 Td [(f)]TJ /F2 1 Tf 1.0121 0 Td [(d)30(\266)469(e\257nies)-415(et)]TJ -34.693 -1.1988 Td [(b)-30(orn)30(\266)469(ees)-427(sur)-433(un)-433(segmen)24(t)-433([)]TJ /F8 1 Tf 11.295 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(].)-728(On)-433(app)-30(elle)-433(alors)]TJ /F1 1 Tf 9.0542 0 Td [(sub)-36(division)]TJ /F2 1 Tf 6.0663 0 Td [(de)-427([)]TJ /F8 1 Tf 1.7048 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])-427(toute)-433(partie)-433(\257nie)]TJ /F8 1 Tf -30.928 -1.1988 Td [(\276)]TJ /F2 1 Tf 0.99398 0 Td [(de)-385([)]TJ /F8 1 Tf 1.6627 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])-391(qui)-385(p)-30(oss)30(\265)469(ede)]TJ /F8 1 Tf 6.1205 0 Td [(a)]TJ /F2 1 Tf 0.91566 0 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(.)-608(On)-361(\266)469(en)30(um)24(\265)469(ere)-391(en)-385(g)24(\266)475(en)24(\266)475(eral)-391(cette)-391(partie)-391(dans)-385(l'ordre)-391(croissan)24(t)]TJ /F8 1 Tf -12.747 -1.1988 Td [(\276)]TJ /F2 1 Tf 0.89157 0 Td [(=)]TJ /F6 1 Tf 1.0663 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(\276)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 11.966 10.504 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8133 7.4906 Td [(;)-162(\276)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.06 10.504 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.277 7.4906 Td [(;)-168(:)-162(:)-168(:)-168(;)-168(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.707 10.504 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.614 7.4906 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(,)-343(a)30(v)24(ec)]TJ /F8 1 Tf 2.8193 0 Td [(\276)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 25.052 10.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.241 7.4906 Td [(=)]TJ /F8 1 Tf 1.0663 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)]TJ /F8 1 Tf 0.62048 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.086 10.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.861 7.4906 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(,)-337(on)-343(p)-30(ose,)-343(p)-30(our)-337(1)]TJ /F11 1 Tf 7.7108 0 Td [(6)]TJ /F8 1 Tf 1.0663 0 Td [(k)]TJ /F11 1 Tf 0.83735 0 Td [(6)]TJ /F8 1 Tf 1.0663 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.086 10.504 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.482 7.4906 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.543 10.504 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.837 7.4906 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.828 10.504 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(\241)]TJ /F4 1 Tf 0.88794 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.265 7.4906 Td [(et)-343(on)]TJ -36.934 -1.2048 Td [(d)30(\266)469(e\257nit)-331(le)]TJ /F1 1 Tf 4.1687 0 Td [(pas)]TJ /F2 1 Tf 1.9819 0 Td [(de)]TJ /F8 1 Tf 1.3374 0 Td [(\276)]TJ /F2 1 Tf 0.93976 0 Td [(comme)-337(le)-337(maxim)24(um)-337(des)]TJ /F8 1 Tf 10.843 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.845 8.7797 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.542 6.2858 Td [(.)]TJ -1.2952 -2.3976 Td [(87)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 297 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 296 0 R /ProcSet 2 0 R >> /Contents 295 0 R >> endobj 296 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 298 0 obj << /Length 23615 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F2 1 Tf 0 -2.4036 Td [(A)-331(une)-331(sub)-30(division)]TJ /F8 1 Tf 8.1928 0 Td [(\276)]TJ /F2 1 Tf 0.93976 0 Td [(on)-331(asso)-24(cie)-337(la)]TJ /F1 1 Tf 5.8072 0 Td [(somme)-379(de)-385(Darb)-30(oux)-379(inf)18(\266)548(erieure)]TJ /F8 1 Tf 15.307 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))-337(et)-331(la)]TJ /F1 1 Tf 2.9879 0 Td [(somme)]TJ -35.687 -1.1988 Td [(de)-385(Darb)-30(oux)-391(sup)-6(\266)548(erieure)]TJ /F8 1 Tf 11.928 0 Td [(S)]TJ /F2 1 Tf 0.66868 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))-337(d)24(\266)469(e\257nies)-337(par)-337(:)]TJ /F8 1 Tf -7.8735 -1.4036 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))-283(=)]TJ /F9 1 Tf 2.2952 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 23.198 83.599 Td [(1)]TJ /F17 1 Tf 0.56896 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.88793 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.964 59.617 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.776 85.099 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.904 59.617 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.741 85.099 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.982 59.617 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))-283(=)]TJ /F9 1 Tf 2.2952 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 46.302 83.599 Td [(1)]TJ /F17 1 Tf 0.57759 0 Td [(6)]TJ /F7 1 Tf 0.88793 0 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.108 59.617 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.879 85.099 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.048 59.617 Td [(M)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.974 85.099 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.512 59.617 Td [(,)]TJ -32.181 -2.6265 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3614 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.836 81.349 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3735 56.991 Td [(\(resp.)]TJ /F8 1 Tf 2.8916 0 Td [(M)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.509 81.349 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.729 56.991 Td [(\))-313(est)-319(la)-307(b)-30(orne)-313(inf)30(\266)469(erieure)-313(\(resp.)-439(sup)0(\266)469(erieure\))-313(de)]TJ /F8 1 Tf 19.898 0 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(sur)-313(le)-313(segmen)18(t)-313([)]TJ /F8 1 Tf 6.8072 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.552 81.349 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.428 56.991 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.172 81.349 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.94 56.991 Td [(].)]TJ -38.608 -1.5361 Td [(On)-403(note)-409(en\257n)]TJ /F8 1 Tf 6.5422 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))-403(\(resp.)]TJ /F8 1 Tf 3.8976 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\)\))-403(la)-403(b)-30(orne)-403(sup)0(\266)475(erieure)-409(\(resp.)-650(inf)24(\266)469(erieure\))-403(des)]TJ /F8 1 Tf 19.91 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))-409(\(resp.)]TJ -35.916 -1.1988 Td [(des)]TJ /F8 1 Tf 1.7289 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\)\))-337(lorsque)]TJ /F8 1 Tf 4.5422 0 Td [(\276)]TJ /F2 1 Tf 0.93976 0 Td [(parcourt)-331(l'ensem)24(ble)-331(des)-337(sub)-30(divisions)-337(de)-331([)]TJ /F8 1 Tf 17.849 0 Td [(a)]TJ /F2 1 Tf 0.52409 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(].)]TJ -29.108 -1.5422 Td [(On)-373(a)-337(\266)469(evidemmen)24(t)]TJ /F8 1 Tf 8.2229 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)12(;)-168(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))]TJ /F11 1 Tf 0.72892 0 Td [(6)]TJ /F8 1 Tf 1.1145 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)12(;)-162(f)]TJ /F2 1 Tf 1.5964 0 Td [(\).)-554(P)24(ar)-367(ailleurs)-373(il)-367(est)-379(facile)-373(de)-367(v)24(oir)-373(que)-373(le)-373(fait)-373(d'a)-60(jouter)]TJ -15.175 -1.1988 Td [(un)-301(p)-30(oin)24(t)-301(\265)499(a)-301(une)-295(sub)-30(division)-301(augmen)24(te)-301(la)-301(somme)-301(inf)30(\266)469(erieure)-307(et)-301(dimin)30(ue)-301(la)-301(somme)-295(sup)0(\266)469(erieure.)]TJ 0 -1.1988 Td [(On)-355(en)-361(d)30(\266)469(eduit)-355(que,)-373(si)]TJ /F8 1 Tf 9.3855 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.922 72.59 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.958 50.316 Td [(et)]TJ /F8 1 Tf 1.1928 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.414 72.59 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.627 50.316 Td [(son)24(t)-361(des)-361(sub)-30(divisions)-361(de)-355([)]TJ /F8 1 Tf 11.211 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(],)-361(en)-361(p)-30(osan)24(t)]TJ /F8 1 Tf 5.5301 0 Td [(\276)]TJ /F2 1 Tf 0.92169 0 Td [(=)]TJ /F8 1 Tf 1.0964 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.379 72.59 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.916 50.316 Td [([)]TJ /F8 1 Tf 0.90362 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.284 72.59 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.94 50.316 Td [(,)-367(on)-355(a)]TJ /F8 1 Tf -36.608 -1.2048 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.7241 70.875 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0783 49.111 Td [(;)-162(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)12(;)-168(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.707 70.875 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.572 49.111 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\).)-451(On)-331(en)-337(tire)-337(ais)24(\266)469(emen)30(t)]TJ /F8 1 Tf 10.096 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(S)]TJ /F2 1 Tf 0.66868 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)]TJ -31.217 -1.5361 Td [(P)30(ar)-283(d)24(\266)469(e\257nition)-283(on)-289(dit)-289(alors)-289(que)]TJ /F8 1 Tf 13.235 0 Td [(f)]TJ /F2 1 Tf 0.88554 0 Td [(est)]TJ /F1 1 Tf 1.512 0 Td [(Riemann-in)30(t)18(\266)548(egrable)]TJ /F2 1 Tf 10.169 0 Td [(lorsque)]TJ /F8 1 Tf 3.3795 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\).)-427(La)-289(v)48(aleur)]TJ -34.018 -1.2048 Td [(comm)24(une)-331(de)]TJ /F8 1 Tf 5.8072 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))-337(et)]TJ /F8 1 Tf 1.8855 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-313(\266)475(etan)18(t)-331(l'in)18(t)30(\266)469(egrale)-337(de)]TJ /F8 1 Tf 9.2892 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(au)-337(sens)-331(de)-337(Riemann.)]TJ -21.024 -1.5361 Td [(On)-542(p)-30(eut)-542(in)24(terpr)24(\266)469(eter)-542(la)-542(construction)-542(pr)24(\266)469(ec)24(\266)475(eden)24(te)-542(a)30(v)18(ec)-542(le)-548(p)-30(oin)30(t)-548(de)-542(vue)-542(que)-548(nous)-542(a)30(v)24(ons)]TJ 0 -1.1988 Td [(d)30(\266)469(ev)24(elopp)0(\266)469(e)-337(jusqu'ici)-337(de)-337(la)-331(fa)-6(\273)439(con)-331(suiv)48(an)30(te.)]TJ 0 -1.5422 Td [(Etan)24(t)-228(donn)24(\266)469(ee)-222(une)-228(sub)-24(division)]TJ /F8 1 Tf 12.898 0 Td [(\276)]TJ /F2 1 Tf 0.60241 0 Td [(,)-246(notons)]TJ /F8 1 Tf 3.6446 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.793 60.021 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.633 42.093 Td [(l'in)24(terv)54(alle)-222([)]TJ /F8 1 Tf 5.0602 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.879 60.021 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.783 42.093 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.5 60.021 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.289 42.093 Td [([)-222(don)24(t)-228(la)-222(mesure)-228(de)-222(Leb)-36(esgue)]TJ -26.958 -1.1988 Td [(v)54(aut)]TJ /F8 1 Tf 2.259 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.5 58.306 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5301 40.894 Td [(.)-469(Sur)]TJ /F8 1 Tf 2.5964 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.552 58.306 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.054 40.894 Td [(,)]TJ /F8 1 Tf 0.6265 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(est)-349(donc)-343(encadr)24(\266)475(ee)-349(en)30(tre)]TJ /F8 1 Tf 10.639 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.966 58.306 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.97 40.894 Td [(et)]TJ /F8 1 Tf 1.1747 0 Td [(M)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.241 58.306 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.608 40.894 Td [(.)-475(P)30(ar)-343(suite,)-349(sur)-349([)]TJ /F8 1 Tf 7.2892 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [([,)]TJ /F8 1 Tf 0.89759 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(est)-349(encadr)24(\266)469(ee)]TJ -33.807 -1.4036 Td [(en)30(tre)-271(les)-271(fonctions)]TJ /F8 1 Tf 8.0361 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)12(;)-162(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.922 56.297 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.705 39.491 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.431 56.297 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 43.241 78.475 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.373 39.491 Td [(et)]TJ /F8 1 Tf 1.1024 0 Td [(\257)]TJ /F2 1 Tf 0.61446 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)12(;)-162(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(M)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.474 56.297 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.873 39.491 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.983 56.297 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 63.578 78.475 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.277 39.491 Td [(.)-421(Ces)-271(fonctions)-271(son)24(t)-240(\266)469(evidem-)]TJ -26.946 -1.3976 Td [(men)24(t)-337(mesurables)-337(et)-331(leur)-337(in)24(t)24(\266)475(egrale)-343(de)-331(Leb)-36(esgue)-331(v)48(aut)-337(resp)-30(ectiv)24(emen)30(t)]TJ /F8 1 Tf 29.669 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)12(;)-162(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))-331(et)]TJ /F8 1 Tf 1.8855 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)12(;)-168(f)]TJ /F2 1 Tf 1.5964 0 Td [(\).)]TJ -36.669 -1.5422 Td [(On)-391(p)-30(eut)-397(alors)-391(c)18(hoisir)-391(une)-391(suite)-391(de)-397(sub)-24(divisions)]TJ /F8 1 Tf 21.235 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.836 52.09 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.072 36.551 Td [(telle)-397(que)]TJ /F8 1 Tf 4.1446 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.155 52.09 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.193 36.551 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-397(tende)-391(v)24(ers)]TJ /F8 1 Tf 5.6867 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)-626(Si)]TJ -37.042 -1.1988 Td [(l'on)-349(note)]TJ /F8 1 Tf 4.1988 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.5 51.185 Td [(0)]TJ /F7 1 Tf -0.051724 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.994 35.352 Td [(la)-343(r)24(\266)469(eunion)-349(des)-349(sub)-30(divisions)]TJ /F8 1 Tf 12.139 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.922 50.375 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.518 35.352 Td [(p)-30(our)]TJ /F8 1 Tf 2.3795 0 Td [(p)]TJ /F11 1 Tf 0.8012 0 Td [(6)]TJ /F8 1 Tf 1.0843 0 Td [(n)]TJ /F2 1 Tf 0.94578 0 Td [(on)-349(obtien)24(t)-349(une)-349(suite)-349(croissan)24(te)-349(de)]TJ -24.398 -1.1988 Td [(sub)-24(divisions)-421(p)-30(our)-421(laquelle)]TJ /F8 1 Tf 11.886 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.69 48.659 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.193 34.153 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F8 1 Tf 1.1988 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.922 49.461 Td [(0)]TJ /F7 1 Tf -0.051727 -0.93966 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.217 34.153 Td [(;)-162(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-421(et,)-439(bien)-421(s)-30(^)530(ur,)]TJ /F8 1 Tf 6.6687 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.784 49.461 Td [(0)]TJ /F7 1 Tf -0.051727 -0.93966 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.904 34.153 Td [(;)-162(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F8 1 Tf 1.1928 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)-704(On)-415(a)-421(donc)]TJ /F8 1 Tf -33.072 -1.2048 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.7241 47.745 Td [(0)]TJ /F7 1 Tf -0.051724 -0.94828 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.3012 32.948 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(!)]TJ /F8 1 Tf 1.2771 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)]TJ -6.4157 -1.5361 Td [(P)30(ar)-228(ailleurs,)-253(la)-228(suite)-228(des)]TJ /F8 1 Tf 10.337 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.293 45.538 Td [(0)]TJ /F7 1 Tf -0.051725 -0.93966 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.018 31.412 Td [(est)-204(\266)475(evidemmen)24(t)-234(croissan)24(te)-228(\(p)-30(our)-234(l'inclusion\),)-253(on)-228(en)-228(tire)-228(ais)24(\266)469(emen)24(t)]TJ -11.687 -1.1988 Td [(que)-337(la)-337(suite)-331(des)-337(fonctions)]TJ /F8 1 Tf 11.355 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.233 43.823 Td [(0)]TJ /F7 1 Tf -0.051723 -0.94827 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.837 30.214 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-337(est)-343(croissan)30(te.)-451(Ces)-337(fonctions)-337(son)24(t)-337(ma)-60(jor)24(\266)475(ees)-337(par)]TJ /F8 1 Tf 21.566 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)-337(elles)]TJ -36.711 -1.2048 Td [(on)30(t)-367(donc)-361(une)-361(limite)-361(simple)]TJ /F8 1 Tf 12.114 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.828 42.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.687 29.009 Td [(qui)-361(est)-367(mesurable,)-367(minore)]TJ /F8 1 Tf 11.717 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(et)-367(qui,)-367(d'apr)24(\265)469(es)-361(le)-361(th)18(\266)475(eor)18(\265)475(eme)-361(de)]TJ -26.03 -1.6566 Td [(con)30(v)24(ergence)-259(monotone,)-271(v)54(\266)469(eri\257e)]TJ /F9 1 Tf 13.139 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.724 39.737 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.349 27.352 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-259(\(on)-253(remarquera)-253(que,)-271(si)]TJ /F8 1 Tf 10.572 0 Td [(M)]TJ /F2 1 Tf 1.3313 0 Td [(est)-253(un)-253(ma)-60(joran)24(t)]TJ -32.319 -1.7108 Td [(de)]TJ /F6 1 Tf 1.2892 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(,)-301(toutes)-295(ces)-289(fonctions)-295(son)24(t)-295(comprises)-289(en)24(tre)]TJ /F6 1 Tf 18.446 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(M)]TJ /F2 1 Tf 1.3675 0 Td [(et)]TJ /F8 1 Tf 1.1205 0 Td [(M)]TJ /F2 1 Tf 1.3735 0 Td [(donc)-289(leur)-295(in)30(t)24(\266)469(egrale)-295(de)-289(Leb)-36(esgue)]TJ -25.53 -1.1988 Td [(est)-337(comprise)-331(en)24(tre)]TJ /F6 1 Tf 8.2711 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F6 1 Tf 0.65663 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(M)]TJ /F2 1 Tf 1.4157 0 Td [(et)-331(\()]TJ /F8 1 Tf 1.5542 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(\).)]TJ -18.627 -1.5422 Td [(Il)-433(est)-439(alors)-439(facile)-439(de)-433(faire)-439(un)-439(raisonnemen)24(t)-433(analogue)-439(a)24(v)24(ec)-433(les)-439(sommes)-439(sup)0(\266)469(erieures)-433(et)-439(les)]TJ 0 -1.1988 Td [(fonctions)]TJ /F8 1 Tf 4.1867 0 Td [(\257)]TJ /F2 1 Tf 0.62048 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))-271(\(d'ailleurs)]TJ /F8 1 Tf 5.2651 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)12(;)-168(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))-283(=)]TJ /F6 1 Tf 1.7229 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)]TJ /F6 1 Tf 0.99398 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\)\).)-427(On)-271(obtien)30(t)-271(ainsi)-265(une)-271(fonction)-271(mesurable)]TJ /F8 1 Tf -20.44 -1.7108 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4828 29.193 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7952 19.991 Td [(qui)-355(ma)-60(jore)]TJ /F8 1 Tf 5.1024 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(et)-355(v)48(\266)475(eri\257e)]TJ /F9 1 Tf 4.1566 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.638 29.193 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.12 19.991 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4096 0 Td [(=)]TJ /F8 1 Tf 1.0904 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\).)-512(On)-361(v)24(oit)-361(donc)-355(que,)-367(lorsque)]TJ /F8 1 Tf 12.916 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(est)-361(Riemann-)]TJ -33.343 -1.7651 Td [(in)24(t)24(\266)475(egrable,)]TJ /F8 1 Tf 4.8434 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.414 26.676 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.313 18.226 Td [(\241)]TJ /F8 1 Tf 0.80723 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.198 26.676 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.235 18.226 Td [(est)-240(une)-234(fonction)-246(p)-24(ositiv)24(e)-240(d'in)24(t)24(\266)469(egrale)-240(n)24(ulle)-234(donc)-240(une)-234(fonction)-246(n)30(ulle)-240(presque)]TJ -7.9036 -1.1988 Td [(partout.)-433(P)30(ar)-283(suite)]TJ /F8 1 Tf 8.1566 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.155 24.952 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.651 17.027 Td [(et)]TJ /F8 1 Tf 1.1145 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.414 24.952 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.157 17.027 Td [(son)24(t)-259(\266)475(egales)-289(presque)-283(partout)-283(et)]TJ /F8 1 Tf 13.217 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(qui)-283(est)-283(comprise)-283(en)24(tre)-283(les)-283(deux)]TJ -25.922 -1.2048 Td [(est)-337(encore)-313(\266)469(egale)-337(presque)-337(partout)-337(\265)500(a)-337(ces)-337(fonctions.)-451(Ainsi)]TJ /F8 1 Tf 24.145 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(in)24(t)24(\266)475(egrable)-337(p)-30(our)-337(la)-337(mesure)-337(de)]TJ -25.078 -1.6566 Td [(Leb)-30(esgue)-337(et)-337(on)-331(a)]TJ /F9 1 Tf 7.7169 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.888 20.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.47 14.165 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\).)]TJ -20.536 -1.994 Td [(On)-337(p)-24(eut)-337(donc)-337(conclure)-337(:)]TJ /F12 1 Tf 0 -1.5361 Td [(une)-259(fonction)-259(Riemann-in)18(t)24(\266)475(egrable)-259(est)-265(in)30(t)24(\266)469(egrable)-259(p)-30(our)-259(la)-259(mesure)-259(de)-259(Leb)-30(esgue)-259(et)-259(son)-265(in)24(t)24(\266)475(egrale)]TJ 0 -1.2048 Td [(de)-331(Riemann)-337(co)108(\304)385(\263ncide)-331(a)24(v)24(ec)-337(son)-337(in)24(t)24(\266)475(egrale)-343(p)-30(our)-331(la)-337(mesure)-337(de)-331(Leb)-36(esgue.)]TJ /F2 1 Tf 0 -2.2289 Td [(On)-337(note)-337(encore)]TJ /F9 1 Tf 7.006 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.086 11.9 Td [(b)]TJ -0.63793 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.741 7.2015 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.4217 0 Td [(cette)-337(in)24(t)24(\266)469(egrale)-337(de)-331(Riemann.)]TJ /F5 1 Tf -11.946 -3.3133 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(88)]TJ ET endstream endobj 300 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 299 0 R /ProcSet 2 0 R >> /Contents 298 0 R >> endobj 299 0 obj << /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F17 97 0 R >> endobj 301 0 obj << /Length 25106 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(VI.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 62.22 Td [(2.)-289(Appro)30(ximation)-391(des)-391(in)36(t)18(\266)548(egrales.)]TJ /F2 1 Tf -2.9458 -1.8434 Td [(Un)]TJ /F1 1 Tf 1.8072 0 Td [(p)-30(oin)30(tage)]TJ /F2 1 Tf 4.7771 0 Td [(de)-506(la)-500(sub)-24(division)]TJ /F8 1 Tf 8.1807 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.578 86.185 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.663 60.376 Td [(est)-506(alors)-506(une)-499(famille)]TJ /F8 1 Tf 9.7651 0 Td [(\273)]TJ /F2 1 Tf 0.98796 0 Td [(de)-499(r)30(\266)469(eels)]TJ /F8 1 Tf 3.9518 0 Td [(\273)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.672 86.185 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.253 60.376 Td [(;)-168(:)-168(:)-168(:)-168(;)-162(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.121 86.185 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.958 60.376 Td [(tels)-500(que)]TJ /F8 1 Tf -35.627 -1.1988 Td [(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.25 84.469 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5361 59.177 Td [(2)]TJ /F2 1 Tf 0.94578 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.922 84.469 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.8494 59.177 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.552 84.469 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.355 59.177 Td [(].)-445(Un)-343(tel)-331(p)-30(oin)24(tage)-337(p)-30(ermet)-337(de)-331(d)24(\266)469(e\257nir)-337(la)]TJ /F1 1 Tf 17.127 0 Td [(somme)-385(de)-391(Riemann)]TJ /F2 1 Tf -9.3855 -1.4036 Td [(\247\()]TJ /F8 1 Tf 1.1145 0 Td [(\276)18(;)-168(\273)-48(;)-168(f)]TJ /F2 1 Tf 2.5181 0 Td [(\))-283(=)]TJ /F9 1 Tf 2.2952 0.95181 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 34.991 80.961 Td [(1)]TJ /F17 1 Tf 0.57759 0 Td [(6)]TJ /F7 1 Tf 0.88793 0 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.205 57.774 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.569 82.469 Td [(k)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.145 57.774 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.31 82.469 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.06 57.774 Td [(\).)]TJ -24.729 -2.9277 Td [(On)-337(a)-301(\266)469(evidemmen)24(t)]TJ /F8 1 Tf 8.1145 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)12(;)-168(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(\247\()]TJ /F8 1 Tf 1.1084 0 Td [(\276)12(;)-168(\273)-48(;)-168(f)]TJ /F2 1 Tf 2.5241 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)12(;)-168(f)]TJ /F2 1 Tf 1.5964 0 Td [(\).)]TJ -20.301 -1.8434 Td [(Le)-337(r)30(\266)469(esultat)-337(fondamen)24(tal)-337(de)-337(la)-331(th)18(\266)475(eorie)-337(de)-337(Riemann)-331(est)-343(alors)-337(donn)24(\266)469(e)-337(par)-337(l')30(\266)469(enonc)30(\266)469(e)-337(suiv)48(an)30(t)-337(:)]TJ /F15 1 Tf 0 -2.259 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8795 0 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F12 1 Tf 0.93976 0 Td [(est)-343(Riemann-in)24(t)24(\266)475(egrable)-343(sur)]TJ /F2 1 Tf 12.127 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(,)-343(p)-30(our)-337(toute)-337(suite)-343(de)-337(sub)-24(divisions)]TJ /F8 1 Tf 14.976 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 72.4 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 49.539 Td [(don)24(t)-259(le)-265(pas)-265(tend)-259(v)18(ers)-259(0)-259(et)-259(toute)-259(suite)-259(de)-259(p)-30(oin)24(tages)]TJ /F8 1 Tf 21.458 0 Td [(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.957 70.676 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 28.03 49.539 Td [(de)]TJ /F8 1 Tf 1.259 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.733 70.676 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 30.663 49.539 Td [(la)-259(suite)-259(des)-265(sommes)-259(de)-259(Riemann)]TJ /F2 1 Tf -25.331 -1.8916 Td [(\247\()]TJ /F8 1 Tf 1.1084 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.034 67.969 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.5602 47.647 Td [(;)-162(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.078 67.969 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.9819 47.647 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(tend)-343(v)24(ers)-331(l'in)18(t)24(\266)475(egrale)]TJ /F9 1 Tf 8.988 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.672 69.78 Td [(b)]TJ -0.62931 -2.9052 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.139 47.647 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F12 1 Tf 1.0904 0 Td [(.)]TJ /F1 1 Tf -19.012 -2.8253 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(P)30(ar)-433(h)24(yp)-30(oth)18(\265)475(ese)]TJ /F8 1 Tf 6.7771 0 Td [(f)]TJ /F2 1 Tf 1.0301 0 Td [(est)-439(b)-30(orn)24(\266)475(ee,)-463(notons)]TJ /F8 1 Tf 8.6506 0 Td [(M)]TJ /F2 1 Tf 1.512 0 Td [(un)-433(ma)-60(joran)30(t)-439(de)]TJ /F6 1 Tf 7.4157 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.2771 0 Td [(.)-746(Consi-)]TJ -35.355 -1.1988 Td [(d)30(\266)469(erons)-409(alors)-409(une)-409(sub)-30(division)]TJ /F8 1 Tf 12.994 0 Td [(\276)]TJ /F2 1 Tf 1.012 0 Td [(et)-409(observ)30(ons)-415(ce)-409(qui)-403(se)-409(passe)-409(p)-30(our)-409(la)-403(somme)-409(de)-403(Darb)-30(oux)]TJ -14.006 -1.2048 Td [(inf)30(\266)469(erieure)]TJ /F8 1 Tf 4.3855 0 Td [(s)]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)12(;)-168(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))-295(lorsque)-295(l'on)-301(a)-54(joute)-295(un)-301(p)-24(oin)24(t)-301(de)-289(sub)-30(division)-295(\265)500(a)]TJ /F8 1 Tf 20.283 0 Td [(\276)]TJ /F2 1 Tf 0.90361 0 Td [(p)-24(our)-301(obtenir)-295(une)-295(nouv)30(elle)]TJ -28.024 -1.1988 Td [(sub)-24(division)]TJ /F8 1 Tf 5.2349 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.991 59.581 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.452 41.22 Td [(.)]TJ -6.1205 -1.8434 Td [(Si)-325(le)-331(p)-30(oin)24(t)-325(de)-325(sub)-30(division)-325(a)-54(jout)24(\266)469(e)-331(est)]TJ /F8 1 Tf 15.988 0 Td [(c)]TJ /F2 1 Tf 0.75904 0 Td [(et)-325(se)-331(trouv)24(e)-325(en)24(tre)]TJ /F8 1 Tf 7.9518 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.595 56.943 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.072 39.376 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.741 56.943 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.5 39.376 Td [(on)-325(v)24(oit)-331(que)-325(l'a)-60(jout)-325(de)]TJ /F8 1 Tf 9.7349 0 Td [(c)]TJ /F2 1 Tf 0.75903 0 Td [(a)]TJ -38.663 -1.1988 Td [(mo)-30(di\257)24(\266)475(e)-355(la)-349(somme)]TJ /F8 1 Tf 8.0301 0 Td [(s)]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)12(;)-162(f)]TJ /F2 1 Tf 1.5964 0 Td [(\))-349(en)-349(rempla)-6(\273)439(can)24(t)-349(un)-355(terme)-349(\()]TJ /F8 1 Tf 12.006 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.431 55.219 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.994 38.177 Td [(\241)]TJ /F8 1 Tf 1.012 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.56 55.219 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.723 38.177 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(m)]TJ /F2 1 Tf 1.2289 0 Td [(par)-349(\()]TJ /F8 1 Tf 2.1867 0 Td [(c)]TJ /F6 1 Tf 0.66265 0 Td [(\241)]TJ /F8 1 Tf 1.012 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.422 55.219 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.916 38.177 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.638 55.219 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.693 38.177 Td [(+)-234(\()]TJ /F8 1 Tf 1.4036 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58 55.219 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.271 38.177 Td [(\241)]TJ /F8 1 Tf 1.0121 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.94 55.219 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 36.973 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3735 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(,)]TJ /F8 1 Tf 0.60241 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.966 53.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6627 36.973 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.733 53.504 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.524 36.973 Td [(son)18(t)-319(les)-325(b)-30(ornes)-325(inf)30(\266)469(erieure)-325(de)]TJ /F8 1 Tf 12.494 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(sur)-325(les)-325(segmen)24(ts)-319([)]TJ /F8 1 Tf 7.6265 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.647 53.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.277 36.973 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.302 53.504 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.663 36.973 Td [(],)-325([)]TJ /F8 1 Tf 1.1566 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.31 53.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.536 36.973 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(])-319(et)-325([)]TJ /F8 1 Tf 2.0301 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.75 53.504 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.265 36.973 Td [(].)-439(En)]TJ -36.934 -1.1988 Td [(particulier)-277(on)-277(a)]TJ /F8 1 Tf 6.8916 0 Td [(m)]TJ /F2 1 Tf 1.1566 0 Td [(=)-277(min)-6(\()]TJ /F8 1 Tf 3.1084 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.853 51.788 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.651 35.774 Td [(;)-162(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.147 51.788 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.482 35.774 Td [(\).)-427(Il)-277(s'ensuit)-277(que)-283(la)-277(nouv)30(elle)-283(somme)-277(de)-277(Darb)-24(oux)-283(inf)24(\266)475(erieure)]TJ -14.151 -1.1988 Td [(est)-271(\266)469(egale)-301(\265)499(a)-295(la)-301(premi)24(\265)475(ere)-301(augmen)24(t)24(\266)469(ee)-301(de)-295(\()]TJ /F8 1 Tf 16.566 0 Td [(c)]TJ /F6 1 Tf 0.57831 0 Td [(\241)]TJ /F8 1 Tf 0.92771 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.112 50.064 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.114 34.575 Td [(\)\()]TJ /F8 1 Tf 0.78313 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.879 50.064 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.205 34.575 Td [(\241)]TJ /F8 1 Tf 0.92169 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\))-301(ou)-301(\()]TJ /F8 1 Tf 2.4277 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.172 50.064 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.53 34.575 Td [(\241)]TJ /F8 1 Tf 0.92169 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(\)\()]TJ /F8 1 Tf 0.77711 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.431 50.064 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 35.205 34.575 Td [(\241)]TJ /F8 1 Tf 0.92771 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\).)-439(De)-301(toute)-295(fa)-6(\273)439(con)]TJ -31.675 -1.1988 Td [(cette)-289(augmen)24(tation)-289(est)-295(p)-30(ositiv)30(e)-295(et)-283(au)-289(plus)-259(\266)469(egale)-289(\265)499(a)-289(\()]TJ /F8 1 Tf 21.994 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.724 48.349 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.398 33.376 Td [(\241)]TJ /F8 1 Tf 0.90362 0 Td [(c)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.552 48.349 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.018 33.376 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.164 48.349 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.97 33.376 Td [(\241)]TJ /F8 1 Tf 0.90964 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.302 48.349 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.265 33.376 Td [(j)]TJ /F2 1 Tf 0.56627 0 Td [(donc)-289(inf)24(\266)475(erieure)-295(\265)499(a)-283(2)]TJ /F8 1 Tf 8.006 0 Td [(M)-108(h)]TJ /F2 1 Tf -37.506 -1.2048 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3855 0 Td [(h)]TJ /F2 1 Tf 0.90964 0 Td [(est)-343(le)-337(pas)-337(de)]TJ /F8 1 Tf 5.741 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(.)]TJ -8.6446 -1.8434 Td [(En)-331(conclusion,)-337(on)-337(a)]TJ /F8 1 Tf 8.9096 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.457 43.995 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.163 30.328 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))-228(+)-222(2)]TJ /F8 1 Tf 2.1145 0 Td [(M)-108(h)]TJ /F2 1 Tf 1.6566 0 Td [(.)]TJ -23.819 -1.8374 Td [(Le)-331(r)24(\266)475(esultat)-337(pr)24(\266)475(ec)24(\266)469(eden)30(t)-337(se)-331(g)30(\266)469(en)30(\266)469(eralise)-331(par)-331(r)24(\266)469(ecurrence)-331(au)-331(cas)-331(o)-30(\265)530(u)-331(l'on)-337(a)-54(joute)-331(un)-337(nom)24(bre)-331(\257ni)-331(de)]TJ 0 -1.2048 Td [(p)-30(oin)30(ts)-355(\265)499(a)-355(la)-355(sub)-30(division.)-512(En)-355(particulier,)-361(en)-355(prenan)24(t)-355(deux)-355(sub)-30(divisions)]TJ /F8 1 Tf 30.602 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.293 39.642 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.181 27.286 Td [(et)]TJ /F8 1 Tf 1.1867 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.776 39.642 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.837 27.286 Td [(et)-361(en)-355(com-)]TJ -34.506 -1.1988 Td [(paran)24(t)-355(les)-361(sommes)-355(inf)24(\266)475(erieures)-361(qu'elles)-355(d)24(\266)475(e\257nissen)24(t)-361(\265)500(a)-349(celle)-361(qui)-355(est)-361(d)30(\266)469(e\257nie)-355(par)-355(la)-355(sub)-30(division)]TJ /F8 1 Tf 0 -1.1988 Td [(\276)]TJ /F2 1 Tf 0.88554 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.276 36.202 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2831 24.888 Td [([)]TJ /F8 1 Tf 0.78916 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.853 36.202 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.193 24.888 Td [(,)-295(on)-283(obtien)24(t)]TJ /F8 1 Tf 5.2771 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.241 36.202 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.217 24.888 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)18(;)-168(f)]TJ /F2 1 Tf 1.5904 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.672 36.202 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.133 24.888 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-126(+)-126(2)]TJ /F8 1 Tf 1.9157 0 Td [(N)-114(M)-108(h)]TJ /F2 1 Tf 2.8554 0 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3373 0 Td [(N)]TJ /F2 1 Tf 1.1988 0 Td [(est)-289(le)-289(nom)24(bre)-283(de)-283(p)-30(oin)24(ts)]TJ -29.151 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.405 34.487 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8855 23.683 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(h)]TJ /F2 1 Tf 0.90964 0 Td [(est)-337(le)-337(pas)-343(de)]TJ /F8 1 Tf 5.7349 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.336 34.487 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.819 23.683 Td [(.)]TJ -11.488 -1.8373 Td [(Ainsi,)-313(a)24(v)24(ec)-307(les)-307(notations)-313(de)-307(l')30(\266)469(enonc)30(\266)469(e,)-319(p)-30(our)-307(toute)-307(sub)-30(division)]TJ /F8 1 Tf 26.337 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.181 31.849 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.861 21.846 Td [(de)-307([)]TJ /F8 1 Tf 1.5843 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43975 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(])-307(on)-307(a)-307(la)-307(ma)-54(joration)]TJ /F8 1 Tf -30.518 -1.2048 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.7241 30.133 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0783 20.641 Td [(;)-162(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.129 29.323 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.813 20.641 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-180(+)-174(2)]TJ /F8 1 Tf 2.0181 0 Td [(N)-108(M)-114(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.957 29.323 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.295 20.641 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.3675 0 Td [(N)]TJ /F2 1 Tf 1.2169 0 Td [(est)-319(le)-313(nom)24(bre)-307(de)-313(p)-30(oin)24(ts)-307(de)]TJ /F8 1 Tf 11.735 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.543 30.133 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.813 20.641 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.845 29.323 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.386 20.641 Td [(est)-313(le)-319(pas)-313(de)]TJ /F8 1 Tf 5.6386 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.957 29.323 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.139 20.641 Td [(.)-439(Il)]TJ -37.807 -1.1988 Td [(s'ensuit)-343(que,)-343(si)]TJ /F8 1 Tf 6.7952 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.172 27.607 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.584 19.442 Td [(tend)-349(v)24(ers)-343(0,)-343(on)-343(a)]TJ /F8 1 Tf 7.7169 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.586 28.418 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.048 19.442 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.68072 0 Td [(6)]TJ /F2 1 Tf 1.0663 0 Td [(lim)-174(inf)]TJ /F8 1 Tf 2.9398 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.233 27.607 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.753 19.442 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\).)-469(Cela)-349(a)30(y)24(an)24(t)-343(lieu)-343(p)-30(our)-343(toute)]TJ -26.464 -1.1988 Td [(sub)-24(division)]TJ /F8 1 Tf 5.2349 0 Td [(\276)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.991 26.693 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.452 18.244 Td [(,)-337(en)-331(passan)24(t)-337(\265)500(a)-337(la)-331(b)-30(orne)-337(sup)0(\266)475(erieure,)-337(on)-331(obtien)24(t)]TJ /F8 1 Tf 20.205 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(lim)-174(inf)]TJ /F8 1 Tf 2.9458 0 Td [(s)]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.103 25.892 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.747 18.244 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\).)]TJ -35.458 -1.8434 Td [(P)30(ar)-427(ailleurs)-427(la)-427(ma)-54(joration)-403(\266)469(eviden)24(te)]TJ /F8 1 Tf 16 0 Td [(s)]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.569 23.254 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.301 16.4 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.8253 0 Td [(6)]TJ /F8 1 Tf 1.2108 0 Td [(s)]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-433(assure)-421(lim)-174(sup)]TJ /F8 1 Tf 7.1566 0 Td [(s)]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.112 23.254 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.958 16.4 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.8253 0 Td [(6)]TJ /F8 1 Tf 1.2108 0 Td [(s)]TJ /F2 1 Tf 0.46386 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\),)-451(d'o)-36(\265)530(u)]TJ -36.157 -1.2048 Td [(l'on)-337(tire)-337(que)]TJ /F8 1 Tf 5.6446 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.75 21.538 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.946 15.195 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-337(tend)-343(v)24(ers)]TJ /F8 1 Tf 5.0663 0 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)]TJ -15.181 -1.8373 Td [(De)-337(la)-331(m)24(^)469(eme)-331(fa)-6(\273)439(con)-331(on)-337(mon)24(tre)-337(que)]TJ /F8 1 Tf 14.855 0 Td [(S)]TJ /F2 1 Tf 0.66868 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.224 18.9 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.361 13.358 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-337(tend)-343(v)24(ers)]TJ /F8 1 Tf 5.0663 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\).)]TJ -24.801 -2.5964 Td [(Lorsque)]TJ /F8 1 Tf 3.9819 0 Td [(f)]TJ /F2 1 Tf 1.1325 0 Td [(est)-548(Riemann-in)24(t)24(\266)469(egrable)-542(on)-542(a)]TJ /F8 1 Tf 13.5 0 Td [(s)]TJ /F2 1 Tf 0.46385 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\))-626(=)]TJ /F8 1 Tf 2.4157 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\))-626(=)]TJ /F9 1 Tf 2.4096 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.06 16.995 Td [(b)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.283 10.762 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.6325 0 Td [(et)-542(l'encadremen)18(t)]TJ /F8 1 Tf -31.699 -1.759 Td [(s)]TJ /F2 1 Tf 0.46988 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.6724 12.668 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.3012 9.0027 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.98193 0 Td [(6)]TJ /F2 1 Tf 1.3675 0 Td [(\247\()]TJ /F8 1 Tf 1.1084 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.707 12.668 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.922 9.0027 Td [(;)-162(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.75 12.668 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.343 9.0027 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))]TJ /F11 1 Tf 0.98193 0 Td [(6)]TJ /F8 1 Tf 1.3675 0 Td [(S)]TJ /F2 1 Tf 0.66868 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.707 12.668 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.91 9.0027 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\),)-572(on)-518(en)-524(tire)-524(que)-524(la)-518(suite)-524(des)-524(sommes)-524(de)-518(Riemann)]TJ -15.62 -1.2048 Td [(\247\()]TJ /F8 1 Tf 1.1084 0 Td [(\276)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.034 10.952 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.5602 7.7979 Td [(;)-162(\273)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.078 10.952 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.9819 7.7979 Td [(;)-168(f)]TJ /F2 1 Tf 1.0422 0 Td [(\))-337(tend)-343(v)24(ers)-331(l'in)18(t)24(\266)475(egrale)-337(de)]TJ /F8 1 Tf 11.048 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)]TJ ET 512.64 92.048 m 519.84 92.048 l 519.84 100.47 l 512.64 100.47 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.247 3.8882 Td [(89)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 303 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 302 0 R /ProcSet 2 0 R >> /Contents 301 0 R >> endobj 302 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F17 97 0 R >> endobj 304 0 obj << /Length 14124 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-8234(VI.)-385(L'in)30(t)24(\266)548(egrale)-385(de)-391(Leb)-30(esgue)-385(sur)-385(I)283(R)]TJ 0 -2.4036 Td [(Remarque.)]TJ /F2 1 Tf 6.7169 0 Td [(La)-457(prop)-30(osition)-457(pr)30(\266)469(ec)24(\266)475(eden)24(te)-451(mon)24(tre)-451(que)-457(les)-451(sommes)-451(de)-451(Riemann)-451(son)24(t)-457(des)]TJ -6.7169 -1.1988 Td [(appro)24(ximations)-439(de)-439(l'in)24(t)24(\266)475(egrale)-439(d'autan)24(t)-445(meilleures)-433(que)-439(le)-445(pas)-439(est)-439(p)-30(etit.)-753(En)-439(fait,)-463(dans)-439(la)]TJ 0 -1.1988 Td [(pratique,)-307(il)-295(est)-295(p)-30(ossible)-301(de)-289(quan)24(ti\257er)-301(cela)-295(d)24(\265)475(es)-301(que)-295(l'on)-295(a)-295(une)-295(b)-30(onne)-295(h)24(yp)-36(oth)24(\265)475(ese)-295(de)-295(r)24(\266)475(egularit)24(\266)475(e)]TJ 0 -1.2048 Td [(sur)]TJ /F8 1 Tf 1.6687 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)-445(Il)-325(est)-331(p)-30(ossible)-301(\266)469(egalemen)30(t)-331(de)-331(c)24(hoisir)-325(le)-331(p)-30(oin)24(tage)-331(judicieusemen)30(t)-331(p)-30(our)-331(am)24(\266)475(eliorer)-331(cette)]TJ -2.2651 -1.1988 Td [(appro)24(ximation.)-433(Ces)-307(r)30(\266)469(esultats)-307(rel)24(\265)469(ev)30(en)24(t)-307(de)-301(la)-301(th)18(\266)475(eorie)-307(de)-307(l'in)24(t)24(\266)475(egration)-313(n)30(um)24(\266)469(erique,)-313(nous)-301(ne)-307(les)]TJ 0 -1.1988 Td [(ab)-30(orderons)-331(pas)-337(ici.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 43.047 Td [(VI.5.)-236(In)30(t)20(\266)537(egrales)-376(semi-con)25(v)30(ergen)25(tes.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 49.894 Td [(Consid)30(\266)469(erons)-397(une)-397(fonction)]TJ /F8 1 Tf 11.56 0 Td [(f)]TJ /F2 1 Tf 0.99397 0 Td [(d)24(\266)475(e\257nie)-397(et)-397(mesurable)-397(sur)-397(un)-397(in)24(terv)54(alle)-397([)]TJ /F8 1 Tf 17.175 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [([)-391(\()]TJ /F6 1 Tf 1.0602 0 Td [(\2411)]TJ /F8 1 Tf 2.1566 0 Td [(<)-385(a)-379(<)-385(b)]TJ /F11 1 Tf 4.0361 0 Td [(6)]TJ /F2 1 Tf -38.386 -1.1988 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\).)-933(Supp)-24(osons)]TJ /F8 1 Tf 6.6867 0 Td [(f)]TJ /F2 1 Tf 1.0904 0 Td [(lo)-30(calemen)24(t)-493(in)24(t)24(\266)469(egrable)-493(sur)-500([)]TJ /F8 1 Tf 12.084 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [([.)-927(Nous)-493(p)-30(ouv)18(ons)-500(donc)-493(d)24(\266)475(e\257nir)-500(p)-30(our)]TJ /F8 1 Tf -23.042 -1.8976 Td [(t)]TJ /F6 1 Tf 0.63855 0 Td [(2)]TJ /F2 1 Tf 0.94578 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [([)-331(les)-331(in)24(t)24(\266)475(egrales)]TJ /F9 1 Tf 6.5361 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.086 68.564 Td [(t)]TJ -0.63793 -2.8966 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.482 46.798 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4217 0 Td [(et)]TJ /F9 1 Tf 1.1627 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.75 68.564 Td [(t)]TJ -0.63793 -2.8966 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.536 46.798 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.0964 0 Td [(.)-439(La)-337(propri)24(\266)469(et)24(\266)475(e)-337(de)-331(Bepp)-30(o)-331(Levi)-331(assure)-331(alors)]TJ /F9 1 Tf -20.97 -1.247 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.0603 64.831 Td [(b)]TJ -0.63793 -2.9052 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7349 44.189 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.4096 0 Td [(=)-349(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.043 62.297 Td [(t)]TJ /F10 1 Tf 0.43104 0 Td [(!)]TJ /F7 1 Tf 1.1379 0 Td [(b)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 13.524 45.551 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.784 64.831 Td [(t)]TJ -0.62931 -2.9052 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.88 44.189 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.0964 0 Td 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8.352 0 0 Tm 10.69 30.323 Td [(c)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2108 20.78 Td [(est)-337(le)-343(compl)24(\266)475(emen)24(taire)-337(de)]TJ /F8 1 Tf 11.151 0 Td [(A)]TJ /F12 1 Tf 0.75301 0 Td [(.)]TJ -14.783 -1.6024 Td [(Alors)]TJ /F6 1 Tf 2.759 0 Td [(T)]TJ /F12 1 Tf 1.241 0 Td [(est)-451(une)-445(tribu)-445(et)-445(la)-445(restriction)-451(de)]TJ /F8 1 Tf 15.139 0 Td [(\271)]TJ /F12 1 Tf 1.0422 0 Td [(\265)500(a)]TJ /F6 1 Tf 0.94578 0 Td [(T)]TJ /F12 1 Tf 1.247 0 Td [(est)-445(une)-445(mesure.)-783(En)-445(outre,)-475(la)-445(mesure)]TJ -22.373 -1.1988 Td [(ext)30(\266)469(erieure)]TJ /F8 1 Tf 4.6747 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.19 26.314 Td [(\244)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 11.44 17.979 Td [(d)30(\266)469(e\257nie)-367(par)-367(cette)-367(mesure)-373(est)-373(sup)0(\266)475(erieure)-373(\265)500(a)]TJ /F8 1 Tf 18.235 0 Td [(\271)]TJ /F12 1 Tf 0.96988 0 Td [(et)-367(l'espace)-367(mesur)30(\266)469(e)]TJ /F2 1 Tf 8.3373 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-168(\271)]TJ /F2 1 Tf 1.0482 0 Td [(\))]TJ /F12 1 Tf 0.75301 0 Td [(est)]TJ -37.934 -1.1988 Td [(complet.)]TJ /F1 1 Tf 0 -1.8976 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.5964 Td [(1.)]TJ /F6 1 Tf 1.5602 0 Td [(T)]TJ /F1 1 Tf 1.1807 0 Td [(est)-385(une)-391(tribu.)]TJ /F2 1 Tf -3.5542 -1.6024 Td [(La)-355(d)24(\266)475(e\257nition)-355(de)]TJ /F6 1 Tf 7.2892 0 Td [(T)]TJ /F2 1 Tf 1.1506 0 Td [(mon)24(tre)-355(imm)24(\266)469(ediatemen)24(t)-349(que)]TJ /F8 1 Tf 12.422 0 Td [(X)]TJ /F2 1 Tf 1.259 0 Td [(appartien)24(t)-355(\265)500(a)]TJ /F6 1 Tf 5.7349 0 Td [(T)]TJ /F2 1 Tf 1.1506 0 Td [(et)-355(que)]TJ /F6 1 Tf 3.0663 0 Td [(T)]TJ /F2 1 Tf 1.1506 0 Td [(est)-355(stable)-355(par)]TJ -33.223 -1.1988 Td [(passage)-331(au)-337(compl)24(\266)469(emen)30(taire.)-451(Prouv)30(ons)-343(la)-331(stabilit)24(\266)475(e)-343(par)-331(r)24(\266)475(eunion)-343(d)30(\266)469(enom)24(brable.)]TJ 0 -1.5964 Td [(Consid)30(\266)469(erons)-295(donc)-295(une)-289(famille)-295(d)24(\266)475(enom)24(brable)-295(\()]TJ /F8 1 Tf 19.259 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.259 12.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.675 8.8882 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.293 12.504 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.633 8.8882 Td [(d')24(\266)469(el)24(\266)469(emen)30(ts)-295(de)]TJ /F6 1 Tf 6.1747 0 Td [(T)]TJ /F2 1 Tf 1.0904 0 Td [(de)-295(r)30(\266)469(eunion)]TJ /F8 1 Tf 4.8675 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(.)-427(Il)-295(s'agit)]TJ -35.187 -1.2048 Td [(de)-331(prouv)24(er)]TJ /F8 1 Tf 5.006 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(T)]TJ /F2 1 Tf 1.741 0 Td [(,)-337(c'est-)0(\265)500(a-dire,)-337(p)-30(our)]TJ /F8 1 Tf 8.3795 0 Td [(E)]TJ /F6 1 Tf 1.0723 0 Td [(\275)]TJ /F8 1 Tf 1.0542 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)]TJ /F8 1 Tf 1.2771 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6145 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.974 11.59 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 36.946 7.6834 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ -33.301 -1.3976 Td [(\(o)-30(\265)530(u)]TJ /F8 1 Tf 1.7771 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.25 9.59 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6024 6.2858 Td [(est)-337(le)-337(compl)24(\266)469(emen)30(taire)-337(de)]TJ /F8 1 Tf 11.151 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\).)]TJ 3.747 -2.3976 Td [(93)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 315 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 314 0 R /ProcSet 2 0 R >> /Contents 313 0 R >> endobj 314 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 316 0 obj << /Length 29010 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4036 Td [(Quitte)-331(\265)500(a)-331(a)-54(jouter)-331(des)-331(parties)-331(vides,)-331(on)-331(p)-30(eut)-331(supp)-30(oser)]TJ /F8 1 Tf 23.114 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(strictemen)24(t)-337(d)30(\266)469(enom)24(brable)-331(et,)-331(quitte)]TJ -23.964 -1.1988 Td [(\265)499(a)-379(faire)-385(une)-385(r)30(\266)469(eindexation,)-391(on)-385(p)-30(eut)-385(supp)-30(oser)]TJ /F8 1 Tf 19.205 0 Td [(I)]TJ /F2 1 Tf 0.87952 0 Td [(=)]TJ /F3 1 Tf 1.1325 0 Td [(N)]TJ /F2 1 Tf 0.72289 0 Td [(.)-590(La)-385(famille)-355(\266)469(etudi)30(\266)469(ee)-385(est)-385(alors)-385(une)-385(suite)]TJ -21.94 -1.1988 Td [(d')24(\266)475(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 4.9217 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.75 85.392 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.886 59.822 Td [(de)]TJ /F6 1 Tf 1.3313 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)]TJ -8.6867 -1.7289 Td [(P)30(osons)]TJ /F8 1 Tf 3.3434 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.5 82.918 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.271 58.093 Td [(=)]TJ /F9 1 Tf 1.2952 0.94578 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.233 81.409 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.072 58.093 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.784 82.918 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.313 58.093 Td [(,)-349(on)-343(obtien)24(t)-343(une)-349(suite)-343(croissan)24(te)-343(de)-349(parties)-343(de)]TJ /F8 1 Tf 20.355 0 Td [(X)]TJ /F2 1 Tf 1.253 0 Td [(de)-343(r)30(\266)469(eunion)]TJ /F8 1 Tf 4.9699 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)-475(P)24(our)]TJ /F8 1 Tf -36.307 -2.2349 Td [(n)]TJ /F2 1 Tf 0.95783 0 Td [(en)24(tier)-361(notons)]TJ /F8 1 Tf 6.0964 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.741 79.719 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.018 55.858 Td [(la)-361(partie)-361(qu'il)-361(faut)-367(a)-54(jouter)-361(\265)499(a)]TJ /F8 1 Tf 12.807 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.905 79.719 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.494 55.858 Td [(p)-24(our)-367(obtenir)]TJ /F8 1 Tf 5.8675 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.681 79.719 Td [(n)]TJ /F4 1 Tf 0.71552 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38 55.858 Td [(=)]TJ /F8 1 Tf 1.0964 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.034 79.719 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 40.645 55.858 Td [([)]TJ /F8 1 Tf 0.90362 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.534 79.719 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 55.858 Td [(;)]TJ -38.886 -1.1988 Td [(on)-289(a)-289(donc)]TJ /F8 1 Tf 4.4759 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.06 78.004 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.361 54.659 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.267 78.004 Td [(n)]TJ /F4 1 Tf 0.71552 0 Td [(+1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.855 54.659 Td [(\\)]TJ /F2 1 Tf 0.8012 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.474 78.004 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.349 54.659 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.81 78.814 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.145 54.659 Td [(.)-427(Si)-295(l'on)-289(p)-30(ose)]TJ /F8 1 Tf 5.9398 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.922 78.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.53 54.659 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.56 78.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.072 54.659 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.181 78.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.614 54.659 Td [(alors)]TJ /F8 1 Tf 2.3494 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.69 78.004 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.56 54.659 Td [(est)-295(encore)-289(la)-289(r)24(\266)475(eunion)]TJ -30.229 -1.3976 Td [(des)]TJ /F8 1 Tf 1.7289 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.121 76.004 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5964 53.262 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(k)]TJ /F11 1 Tf 0.83133 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.93373 0 Td [(et)-337(donc)]TJ /F8 1 Tf 3.5602 0 Td [(A)]TJ /F2 1 Tf 1.0241 0 Td [(=)]TJ /F9 1 Tf 1.247 0.94579 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.802 74.495 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.078 53.262 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.19 76.004 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.289 53.262 Td [(.)]TJ -16.958 -2.9398 Td [(Prouv)30(ons,)-337(p)-30(our)-337(tout)-337(en)30(tier)]TJ /F8 1 Tf 12.03 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7831 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.603 70.297 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.813 50.322 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.526 71.806 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.235 50.322 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.81326 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.422 71.806 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.289 50.322 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.759 72.607 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.307 50.322 Td [(\\)]TJ /F8 1 Tf 0.89156 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.81326 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(.)]TJ -30.12 -2.8133 Td [(Lorsque)]TJ /F8 1 Tf 3.7711 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)-277(0,)-337(cela)-337(s')24(\266)469(ecrit)]TJ /F8 1 Tf 7.1205 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32 67.771 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.03 47.509 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.81326 Td [(\241)]TJ /F2 1 Tf 0.46386 -0.81326 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 40.828 67.771 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.976 47.509 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.026 68.581 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.994 47.509 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.81326 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(,)-337(ce)-337(qui)-331(r)24(\266)475(esulte)-337(de)]TJ /F8 1 Tf 8.0964 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 58.655 67.771 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.711 47.509 Td [(2)-283(T)]TJ /F2 1 Tf 1.747 0 Td [(.)]TJ -38.127 -1.6265 Td [(Remarquons)-283(alors)-295(que)-289(l'on)-289(a)]TJ /F8 1 Tf 12.572 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.707 65.443 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.5 45.882 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.922 65.443 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.988 45.882 Td [([)]TJ /F8 1 Tf 0.79518 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.552 65.443 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.373 45.882 Td [(donc)-289(\()]TJ /F8 1 Tf 2.7289 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.733 65.443 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.422 45.882 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.517 66.254 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.494 45.882 Td [(=)-277(\()]TJ /F8 1 Tf 1.4458 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.647 65.443 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.247 45.882 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.431 66.254 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.169 45.882 Td [(\\)]TJ /F2 1 Tf 0.79518 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.974 65.443 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.657 45.882 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.31 66.254 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.452 45.882 Td [(.)-433(Comme)]TJ /F8 1 Tf -35.12 -1.1988 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 63.728 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1386 44.683 Td [(2)-506(T)]TJ /F2 1 Tf 1.9639 0 Td [(,)-499(on)-469(a)]TJ /F8 1 Tf 3.2711 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.80722 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.80722 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.302 63.728 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.127 44.683 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.638 64.53 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.235 44.683 Td [(\\)]TJ /F8 1 Tf 0.9759 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.80722 Td [(\242)]TJ /F2 1 Tf 0.96385 -0.80722 Td [(=)]TJ /F8 1 Tf 1.2771 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.80722 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80722 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33 63.728 Td [(n)]TJ /F4 1 Tf 0.70689 0 Td [(+1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.928 44.683 Td [(\\)]TJ /F2 1 Tf 0.9759 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.716 63.728 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.596 44.683 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.052 64.53 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.705 44.683 Td [(\\)]TJ /F8 1 Tf 0.9759 0 Td [(E)]TJ /F9 1 Tf 0.80121 0.80722 Td [(\242)]TJ /F2 1 Tf 0.76506 -0.80722 Td [(+)]TJ /F8 1 Tf 1.0904 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.80722 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.80722 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.431 63.728 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.096 44.683 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.216 64.53 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.205 44.683 Td [(\\)]TJ /F2 1 Tf 0.9759 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.284 63.728 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.873 44.683 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.621 64.53 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 40.982 44.683 Td [(\\)]TJ /F8 1 Tf 0.98193 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.80722 Td [(\242)]TJ /F2 1 Tf 0.95783 -0.80722 Td [(=)]TJ /F8 1 Tf -38.386 -1.3012 Td [(\271)]TJ /F9 1 Tf 0.60241 0.80722 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80722 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.172 61.866 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8855 43.382 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.80722 Td [(\242)]TJ /F2 1 Tf 0.68072 -0.80722 Td [(+)]TJ /F8 1 Tf 1 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.80722 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.80722 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.698 61.866 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.018 43.382 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.483 62.668 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.036 43.382 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.80722 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.80722 Td [(.)]TJ -13.849 -1.6265 Td [(Si)-337(l'on)-331(supp)-30(ose)-337(la)-337(relation)-331(au)-337(rang)]TJ /F8 1 Tf 15.325 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)-337(on)-331(a)-337(donc)]TJ /F8 1 Tf -15.928 -1.7771 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7771 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.655 55.495 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.572 39.979 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.569 56.995 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.825 39.979 Td [(\\)]TJ /F8 1 Tf 0.72289 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-60(+)]TJ /F8 1 Tf 1.2831 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.81326 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.905 56.995 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.018 39.979 Td [(\\)]TJ /F8 1 Tf 0.72289 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.81326 Td [(\242)]TJ /F2 1 Tf 0.51807 -0.81326 Td [(+)]TJ /F8 1 Tf 0.83133 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.81326 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.483 56.995 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.657 39.979 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.267 57.806 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.506 39.979 Td [(\\)]TJ /F8 1 Tf 0.72289 0 Td [(E)]TJ /F9 1 Tf 0.8012 0.81326 Td [(\242)]TJ /F2 1 Tf 0.73494 -0.81326 Td [(=)]TJ /F9 1 Tf 1.6205 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.672 55.495 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.88793 0 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.56 39.979 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.043 56.995 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.819 39.979 Td [(\\)]TJ /F8 1 Tf 0.72289 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-60(+)]TJ /F8 1 Tf 1.2831 0 Td [(\271)]TJ /F9 1 Tf 0.60241 0.81326 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.991 56.995 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.386 39.979 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.784 57.806 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.241 39.979 Td [(\\)]TJ /F8 1 Tf 0.72289 0 Td [(E)]TJ /F9 1 Tf 0.79518 0.81326 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(,)]TJ -38.886 -2.6566 Td [(ce)-337(qui)-331(prouv)24(e)-337(la)-337(relation)-331(annonc)24(\266)469(ee)-337(par)-331(r)24(\266)475(ecurrence.)]TJ 0 -1.9398 Td [(P)30(our)-325(c)18(haque)-325(en)30(tier)]TJ /F8 1 Tf 8.5542 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.681 50.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.717 35.382 Td [(est)-331(con)30(ten)24(u)-325(dans)-325(la)-325(r)24(\266)475(eunion)]TJ /F8 1 Tf 12.307 0 Td [(A)]TJ /F2 1 Tf 1.0783 0 Td [(des)]TJ /F8 1 Tf 1.7169 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.612 50.418 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.386 35.382 Td [(donc)-331(\()]TJ /F8 1 Tf 2.7711 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.828 50.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.464 35.382 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.164 51.219 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.464 35.382 Td [(\\)]TJ /F8 1 Tf 0.86747 0 Td [(E)]TJ /F6 1 Tf 1.0783 0 Td [(\276)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.414 51.219 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.825 35.382 Td [(\\)]TJ /F8 1 Tf 0.8735 0 Td [(E)]TJ /F2 1 Tf -38.367 -1.4036 Td [(et)-319(la)-325(croissance)-325(de)]TJ /F8 1 Tf 8.241 0 Td [(\271)]TJ /F2 1 Tf 0.92169 0 Td [(assure)]TJ /F8 1 Tf 3 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F9 1 Tf 1.1145 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.06 46.909 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.88793 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.735 33.979 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.983 48.409 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.133 33.979 Td [(\\)]TJ /F8 1 Tf 0.86144 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-198(+)]TJ /F8 1 Tf 1.5602 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.06 49.219 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.693 33.979 Td [(\\)]TJ /F8 1 Tf 0.86145 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)-445(Cela)-325(a)30(y)24(an)24(t)-325(lieu)-319(p)-30(our)-325(tout)]TJ /F8 1 Tf -27.018 -2.5843 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(,)-337(en)-331(passan)24(t)-337(\265)500(a)-337(la)-331(b)-30(orne)-337(sup)0(\266)475(erieure)-343(on)-331(conclut)]TJ /F8 1 Tf 20.012 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F9 1 Tf 1.0783 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.138 43.202 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.102 31.394 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.948 44.711 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.524 31.394 Td [(\\)]TJ /F8 1 Tf 0.89156 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.19 45.512 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.193 31.394 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)]TJ -35.548 -2.5783 Td [(Cep)-24(endan)24(t)-337(la)-337(sous-additivit)24(\266)475(e)-343(d)30(\266)469(enom)24(brable)-337(de)]TJ /F8 1 Tf 20.133 0 Td [(\271)]TJ /F2 1 Tf 0.93976 0 Td [(assure)]TJ /F9 1 Tf -16.036 -0.49397 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.81 37.443 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12 27.37 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.612 38.952 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.422 27.37 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F9 1 Tf 0.60241 1.1084 Td [(\263)]TJ 0.95181 -0.15663 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.474 37.443 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.687 27.37 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.181 38.952 Td [(k)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.506 27.37 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F9 1 Tf 0.39157 1.1084 Td [(\264)]TJ /F2 1 Tf 0.8735 -1.1084 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(\271)]TJ /F9 1 Tf 0.60241 1.1084 Td [(\263)]TJ 0.59639 -0.30121 Td [(\241)]TJ 0.81325 0.14458 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.552 37.443 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.988 27.37 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.94 38.952 Td [(k)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 31.199 28.177 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.80723 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F9 1 Tf 0.79518 1.1084 Td [(\264)]TJ /F2 1 Tf 0.87952 -1.1084 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89156 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)]TJ -33.813 -2.6325 Td [(On)-337(a)-331(donc)]TJ /F8 1 Tf 5.8916 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.80121 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(>)]TJ /F8 1 Tf 1.0602 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.103 35.987 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.367 24.738 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)]TJ -18.723 -1.6325 Td [(Comme)-307(la)-307(sous-additivit)24(\266)475(e)-313(assure)-307(l'in)24(\266)469(egalit)24(\266)475(e)-313(en)-307(sens)-307(in)30(v)24(erse,)-313(l')24(\266)469(egalit)24(\266)475(e)-313(souhait)24(\266)469(ee)-307(est)-313(prouv)54(\266)475(ee.)]TJ /F1 1 Tf 0.81325 -1.9398 Td [(2.)]TJ /F8 1 Tf 1.5602 0 Td [(\271)]TJ /F1 1 Tf 0.98795 0 Td [(est)-385(une)-385(mesure)-385(sur)]TJ /F6 1 Tf 9.9157 0 Td [(T)]TJ /F1 1 Tf 0.79518 0 Td [(.)]TJ /F2 1 Tf -14.072 -1.6265 Td [(P)30(our)-367(mon)24(trer)-367(que)]TJ /F8 1 Tf 8.2048 0 Td [(\271)]TJ /F2 1 Tf 0.96386 0 Td [(d)24(\266)475(e\257nit)-367(une)-367(mesure)-367(sur)]TJ /F6 1 Tf 10.205 0 Td [(T)]TJ /F2 1 Tf 1.1627 0 Td [(il)-367(reste)-367(\265)499(a)-367(prouv)24(er)-367(l'additivit)30(\266)469(e)-367(d)24(\266)469(enom)24(brable)]TJ -20.536 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(\271)]TJ /F2 1 Tf 0.93976 0 Td [(sur)-337(les)-307(\266)469(el)24(\266)469(emen)30(ts)-337(de)]TJ /F6 1 Tf 8.5482 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)]TJ -11.62 -1.6325 Td [(P)30(our)]TJ /F8 1 Tf 2.3373 0 Td [(A)]TJ /F2 1 Tf 0.98193 0 Td [(et)]TJ /F8 1 Tf 1.0663 0 Td [(B)]TJ /F2 1 Tf 1.0482 0 Td [(disjoin)30(ts)-234(dans)]TJ /F8 1 Tf 6.0964 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-253(si)]TJ /F8 1 Tf 1.4398 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(T)]TJ /F2 1 Tf 1.741 0 Td [(,)-259(en)-234(appliquan)30(t)-240(la)-234(d)24(\266)475(e\257nition)-240(de)]TJ /F6 1 Tf 13.271 0 Td [(T)]TJ /F2 1 Tf 1.0301 0 Td [(\265)499(a)]TJ /F8 1 Tf 0.74096 0 Td [(A)]TJ /F2 1 Tf 0.98193 0 Td [(et)-234(\265)500(a)]TJ /F8 1 Tf 1.8072 0 Td [(E)]TJ /F2 1 Tf 1.0723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F6 1 Tf 0.77711 0 Td [([)]TJ /F8 1 Tf 0.69277 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(,)]TJ -38.886 -1.1988 Td [(comme)]TJ /F8 1 Tf 3.3072 0 Td [(A)]TJ /F6 1 Tf 0.8012 0 Td [(\\)]TJ /F8 1 Tf 0.72289 0 Td [(E)]TJ /F2 1 Tf 1.0723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F2 1 Tf 1 0 Td [(et)]TJ /F8 1 Tf 1.0843 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.638 22.788 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.584 15.509 Td [(\\)]TJ /F8 1 Tf 0.72289 0 Td [(E)]TJ /F2 1 Tf 1.0723 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(B)]TJ /F2 1 Tf 0.81325 0 Td [(,)-265(on)-246(trouv)24(e)-259(la)-246(form)24(ule)-246(d'additivit)24(\266)469(e)]TJ /F8 1 Tf 14.717 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.80723 0 Td [([)]TJ /F8 1 Tf 0.71687 0 Td [(B)]TJ /F2 1 Tf 0.81326 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-54(+)]TJ /F8 1 Tf 1.2711 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\).)]TJ -38.494 -1.1988 Td [(P)30(ar)-361(r)24(\266)475(ecurrence)-367(on)-361(conclut)-367(que)-361(si)-367(l'on)-361(a)-361(des)-337(\266)469(el)24(\266)475(emen)24(ts)]TJ /F8 1 Tf 23.349 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 42.121 20.262 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.88 14.31 Td [(;)-168(:)-168(:)-168(:)-162(;)-168(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.017 20.262 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.759 14.31 Td [(deux)-361(\265)500(a)-361(deux)-361(disjoin)30(ts)-361(de)]TJ /F6 1 Tf -28.428 -1.3434 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(on)-331(a)]TJ /F8 1 Tf 2.2229 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F9 1 Tf 0.38554 0.94578 Td [([)]TJ /F8 1 Tf 1.2831 -0.94578 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.75 18.34 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.199 12.967 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.724 18.34 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.771 12.967 Td [(\).)]TJ -12.44 -1.8313 Td [(Si)-373(l'on)-373(a)-373(une)-373(famille)-373(d)24(\266)475(enom)24(brable)-379(\()]TJ /F8 1 Tf 15.627 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.069 15.719 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.042 11.135 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.103 15.719 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.084 11.135 Td [(d')24(\266)469(el)24(\266)475(emen)24(ts)-373(de)]TJ /F6 1 Tf 6.3374 0 Td [(T)]TJ /F2 1 Tf 1.1687 0 Td [(deux)-373(\265)500(a)-373(deux)-367(disjoin)24(ts)-373(don)24(t)-373(la)]TJ -26.259 -1.3976 Td [(r)30(\266)469(eunion)-355(est)]TJ /F8 1 Tf 5.2108 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-355(p)-30(our)-349(toute)-355(partie)-349(\257nie)]TJ /F8 1 Tf 10.735 0 Td [(J)]TJ /F2 1 Tf 1 0 Td [(de)]TJ /F8 1 Tf 1.3494 0 Td [(I)]TJ /F2 1 Tf 0.86747 0 Td [(on)-355(a)]TJ /F9 1 Tf 2.259 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.431 12.219 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.108 9.7376 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.155 13.719 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.187 9.7376 Td [(\))-307(=)]TJ /F8 1 Tf 1.7771 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F9 1 Tf 0.5 0.94578 Td [([)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.586 12.219 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.452 9.7376 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.81 13.719 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.536 9.7376 Td [(\))]TJ /F11 1 Tf 0.69277 0 Td [(6)]TJ /F8 1 Tf 1.0843 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-355(\(puisque)]TJ /F8 1 Tf -34.723 -2.5 Td [(\271)]TJ /F2 1 Tf 0.98193 0 Td [(est)-379(croisan)18(te\).)-578(On)-379(en)-379(d)30(\266)469(eduit)]TJ /F9 1 Tf 12.97 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.75 8.6418 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.898 7.2376 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.397 10.142 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.97 7.2376 Td [(\))]TJ /F11 1 Tf 0.74096 0 Td [(6)]TJ /F8 1 Tf 1.1325 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)-584(Comme)-379(la)-379(sous-additivit)24(\266)469(e)-379(de)]TJ /F8 1 Tf 14.247 0 Td [(\271)]TJ /F2 1 Tf 0.98193 0 Td [(assure)]TJ /F5 1 Tf -36.482 -3.3494 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(94)]TJ ET endstream endobj 318 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 317 0 R /ProcSet 2 0 R >> /Contents 316 0 R >> endobj 317 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F16 81 0 R /F17 97 0 R >> endobj 319 0 obj << /Length 22107 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4518 Td [(l'in)24(\266)475(egalit)18(\266)475(e)-361(in)30(v)24(erse,)-361(on)-355(a)-349(l')24(\266)469(egalit)24(\266)475(e)]TJ /F9 1 Tf 14.199 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.095 87.254 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.139 62.171 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.75 88.754 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.217 62.171 Td [(\))-313(=)]TJ /F8 1 Tf 1.7831 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\),)-361(autremen)30(t)-355(dit)]TJ /F8 1 Tf 7.4277 0 Td [(\271)]TJ /F2 1 Tf 0.95783 0 Td [(est)-355(d)24(\266)475(enom)24(brablemen)24(t)]TJ -29.801 -2.1446 Td [(additiv)30(e)-337(sur)]TJ /F6 1 Tf 5.512 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(.)]TJ /F1 1 Tf -5.494 -1.506 Td [(3.)]TJ /F8 1 Tf 1.5602 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.888 84.331 Td [(\244)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0482 58.521 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F1 1 Tf 0.60241 0 Td [(.)]TJ /F2 1 Tf -5.3735 -1.512 Td [(P)30(our)]TJ /F8 1 Tf 2.5964 0 Td [(A)]TJ /F6 1 Tf 1.2952 0 Td [(2)]TJ /F8 1 Tf 1.2108 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-530(on)-500(a)-493(vu)-500(\()]TJ /F1 1 Tf 5.3193 0 Td [(IV.4.4.)]TJ /F2 1 Tf 3.4157 0 Td [(\))-500(que)]TJ /F8 1 Tf 2.9096 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.759 82.176 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.048 57.009 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-499(est)-500(la)-493(b)-30(orne)-493(inf)30(\266)469(erieure)-499(des)]TJ /F8 1 Tf 13.325 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-493(p)-30(our)]TJ /F8 1 Tf 3.4096 0 Td [(U)]TJ /F2 1 Tf -38.373 -1.1988 Td [(mesurables)-379(con)30(tenan)18(t)]TJ /F8 1 Tf 9.8193 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(.)-566(La)-379(croissance)-385(de)]TJ /F8 1 Tf 8.4458 0 Td [(\271)]TJ /F2 1 Tf 0.9759 0 Td [(assure)-379(que)-373(ces)]TJ /F8 1 Tf 6.6205 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-379(son)24(t)-379(sup)0(\266)475(erieurs)-379(\265)500(a)]TJ /F8 1 Tf 8.6325 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ -38.777 -1.1988 Td [(ainsi)-331(donc)-337(que)-337(leur)-337(b)-30(orne)-331(inf)24(\266)475(erieure.)]TJ /F1 1 Tf 0.81325 -1.512 Td [(4.)]TJ /F2 1 Tf 1.5602 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F8 1 Tf 0.79518 0 Td [(;)-168(\271)]TJ /F2 1 Tf 1.0482 0 Td [(\))]TJ /F1 1 Tf 0.77711 0 Td [(est)-379(complet.)]TJ /F2 1 Tf -6.6807 -1.506 Td [(Il)-397(s'agit)-403(de)-403(prouv)24(er,)-421(p)-30(our)]TJ /F8 1 Tf 11.651 0 Td [(A)]TJ /F2 1 Tf 1.1506 0 Td [(partie)-403(n)24(\266)469(egligeable,)-415(que)]TJ /F8 1 Tf 10.367 0 Td [(A)]TJ /F2 1 Tf 1.1506 0 Td [(appartien)24(t)-403(\265)500(a)]TJ /F6 1 Tf 5.8313 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(,)-421(c'est-)0(\265)500(a-dire,)-421(p)-30(our)]TJ /F8 1 Tf -30.952 -1.1988 Td [(E)]TJ /F6 1 Tf 1.1024 0 Td [(\275)]TJ /F8 1 Tf 1.0904 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)]TJ /F8 1 Tf 0.63253 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-313(=)]TJ /F8 1 Tf 1.7831 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.98795 0 Td [(\\)]TJ /F8 1 Tf 0.89759 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-240(+)]TJ /F8 1 Tf 1.6386 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.172 72.702 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.331 50.394 Td [(\\)]TJ /F8 1 Tf 0.90361 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)-500(A)-355(cause)-355(de)-355(la)-349(sous-additivit)24(\266)469(e)-355(de)]TJ /F8 1 Tf 15.518 0 Td [(\271)]TJ /F2 1 Tf 0.95783 0 Td [(il)-355(su\261t)-355(m)24(^)475(eme)]TJ -33.175 -1.2048 Td [(de)-331(mon)24(trer)]TJ /F8 1 Tf 5.1446 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89156 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.043 70.987 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 21.62 49.189 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)]TJ -17.976 -1.506 Td [(La)-331(croissance)-337(de)]TJ /F8 1 Tf 7.4639 0 Td [(\271)]TJ /F2 1 Tf 0.92771 0 Td [(assure)]TJ /F8 1 Tf 3.012 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96386 0 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.431 68.823 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.62 47.683 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-331(\(d'apr)24(\265)475(es)]TJ /F1 1 Tf 4.5542 0 Td [(3.)]TJ /F2 1 Tf 0.89759 0 Td [(\).)-445(Comme)]TJ /F8 1 Tf 4.7711 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(est)-331(supp)-30(os)24(\266)469(ee)]TJ -33.735 -1.1988 Td [(n)30(\266)469(egligeable,)]TJ /F8 1 Tf 5.4639 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.31 67.107 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.855 46.485 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-391(est)-403(n)30(ul)-397(ainsi)-391(donc)-397(que)]TJ /F8 1 Tf 10.928 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 1.012 0 Td [(\\)]TJ /F8 1 Tf 0.92771 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(\).)-626(La)-397(relation)-391(v)24(oulue)-397(d)30(\266)469(ecoule)-397(alors)-397(de)]TJ /F8 1 Tf -22.325 -1.2048 Td [(E)]TJ /F6 1 Tf 1.0723 0 Td [(\276)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.75 65.392 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8373 45.28 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 1.1325 0 Td [(et)-331(de)-337(la)-331(croissance)-343(de)]TJ /F8 1 Tf 9.6265 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(.)]TJ ET 512.64 540.03 m 519.84 540.03 l 519.84 548.46 l 512.64 548.46 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 35.404 Td [(VI)-35(I.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8193 42.442 Td [(3.)-289(L'exemple)-385(de)-385(Carath)18(\266)548(eo)-30(dory)90(.)]TJ /F2 1 Tf -3.488 -1.506 Td [(Soien)24(t)]TJ /F8 1 Tf 2.9819 0 Td [(X)]TJ /F2 1 Tf 1.1988 0 Td [(un)-289(ensem)18(ble,)]TJ /F6 1 Tf 5.8976 0 Td [(S)]TJ /F2 1 Tf 0.96988 0 Td [(un)-295(ensem)24(ble)-289(de)-289(parties)-295(de)]TJ /F8 1 Tf 11.44 0 Td [(X)]TJ /F2 1 Tf 1.1928 0 Td [(et)]TJ /F8 1 Tf 1.1265 0 Td [(m)]TJ /F2 1 Tf 1.1687 0 Td [(une)-289(application)-295(de)]TJ /F6 1 Tf 8.259 0 Td [(S)]TJ /F2 1 Tf 0.96988 0 Td [(dans)]TJ ET 0.504 w 511.92 499.17 m 520.56 499.17 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 42.831 40.936 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 62.328 58.366 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 40.936 Td [(.)]TJ -38.886 -1.512 Td [(P)30(our)]TJ /F8 1 Tf 2.4699 0 Td [(A)]TJ /F6 1 Tf 1.0843 0 Td [(\275)]TJ /F8 1 Tf 1.1145 0 Td [(X)]TJ /F2 1 Tf 1.2771 0 Td [(on)-373(app)-30(elle)]TJ /F1 1 Tf 4.8795 0 Td [(recouvremen)30(t)-427(d)24(\266)548(enom)30(brable)]TJ /F2 1 Tf 14.139 0 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(A)]TJ /F2 1 Tf 1.1205 0 Td [(toute)-367(famille)-373(\()]TJ /F8 1 Tf 6.3193 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.94 56.202 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.12 39.424 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.974 56.202 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.163 39.424 Td [(d')30(\266)469(el)24(\266)469(e-)]TJ -36.831 -1.3976 Td [(men)24(ts)-361(de)]TJ /F6 1 Tf 4.3193 0 Td [(S)]TJ /F2 1 Tf 1.0422 0 Td [(v)54(\266)469(eri\257an)24(t)-367(:)]TJ /F8 1 Tf 4.7349 0 Td [(A)]TJ /F6 1 Tf 1.0783 0 Td [(\275)]TJ /F9 1 Tf 1.1084 0.94578 Td [([)]TJ /F8 1 Tf 1.2771 -0.94578 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.017 54.202 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.271 38.027 Td [(et)]TJ /F8 1 Tf 1.1988 0 Td [(I)]TJ /F2 1 Tf 0.87952 0 Td [(d)24(\266)469(enom)24(brable.)-536(La)]TJ /F1 1 Tf 7.8976 0 Td [(taille)]TJ /F2 1 Tf 2.8554 0 Td [(du)-367(recouvremen)24(t)-367(est)-367(alors)]TJ -27.771 -1.7048 Td [(la)-475(somme)]TJ /F9 1 Tf 4.7349 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.509 51.762 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.964 36.322 Td [(\))-481(et)-475(on)-481(note)]TJ /F8 1 Tf 6.0723 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-475(la)-481(b)-24(orne)-481(inf)30(\266)469(erieure)-481(des)-481(tailles)-475(des)-481(recouvremen)24(ts)]TJ -16.452 -1.3434 Td [(d)30(\266)469(enom)24(brables)-331(de)]TJ /F8 1 Tf 7.6506 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)-445(Mon)24(trons)-337(que)]TJ /F8 1 Tf 7.0964 0 Td [(\271)]TJ /F2 1 Tf 0.93976 0 Td [(est)-337(une)-337(pseudo-mesure)-331(sur)]TJ /F8 1 Tf 11.867 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(.)]TJ /F1 1 Tf -28.392 -1.7169 Td [(1.)]TJ /F8 1 Tf 1.5964 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(\))-313(=)-313(0)]TJ /F1 1 Tf 2.2892 0 Td [(.)]TJ /F2 1 Tf 1.3193 0 Td [(En)-355(e\256et,)-361(la)-349(famille)-355(vide)-361(d')24(\266)475(el)18(\266)475(emen)24(ts)-355(de)]TJ /F6 1 Tf 16.946 0 Td [(S)]TJ /F2 1 Tf 1.0361 0 Td [(est)-355(un)-355(recouvremen)18(t)-355(de)-355(taille)-355(0)]TJ -25.494 -1.1988 Td [(de)-331(l'ensem)24(ble)-331(vide.)]TJ /F1 1 Tf 0.81325 -1.7169 Td [(2.)]TJ /F8 1 Tf 1.5301 0 Td [(\271)]TJ /F1 1 Tf 0.96988 0 Td [(est)-367(croissan)30(te.)]TJ /F2 1 Tf 8.0422 0 Td [(En)-319(e\256et,)-325(p)-30(our)]TJ /F8 1 Tf 6.3675 0 Td [(A)]TJ /F6 1 Tf 1.0241 0 Td [(\275)]TJ /F8 1 Tf 1.0602 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(,)-325(tout)-319(recouvremen)24(t)-325(d)30(\266)469(enom)24(brable)-319(de)]TJ /F8 1 Tf 16.193 0 Td [(B)]TJ /F2 1 Tf 1.1265 0 Td [(est)]TJ -37.934 -1.1988 Td [(un)-361(recouvremen)24(t)-361(d)24(\266)475(enom)24(brable)-361(de)]TJ /F8 1 Tf 15.03 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-367(il)-361(a)-355(une)-361(donc)-361(taille)-361(sup)0(\266)469(erieure)-361(ou)-331(\266)469(egale)-361(\265)500(a)]TJ /F8 1 Tf 18.916 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)-524(P)30(ar)]TJ -36.434 -1.2048 Td [(suite)]TJ /F8 1 Tf 2.3916 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\).)]TJ /F1 1 Tf -6.8494 -1.7108 Td [(3.)]TJ /F8 1 Tf 1.8193 0 Td [(\271)]TJ /F1 1 Tf 1.1386 0 Td [(est)-530(d)24(\266)548(enom)30(brablemen)30(t)-536(sous-additiv)30(e.)]TJ /F2 1 Tf 19.349 0 Td [(Consid)24(\266)475(erons)-469(\()]TJ /F8 1 Tf 6.1506 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.586 37.323 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.771 26.232 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.75 37.323 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.096 26.232 Td [(une)-463(famille)-463(de)]TJ -32.765 -1.4398 Td [(parties)-259(de)]TJ /F8 1 Tf 4.4759 0 Td [(X)]TJ /F2 1 Tf 1.1627 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1265 0 Td [(J)]TJ /F2 1 Tf 0.90964 0 Td [(d)30(\266)469(enom)24(brable,)-277(on)-265(doit)-259(prouv)24(er)]TJ /F8 1 Tf 13.042 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F9 1 Tf 0.38554 0.95181 Td [([)]TJ /F8 1 Tf 1.2831 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.026 35.262 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.488 24.792 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.897 35.262 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.988 24.792 Td [(\).)-421(On)-265(p)-30(eut)-265(toujours)]TJ -30.657 -1.6988 Td [(supp)-30(oser)]TJ /F9 1 Tf 4.2771 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.552 32.831 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.386 23.093 Td [(\))-457(\257ni)-457(\(sinon)-463(c'est)-427(\266)469(eviden)24(t\))-457(et)-457(donc)-457(c)18(haque)]TJ /F8 1 Tf 19.494 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.552 32.831 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.048 23.093 Td [(\))-457(\257ni.)-813(Comme)]TJ /F8 1 Tf 7.1145 0 Td [(J)]TJ /F2 1 Tf 1.1024 0 Td [(est)]TJ -37.934 -1.3494 Td [(d)30(\266)469(enom)24(brable)-475(on)-469(p)-30(eut)-475(aussi)-475(trouv)24(er)-475(une)-475(famille)-469(de)-475(r)30(\266)469(eels)]TJ /F8 1 Tf 25.584 0 Td [(")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.914 30.909 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.277 21.744 Td [(strictemen)24(t)-475(p)-30(ositifs)-475(don)24(t)-475(la)]TJ -26.946 -1.4337 Td [(somme)]TJ /F8 1 Tf 3.3373 0 Td [(")]TJ /F2 1 Tf 0.74699 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.957 28.849 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.307 20.31 Td [(soit)-331(\257nie.)]TJ -7.9759 -1.6506 Td [(P)30(our)]TJ /F8 1 Tf 2.4096 0 Td [(\270)-283(>)]TJ /F2 1 Tf 1.9157 0 Td [(0)-307(et)-313(c)24(haque)]TJ /F8 1 Tf 5.2651 0 Td [(j)]TJ /F6 1 Tf 0.74699 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(J)]TJ /F2 1 Tf 0.95783 0 Td [(on)-313(p)-24(eut)-313(trouv)24(er)-319(un)-307(recouvremen)18(t)-313(d)30(\266)469(enom)24(brable)-313(de)]TJ /F8 1 Tf 21.964 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.655 26.487 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.018 18.659 Td [(de)-307(taille)]TJ -35.687 -1.2048 Td [(inf)30(\266)469(erieure)-367(\265)500(a)]TJ /F8 1 Tf 5.3133 0 Td [(\270")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.733 24.762 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.355 17.454 Td [(+)]TJ /F8 1 Tf 1.0181 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.629 24.762 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.536 17.454 Td [(\);)-379(soit)-361(\()]TJ /F8 1 Tf 3.3554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.009 24.762 Td [(j)-8(;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.476 17.454 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.853 24.762 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 44.084 34.415 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.825 17.454 Td [(un)-361(tel)-367(recouvremen)24(t.)-530(Alors)-367(la)-361(famille)-361(des)]TJ /F8 1 Tf 18.5 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.112 24.762 Td [(j)-8(;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 17.454 Td [(est)]TJ -37.934 -1.5964 Td [(un)-397(recouvremen)18(t)-397(d)24(\266)475(enom)24(brable)-403(de)]TJ /F9 1 Tf 15.187 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.267 22.478 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.367 15.858 Td [(de)-397(taille)]TJ /F9 1 Tf 4.8012 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.112 20.987 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)25(;i)]TJ /F10 1 Tf 1.3448 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 60.157 29.126 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.614 15.858 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.603 22.478 Td [(j)-8(;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.464 15.858 Td [(\))-391(=)]TJ /F9 1 Tf 1.9398 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.681 20.987 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 37.012 16.967 Td [(\263)]TJ 0.81927 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.06 20.987 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 77.916 29.126 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.494 15.858 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.31 22.478 Td [(j)-8(;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.343 15.858 Td [(\))]TJ /F9 1 Tf 0.39156 1.1084 Td [(\264)]TJ /F11 1 Tf 0.98193 -1.1084 Td [(6)]TJ /F9 1 Tf -38.386 -1.6928 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.6466 17.202 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 6.9398 14.027 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(\270")]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.095 18.693 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1325 13.214 Td [(+)]TJ /F8 1 Tf 1.0361 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.043 18.693 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.331 13.214 Td [(\))]TJ /F9 1 Tf 0.38554 0.81325 Td [(\242)]TJ /F2 1 Tf 0.83132 -0.81325 Td [(=)]TJ /F8 1 Tf 1.1446 0 Td [(\270")]TJ /F2 1 Tf 1.3072 0 Td [(+)]TJ /F9 1 Tf 1.0361 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.397 17.202 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.651 13.214 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.181 18.693 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.813 13.214 Td [(\))-391(.)-608(On)-391(a)-385(donc)]TJ /F8 1 Tf 6.7169 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F9 1 Tf 0.38554 0.95181 Td [([)]TJ /F8 1 Tf 1.2831 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.716 18.693 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.97 13.214 Td [(\))]TJ /F11 1 Tf 0.75904 0 Td [(6)]TJ /F8 1 Tf 1.1446 0 Td [(\270")]TJ /F2 1 Tf 1.3072 0 Td [(+)]TJ /F9 1 Tf 1.0361 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.422 17.202 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.831 13.214 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.198 18.693 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.994 13.214 Td [(\))-391(Cela)-391(a)24(y)24(an)24(t)]TJ -33.663 -2.5843 Td [(lieu)-331(p)-30(our)-337(tout)]TJ /F8 1 Tf 6.4217 0 Td [(\270)-283(>)]TJ /F2 1 Tf 1.9157 0 Td [(0)-337(on)-331(conclut)]TJ /F8 1 Tf 5.7289 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F9 1 Tf 0.38554 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.078 15.004 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.837 10.629 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.164 13.504 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.175 10.629 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.948 15.004 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.337 10.629 Td [(\),)-337(comme)-337(annonc)24(\266)475(e.)]TJ -23.006 -2.9458 Td [(La)-433(pseudo-mesure)]TJ /F8 1 Tf 8.3916 0 Td [(\271)]TJ /F2 1 Tf 1.0361 0 Td [(p)-30(ermet)-433(donc)-433(de)-433(d)24(\266)475(e\257nir)-439(une)-433(tribu)]TJ /F6 1 Tf 15.247 0 Td [(T)]TJ /F2 1 Tf 1.2289 0 Td [(sur)-433(laquelle)]TJ /F8 1 Tf 5.5121 0 Td [(\271)]TJ /F2 1 Tf 1.0361 0 Td [(est)-433(une)-433(mesure)]TJ -32.452 -1.1988 Td [(compl)24(\265)475(ete.)-451(On)-331(a)-337(\265)500(a)-331(ce)-337(sujet)-337(le)-343(r)30(\266)469(esultat)-337(suiv)54(an)24(t)-337(:)]TJ 18.916 -2.5964 Td [(95)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 321 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 320 0 R /ProcSet 2 0 R >> /Contents 319 0 R >> endobj 320 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F13 39 0 R >> endobj 322 0 obj << 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[(p)-30(our)-313(qu'une)-307(partie)]TJ /F8 1 Tf 8.4337 0 Td [(A)]TJ /F12 1 Tf 1.0602 0 Td [(de)]TJ /F8 1 Tf 1.3072 0 Td [(X)]TJ /F12 1 Tf 1.2169 0 Td [(soit)]TJ /F6 1 Tf 1.8675 0 Td [(T)]TJ /F12 1 Tf 0.80121 0 Td [(-mesurable)-313(il)-307(faut)-313(et)-307(il)-313(su\261t)-313(que,)-319(p)-24(our)-313(tout)]TJ /F8 1 Tf 19.048 0 Td [(U)]TJ /F12 1 Tf 1.1024 0 Td [(dans)]TJ /F6 1 Tf -37.157 -1.2048 Td [(S)]TJ /F12 1 Tf 0.68072 0 Td [(,)-331(on)-337(ait)]TJ /F8 1 Tf 4.5 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.06 83.452 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.145 57.9 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(;)]TJ /F1 1 Tf -18.187 -1.5602 Td [(3.)]TJ /F12 1 Tf 1.5602 0 Td [(si)-337(les)-307(\266)469(el)24(\266)469(emen)30(ts)-337(de)]TJ /F6 1 Tf 7.8795 0 Td [(S)]TJ /F12 1 Tf 1.0181 0 Td [(son)24(t)]TJ /F6 1 Tf 2.1446 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurables)-331(alors)]TJ /F8 1 Tf 7.8554 0 Td [(\271)]TJ /F12 1 Tf 0.93976 0 Td [(est)-337(la)-331(mesure)-343(ext)30(\266)469(erieure)]TJ /F8 1 Tf 10.717 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.759 81.219 Td [(\244)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 40.12 56.34 Td [(.)]TJ /F1 1 Tf -34.789 -1.8133 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.5602 Td [(1.)]TJ /F2 1 Tf 1.6566 0 Td [(La)-385(famille)-379(r)24(\266)475(eduite)-385(\265)500(a)]TJ /F8 1 Tf 9.1205 0 Td [(U)]TJ /F2 1 Tf 1.1747 0 Td [(est)-385(un)-385(recouvremen)24(t)-385(d)24(\266)475(enom)24(brable)-385(de)]TJ /F8 1 Tf 16.729 0 Td [(U)]TJ /F2 1 Tf 1.1747 0 Td 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Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.98193 0 Td [(\\)]TJ /F8 1 Tf 0.90361 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-234(+)]TJ /F8 1 Tf 1.6325 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.819 70.719 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.59 49.009 Td [(\\)]TJ /F8 1 Tf 0.89759 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(\).)-500(Comme)-349(la)-355(sous-additivit)24(\266)475(e)-355(de)]TJ /F8 1 Tf 14.054 0 Td [(\271)]TJ /F2 1 Tf 0.95181 0 Td [(assure)-349(toujours)-355(l'in)24(\266)469(egalit)24(\266)475(e)]TJ /F8 1 Tf -27.964 -1.2048 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F2 1 Tf 0.80121 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.672 69.004 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.476 47.804 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\),)-337(l')30(\266)469(egalit)24(\266)469(e)-337(est)-337(en)-337(fait)-307(\266)469(equiv)54(alen)30(te)-337(\265)499(a)-337(l'in)24(\266)475(egalit)24(\266)469(e)-337(in)24(v)24(erse)]TJ /F8 1 Tf 0 -1.1988 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.017 67.28 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.795 46.605 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)]TJ -25.663 -1.5602 Td [(En)-277(particulier)-277(cette)-277(relation)-277(doit)-277(a)30(v)24(oir)-277(lieu)-277(p)-30(our)]TJ /F8 1 Tf 20.849 0 Td [(E)]TJ /F2 1 Tf 1.0723 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(U)]TJ /F6 1 Tf 1.0663 0 Td [(2)-283(S)]TJ /F2 1 Tf 1.9036 0 Td [(et)-277(alors)]TJ /F8 1 Tf 3.4458 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-277(en)30(tra)102(^)391(\263ne)]TJ /F8 1 Tf -34.97 -1.1988 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.95783 0 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-210(+)]TJ /F8 1 Tf 1.5843 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.56 63.331 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.886 43.846 Td [(\\)]TJ /F8 1 Tf 0.87349 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\).)-445(Mon)24(trons)-325(que,)-331(r)24(\266)475(ecipro)-30(quemen)24(t,)-331(cette)-331(ma)-54(joration)-331(p)-30(our)-331(les)]TJ -13.036 -1.1988 Td [(\266)469(el)24(\266)469(emen)30(ts)-337(de)]TJ /F6 1 Tf 5.4518 0 Td [(S)]TJ /F2 1 Tf 1.0181 0 Td [(en)30(tra)108(^)385(\263ne)-331(la)-337(ma)-60(joration)-337(pr)24(\266)475(ec)24(\266)469(eden)30(te)-337(p)-30(our)-337(toutes)-337(les)-337(parties)-331(de)]TJ /F8 1 Tf 26.542 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -33.892 -1.5602 Td [(P)30(ar)-349(d)24(\266)475(e\257nition)-349(de)]TJ /F8 1 Tf 7.6988 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-349(comme)-349(b)-30(orne)-349(inf)30(\266)469(erieure,)-349(cela)-355(revien)24(t)-349(\265)500(a)-349(prouv)30(er)-355(que)-349(tout)-349(recouvre-)]TJ -9.488 -1.2048 Td [(men)24(t)-234(d)24(\266)475(enom)24(brable)-240(de)]TJ /F8 1 Tf 9.488 0 Td [(E)]TJ /F2 1 Tf 1.0301 0 Td [(par)-240(des)-210(\266)475(el)24(\266)469(emen)24(ts)-234(de)]TJ /F6 1 Tf 8.5422 0 Td [(S)]TJ /F2 1 Tf 0.91566 0 Td [(a)-234(une)-234(taille)-240(sup)0(\266)469(erieure)-240(\265)500(a)]TJ /F8 1 Tf 10.386 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.77711 0 Td [(\\)]TJ /F8 1 Tf 0.69277 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-30(+)]TJ /F8 1 Tf 1.2169 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.974 57.668 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.337 39.882 Td [(\\)]TJ /F8 1 Tf 0.69277 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)]TJ -38.494 -1.3976 Td [(Or,)-325(p)-30(our)-331(un)-325(tel)-331(recouvremen)24(t)-331(\()]TJ /F8 1 Tf 13.62 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.095 54.857 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.964 38.485 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.129 54.857 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.639 38.485 Td [(,)-325(on)-325(a)]TJ /F8 1 Tf 2.8133 0 Td [(A)]TJ /F6 1 Tf 0.95783 0 Td [(\\)]TJ /F8 1 Tf 0.87349 0 Td [(E)]TJ /F6 1 Tf 1.0723 0 Td [(\275)]TJ /F9 1 Tf 1.0542 0.95181 Td [([)]TJ /F2 1 Tf 1.1145 -0.95181 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.95783 0 Td [(\\)]TJ /F8 1 Tf 0.8735 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.405 54.857 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.759 38.485 Td [(\))-331(et)-325(donc)-331(la)-325(sous-additivit)24(\266)475(e)]TJ -27.428 -1.6988 Td [(d)30(\266)469(enom)24(brable)-373(et)-373(la)-373(croissance)-379(de)]TJ /F8 1 Tf 14.404 0 Td [(\271)]TJ /F2 1 Tf 0.9759 0 Td [(impliquen)24(t)]TJ /F8 1 Tf 5.0422 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 1 0 Td [(\\)]TJ /F8 1 Tf 0.91566 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.72892 0 Td [(6)]TJ /F9 1 Tf 1.1205 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 1 0 Td [(\\)]TJ /F8 1 Tf 0.91566 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.233 52.426 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.831 36.786 Td [(\).)-566(De)-373(m)24(^)469(eme)-367(on)-373(a)]TJ /F8 1 Tf -31.5 -1.7048 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.121 50.797 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7048 35.081 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.698 50.797 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.096 35.081 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.853 49.987 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17 35.081 Td [(\))-337(et,)-337(en)-331(somman)24(t,)]TJ /F8 1 Tf -11.271 -1.6988 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.216 48.357 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.361 33.382 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.362 47.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.253 33.382 Td [(\))-222(+)]TJ /F9 1 Tf 1.6084 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.948 48.357 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.849 33.382 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.103 47.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.753 33.382 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ 1.6145 -0.14458 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.069 47.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.416 33.382 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.345 48.357 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.404 33.382 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.5 47.556 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.307 33.382 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F2 1 Tf 0.46386 -0.80723 Td [(.)]TJ -38.825 -1.759 Td [(Si)-295(l'on)-295(supp)-30(ose)]TJ /F8 1 Tf 6.747 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.89157 0 Td [(\\)]TJ /F8 1 Tf 0.80121 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.112 45.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.783 31.623 Td [(\))-144(+)]TJ /F8 1 Tf 1.4458 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.147 45.84 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.518 31.623 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.06 45.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.337 31.623 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.793 45.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.343 31.623 Td [(\))-295(p)-30(our)-295(c)24(haque)]TJ /F8 1 Tf 6.3012 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.259 45.038 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.657 31.623 Td [(,)-301(on)-295(obtien)24(t)-295(bien)-295(la)-289(relation)]TJ -27.325 -1.4036 Td [(souhait)24(\266)469(ee)]TJ /F8 1 Tf 4.3976 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F2 1 Tf 0.80121 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.94 43.84 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.361 30.22 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.819 43.03 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.663 30.22 Td [(\).)]TJ /F1 1 Tf -18.518 -1.759 Td [(3.)]TJ /F2 1 Tf 1.6446 0 Td [(P)24(our)]TJ /F8 1 Tf 2.4759 0 Td [(E)]TJ /F6 1 Tf 1.1506 0 Td [(\275)]TJ /F8 1 Tf 1.1265 0 Td [(X)]TJ /F2 1 Tf 1.2831 0 Td [(et)-379(\()]TJ /F8 1 Tf 1.5964 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.052 40.512 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.44 28.461 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.086 40.512 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.488 28.461 Td [(recouvremen)24(t)-379(d)24(\266)469(enom)30(brable)-379(de)]TJ /F8 1 Tf 13.614 0 Td [(E)]TJ /F2 1 Tf 1.1747 0 Td [(par)-379(des)-349(\266)469(el)24(\266)469(emen)30(ts)-379(de)]TJ /F6 1 Tf 9.1024 0 Td [(S)]TJ /F2 1 Tf 1.0602 0 Td [(on)]TJ -38.108 -1.3976 Td [(a)]TJ /F8 1 Tf 0.87349 0 Td [(E)]TJ /F6 1 Tf 1.1446 0 Td [(\275)]TJ /F9 1 Tf 1.1265 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.931 38.512 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.771 27.063 Td [(,)-385(d'o)-30(\265)530(u)]TJ /F8 1 Tf 2.9277 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.466 39.314 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.759 27.063 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.73494 0 Td [(6)]TJ /F9 1 Tf 1.1265 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.655 39.314 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.482 27.063 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.845 38.512 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.886 27.063 Td [(\).)-578(Alors,)-385(si)-379(les)-349(\266)469(el)24(\266)469(emen)30(ts)-379(de)]TJ /F6 1 Tf 12.265 0 Td [(S)]TJ /F2 1 Tf 1.0542 0 Td [(son)18(t)-373(mesurables)-385(on)-373(a)]TJ /F8 1 Tf -29.873 -1.7048 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4914 36.883 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.3916 25.358 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.681 36.081 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.7952 25.358 Td [(\))-409(=)]TJ /F8 1 Tf 1.9819 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.388 36.081 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.783 25.358 Td [(\))]TJ /F11 1 Tf 0.79518 0 Td [(6)]TJ /F8 1 Tf 1.1807 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.483 36.081 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.042 25.358 Td [(\),)-433(d'o)-30(\265)524(u)]TJ /F8 1 Tf 3.3976 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.681 36.883 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.506 25.358 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.79518 0 Td [(6)]TJ /F9 1 Tf 1.1867 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.966 36.081 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.56 25.358 Td [(\).)-680(La)-415(taille)-415(de)-409(tous)-409(les)-415(recouvremen)24(ts)]TJ -22.229 -1.3976 Td [(d)30(\266)469(enom)24(brables)-361(de)]TJ /F8 1 Tf 7.6988 0 Td [(E)]TJ /F2 1 Tf 1.1566 0 Td [(est)-367(donc)-361(sup)0(\266)469(erieure)-361(ou)-331(\266)469(egale)-367(\265)499(a)]TJ /F8 1 Tf 13.651 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.698 34.875 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.904 23.961 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\),)-373(c'est)-361(encore)-367(vrai)-361(de)-361(leur)-361(b)-30(orne)]TJ -24.753 -1.1988 Td [(inf)30(\266)469(erieure)]TJ /F8 1 Tf 4.5482 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\).)-831(Ainsi)-463(on)-463(a)]TJ /F8 1 Tf 6.6928 0 Td [(\271)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.138 33.159 Td [(\244)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 19.91 22.762 Td [(6)]TJ /F8 1 Tf 1.2711 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(,)-493(l'in)24(\266)469(egalit)24(\266)475(e)-469(in)24(v)24(erse)-457(a)-463(d)30(\266)469(ej)-6(\265)500(a)-433(\266)475(et)24(\266)469(e)-463(prouv)54(\266)469(ee)-463(dans)-457(le)-469(cas)]TJ -16.452 -1.2048 Td [(g)30(\266)469(en)30(\266)469(eral.)]TJ ET 512.64 256.5 m 519.84 256.5 l 519.84 264.92 l 512.64 264.92 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 18.967 Td [(Le)-367(probl)24(\265)475(eme)-367(qui)-361(se)-367(p)-30(ose)-367(alors)-367(est)-367(de)-361(sa)24(v)24(oir)-367(si)-367(la)-361(tribu)]TJ /F6 1 Tf 24.012 0 Td [(T)]TJ /F2 1 Tf 1.1626 0 Td [(con)24(tien)24(t)-367(l'ensem)30(ble)]TJ /F6 1 Tf 8.7108 0 Td [(S)]TJ /F2 1 Tf 1.0422 0 Td [(sur)-367(lequel)]TJ /F8 1 Tf -34.928 -1.1988 Td [(m)]TJ /F2 1 Tf 1.1928 0 Td [(\266)475(etait)-349(d)30(\266)469(e\257nie)-343(et)-343(si)]TJ /F8 1 Tf 7.741 0 Td [(\271)]TJ /F2 1 Tf 0.94578 0 Td [(est)-349(un)-343(prolongemen)24(t)-343(de)]TJ /F8 1 Tf 10.633 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-469(Un)-349(r)24(\266)475(esultat)-349(dans)-343(ce)-343(sens)-343(est)-349(donn)18(\266)475(e)-349(par)]TJ -21.392 -1.2048 Td [(la)-331(prop)-36(osition)-337(suiv)54(an)24(te)-331(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 11.364 Td [(VI)-35(I.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8193 13.623 Td [(5.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)18(oposition.)]TJ /F12 1 Tf 6.9398 0 Td [(Av)30(ec)-361(les)-355(notations)-355(pr)24(\266)469(ec)24(\266)475(eden)24(tes,)-361(si)]TJ /F6 1 Tf 15.084 0 Td [(S)]TJ /F12 1 Tf 1.0361 0 Td [(est)-361(un)-355(semi-anneau)-361(et)]TJ /F8 1 Tf 9.988 0 Td [(m)]TJ /F12 1 Tf -38.283 -1.1988 Td [(est)-337(d)24(\266)475(enom)24(brablemen)24(t)-337(additiv)30(e)-337(alors)]TJ /F6 1 Tf 15.91 0 Td [(T)]TJ /F12 1 Tf 1.1325 0 Td [(con)24(tien)24(t)]TJ /F6 1 Tf 3.8374 0 Td [(S)]TJ /F12 1 Tf 1.012 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(\271)]TJ /F12 1 Tf 0.93976 0 Td [(co)108(\304)391(\263ncide)-337(a)24(v)24(ec)]TJ /F8 1 Tf 6.0301 0 Td [(m)]TJ /F12 1 Tf 1.2108 0 Td [(sur)]TJ /F6 1 Tf 1.6747 0 Td [(S)]TJ /F12 1 Tf 0.68073 0 Td [(.)]TJ /F1 1 Tf -33.596 -1.8193 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.5602 Td [(1.)]TJ /F6 1 Tf 1.5602 0 Td [(T)-536(\276)-277(S)]TJ /F1 1 Tf 2.8133 0 Td [(.)]TJ /F2 1 Tf -5.1867 -1.5542 Td [(Soit)]TJ /F8 1 Tf 2.0422 0 Td [(A)]TJ /F2 1 Tf 1.0663 0 Td [(appartenan)24(t)-325(\265)500(a)]TJ /F6 1 Tf 6.4518 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(,)-319(prouv)24(ons)]TJ /F8 1 Tf 4.8735 0 Td [(A)]TJ /F6 1 Tf 1.0241 0 Td [(2)-283(T)]TJ /F2 1 Tf 1.747 0 Td [(.)-439(Comme)-325(nous)-319(v)24(enons)-325(de)-319(le)-325(v)24(oir)-319(dans)-319(la)-319(prop)-30(osi-)]TJ -17.886 -1.2048 Td [(tion)-331(pr)24(\266)475(ec)24(\266)469(eden)24(te)-331(il)-337(su\261t)-337(de)-337(prouv)30(er)]TJ /F8 1 Tf 15.434 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.741 9.59 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.398 6.2858 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-337(p)-30(our)]TJ /F8 1 Tf 3.0904 0 Td [(U)]TJ /F2 1 Tf 1.1265 0 Td [(dans)]TJ /F6 1 Tf 2.3374 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)]TJ /F5 1 Tf -35.765 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(96)]TJ ET endstream endobj 324 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 323 0 R /ProcSet 2 0 R >> /Contents 322 0 R >> endobj 323 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 325 0 obj << /Length 28194 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4036 Td [(Lorsque)]TJ /F6 1 Tf 3.7349 0 Td [(S)]TJ /F2 1 Tf 0.9759 0 Td [(est)-307(un)-295(semi-anneau,)]TJ /F8 1 Tf 8.9157 0 Td [(A)]TJ /F6 1 Tf 0.89759 0 Td [(\\)]TJ /F8 1 Tf 0.81928 0 Td [(U)]TJ /F2 1 Tf 1.0904 0 Td [(appartien)30(t)-301(\265)499(a)]TJ /F6 1 Tf 5.6265 0 Td [(S)]TJ /F2 1 Tf 0.9759 0 Td [(et)]TJ /F8 1 Tf 1.1325 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.284 89.633 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.807 62.22 Td [(\\)]TJ /F8 1 Tf 0.81325 0 Td [(U)]TJ /F2 1 Tf 1.0904 0 Td [(est)-301(la)-301(r)30(\266)469(eunion)-301(d'une)-301(famille)]TJ -27.38 -1.1988 Td [(\257nie)-319(d')24(\266)475(el)18(\266)475(emen)24(ts)]TJ /F8 1 Tf 7.0602 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.707 87.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.723 61.021 Td [(de)]TJ /F6 1 Tf 1.3193 0 Td [(S)]TJ /F2 1 Tf 1 0 Td [(qui)-319(formen)30(t)-325(ainsi)-319(un)-319(recouvremen)24(t)-325(d)30(\266)469(enom)24(brable)-319(de)]TJ /F8 1 Tf 22.584 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.353 87.909 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.976 61.021 Td [(\\)]TJ /F8 1 Tf 0.85542 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(.)-439(On)-319(a)]TJ -36.295 -1.3434 Td [(donc)]TJ /F8 1 Tf 2.3855 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-337(et)]TJ /F8 1 Tf 1.8855 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.543 85.987 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 21.271 59.677 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.414 85.185 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.572 59.677 Td [(\).)]TJ -23.241 -1.8012 Td [(P)30(ar)-385(ailleurs)]TJ /F8 1 Tf 5.4277 0 Td [(U)]TJ /F2 1 Tf 1.1747 0 Td [(est)-385(la)-385(r)30(\266)469(eunion)-385(disjoin)24(te)-385(de)]TJ /F8 1 Tf 11.873 0 Td [(A)]TJ /F6 1 Tf 1.006 0 Td [(\\)]TJ /F8 1 Tf 0.92169 0 Td [(U)]TJ /F2 1 Tf 1.1747 0 Td [(et)-385(des)]TJ /F8 1 Tf 2.994 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.767 82.607 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.916 57.876 Td [(,)-397(l'additivit)30(\266)469(e)-385(de)]TJ /F8 1 Tf 7.1928 0 Td [(m)]TJ /F2 1 Tf 1.259 0 Td [(assure)-379(alors)]TJ /F8 1 Tf -34.036 -1.3494 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.89157 0 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-144(+)]TJ /F9 1 Tf 1.4458 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.586 80.685 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.211 56.527 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.88554 0 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-138(+)]TJ /F8 1 Tf 1.4458 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.069 81.487 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.151 56.527 Td [(\\)]TJ /F8 1 Tf 0.80723 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\),)-307(ce)-295(qui)-289(donne)-295(la)-295(relation)-295(souhait)24(\266)475(ee.)]TJ -23.416 -1.8554 Td [(Remarquons)-331(que)-337(l')24(\266)469(egalit)24(\266)469(e)-337(dans)-331(la)-337(relation)-331(pr)24(\266)475(ec)24(\266)469(eden)24(te)-331(implique)]TJ /F8 1 Tf 27.548 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78915 0 Td [(\).)]TJ /F1 1 Tf -34.434 -1.6506 Td [(2.)]TJ /F8 1 Tf 1.5602 0 Td [(\271)]TJ /F1 1 Tf 0.98795 0 Td [(co)126(\304)445(\263ncide)-391(a)30(v)24(ec)]TJ /F8 1 Tf 6.9518 0 Td [(m)]TJ /F1 1 Tf 1.259 0 Td [(sur)]TJ /F6 1 Tf 1.9518 0 Td [(S)]TJ /F1 1 Tf 0.68072 0 Td [(.)]TJ /F2 1 Tf -14.205 -1.6566 Td [(On)-337(a)-331(d)24(\266)475(ej)-6(\265)499(a)-331(observ)54(\266)469(e)]TJ /F8 1 Tf 8.2048 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-337(p)-30(our)]TJ /F8 1 Tf 3.0904 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-277(S)]TJ /F2 1 Tf 1.6205 0 Td [(.)]TJ -19.428 -1.6566 Td [(P)30(our)-385(prouv)30(er)-385(l'in)24(\266)469(egalit)24(\266)469(e)-385(in)24(v)24(erse)-379(nous)-379(dev)24(ons)-385(mon)24(trer)-385(que)-385(si)-385(\()]TJ /F8 1 Tf 27.096 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.388 70.918 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.446 49.707 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.422 70.918 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.494 49.707 Td [(est)-391(un)-379(recouvremen)18(t)]TJ -30.163 -1.4036 Td [(d)30(\266)469(enom)24(brable)-409(de)]TJ /F8 1 Tf 7.4036 0 Td [(A)]TJ /F2 1 Tf 1.1566 0 Td [(alors)]TJ /F8 1 Tf 2.4759 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F11 1 Tf 0.78916 0 Td [(6)]TJ /F9 1 Tf 1.1807 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.224 68.909 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.247 48.304 Td [(\).)-674(Quitte)-409(\265)500(a)-409(a)-54(jouter)-409(des)-415(parties)-409(vides)-409(on)-409(p)-30(eut)]TJ -18.916 -1.3434 Td [(supp)-30(oser)-331(que)]TJ /F8 1 Tf 6.0181 0 Td [(I)]TJ /F2 1 Tf 0.8494 0 Td [(est)-337(in\257ni)-337(et,)-337(apr)30(\265)469(es)-337(un)-331(c)18(hangemen)24(t)-337(d'indices,)-337(on)-331(p)-30(eut)-337(supp)-30(oser)]TJ /F8 1 Tf 27.38 0 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F3 1 Tf 1.0602 0 Td [(N)]TJ /F2 1 Tf 0.71687 0 Td [(.)]TJ -36.819 -1.6566 Td [(Notons)-343(alors)]TJ /F8 1 Tf 5.8494 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.828 64.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.651 45.304 Td [(la)-349(r)30(\266)469(eunion)-349(des)]TJ /F8 1 Tf 6.494 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.379 64.616 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.669 45.304 Td [(p)-30(our)]TJ /F8 1 Tf 2.3795 0 Td [(k)]TJ /F11 1 Tf 0.8494 0 Td [(6)]TJ /F8 1 Tf 1.0783 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(.)-481(Les)]TJ /F8 1 Tf 2.5723 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.121 64.616 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.62 45.304 Td [(formen)24(t)-349(une)-349(suite)-349(croissan)24(te)-349(qui)-343(a)]TJ -24.289 -1.1988 Td [(la)-337(m)24(^)475(eme)-337(r)30(\266)469(eunion)-343(que)-337(les)]TJ /F8 1 Tf 10.946 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.267 62.9 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.5 44.105 Td [(.)-457(Nous)-337(allons)-337(tout)-337(d'ab)-30(ord)-343(prouv)30(er,)-343(par)-337(r)24(\266)475(ecurrence)-343(sur)]TJ /F8 1 Tf 24.253 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(,)-337(que)]TJ /F8 1 Tf -37.018 -1.1988 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6034 61.185 Td [(p)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6867 42.906 Td [(\\)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.216 61.185 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0783 42.906 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.552 61.987 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.205 42.906 Td [(p)-30(eut)-331(s')24(\266)475(ecrire)-331(comme)-331(r)24(\266)475(eunion)-331(disjoin)24(te)-331(d'un)-331(nom)24(bre)-331(\257ni)-331(d')30(\266)469(el)24(\266)469(emen)24(ts)-325(de)]TJ /F6 1 Tf 30.506 0 Td [(S)]TJ /F2 1 Tf 1.0121 0 Td [(et)-331(ceci)]TJ -36.392 -1.2048 Td [(p)-30(our)-331(un)-337(en)24(tier)]TJ /F8 1 Tf 6.6205 0 Td [(p)]TJ /F2 1 Tf 0.83735 0 Td [(arbitraire.)]TJ -7.4578 -1.6506 Td [(Lorsque)]TJ /F8 1 Tf 3.7711 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)-277(0,)]TJ /F8 1 Tf 2.1687 0 Td [(V)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.224 57.099 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.458 40.051 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(U)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 21.75 57.099 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.928 40.051 Td [(2)-283(S)]TJ /F2 1 Tf 1.9578 0 Td [(et)-337(cela)-337(r)30(\266)469(esulte)-331(du)-337(fait)-337(que)]TJ /F6 1 Tf 11.512 0 Td [(S)]TJ /F2 1 Tf 1.0181 0 Td [(est)-337(un)-337(semi-anneau.)]TJ -25.084 -1.6566 Td [(Si)-397(l'on)-403(supp)-30(ose)-403(que)]TJ /F8 1 Tf 9 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.483 54.728 Td [(p)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.741 38.394 Td [(\\)]TJ /F2 1 Tf 0.93374 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.259 54.728 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.193 38.394 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.595 55.53 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.386 38.394 Td [(est)-409(la)-397(r)30(\266)469(eunion)-403(des)]TJ /F8 1 Tf 8.2831 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.681 54.728 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.223 38.394 Td [(\266)475(el)18(\266)475(emen)24(ts)-397(de)]TJ /F6 1 Tf 5.5843 0 Td [(S)]TJ /F2 1 Tf 1.0783 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.4277 0 Td [(j)]TJ /F2 1 Tf 0.87349 0 Td [(appartenan)24(t)]TJ -33.855 -1.2048 Td [(\265)499(a)-403(un)-403(ensem)24(ble)-403(\257ni)]TJ /F8 1 Tf 8.5361 0 Td [(J)]TJ /F2 1 Tf 1.0542 0 Td [(alors)]TJ /F8 1 Tf 2.4639 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.862 53.012 Td [(p)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.801 37.189 Td [(\\)]TJ /F2 1 Tf 0.93373 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.638 53.012 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.265 37.189 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.422 53.814 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.458 37.189 Td [(=)]TJ /F8 1 Tf 1.1687 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.224 53.012 Td [(p)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.042 37.189 Td [(\\)]TJ /F2 1 Tf 0.93373 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40 53.012 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.494 37.189 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.336 53.814 Td [(c)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.56 37.189 Td [(\\)]TJ /F2 1 Tf 0.93373 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.172 53.012 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.127 37.189 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.957 53.814 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.325 37.189 Td [(est)-409(la)-403(r)30(\266)469(eunion)-403(disjoin)24(te)]TJ -28.994 -1.1988 Td [(des)]TJ /F8 1 Tf 1.741 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.207 51.288 Td [(j)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4819 35.991 Td [(\\)]TJ /F2 1 Tf 0.89759 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.957 51.288 Td [(n)]TJ /F4 1 Tf 0.7069 0 Td [(+1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.012 35.991 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.741 52.099 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.151 35.991 Td [(don)24(t)-349(c)18(hacun)-349(est)-349(r)24(\266)475(eunion)-349(disjoin)24(te)-349(d'un)-349(nom)24(bre)-349(\257ni)-349(d')24(\266)475(el)18(\266)475(emen)24(ts)-349(de)]TJ /F6 1 Tf 28.831 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-487(Le)]TJ -37.331 -1.1988 Td [(r)30(\266)469(esultat)-337(annonc)24(\266)469(e)-337(en)-331(d)24(\266)469(ecoule.)]TJ 0 -1.6566 Td [(En)-379(particulier,)-391(p)-30(our)-379(c)24(haque)]TJ /F8 1 Tf 12.578 0 Td [(n)]TJ /F11 1 Tf 0.95181 0 Td [(>)]TJ /F2 1 Tf 1.1325 0 Td [(1,)]TJ /F8 1 Tf 1.1687 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.121 47.202 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.542 33.135 Td [(\\)]TJ /F2 1 Tf 0.91566 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.966 47.202 Td [(n)]TJ /F10 1 Tf 0.70689 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26 33.135 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.767 48.012 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.151 33.135 Td [(=)]TJ /F8 1 Tf 1.1325 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.448 47.202 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.759 33.135 Td [(\\)]TJ /F2 1 Tf 0.92169 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.302 47.202 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.223 33.135 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.103 48.012 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.398 33.135 Td [(est)-385(la)-379(r)30(\266)469(eunion)-379(disjoin)24(te)]TJ -29.066 -1.1988 Td [(d'une)-337(famille)-337(\257nie)-331(\()]TJ /F8 1 Tf 8.5301 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.922 45.487 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.687 31.936 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23 45.487 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 34.506 63.367 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.127 31.936 Td [(d')24(\266)475(el)18(\266)475(emen)24(ts)-331(de)]TJ /F6 1 Tf 6.259 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-445(P)30(osons)-307(\266)469(egalemen)24(t)]TJ /F8 1 Tf 8.753 0 Td [(I)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.026 45.487 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.982 31.936 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(0)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.83133 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(B)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.957 45.487 Td [(0)]TJ /F7 1 Tf 0.56897 0 Td [(;)]TJ /F4 1 Tf 0.33621 0 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.163 31.936 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(V)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 61.25 45.487 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.253 31.936 Td [(.)]TJ -37.922 -1.6566 Td [(Alors)-253(les)]TJ /F8 1 Tf 3.9398 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.103 43.116 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.458 30.28 Td [(\\)]TJ /F2 1 Tf 0.72892 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.405 43.116 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.729 30.28 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.207 43.918 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.783 30.28 Td [(son)24(t)-259(clairemen)30(t)-259(deux)-253(\265)500(a)-253(deux)-253(disjoin)24(ts)-253(et)-259(son)24(t)-253(disjoin)24(ts)-253(de)]TJ /F8 1 Tf 24.44 0 Td [(V)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.966 43.116 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.259 30.28 Td [(.)-415(Les)]TJ /F8 1 Tf 2.4096 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.147 43.116 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 29.081 Td [(son)24(t)-343(donc)-337(des)-313(\266)469(el)24(\266)469(emen)30(ts)-343(de)]TJ /F6 1 Tf 11.711 0 Td [(S)]TJ /F2 1 Tf 1.0181 0 Td [(deux)-337(\265)500(a)-337(deux)-337(disjoin)24(ts.)-457(P)24(ar)-337(ailleurs,)-337(c)24(haque)]TJ /F8 1 Tf 19.006 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.879 41.4 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.53 29.081 Td [(est)-343(la)-337(r)24(\266)475(eunion)]TJ -33.199 -1.2048 Td [(des)]TJ /F8 1 Tf 1.7349 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.198 39.676 Td [(p;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1566 27.876 Td [(p)-24(our)]TJ /F8 1 Tf 2.3735 0 Td [(p)]TJ /F11 1 Tf 0.79518 0 Td [(6)]TJ /F8 1 Tf 1.0783 0 Td [(n)]TJ /F2 1 Tf 0.93976 0 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(i)]TJ /F6 1 Tf 0.63855 0 Td [(2)]TJ /F8 1 Tf 0.96386 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.138 39.676 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.024 27.876 Td [(.)-475(P)24(ar)-343(suite)]TJ /F8 1 Tf 5.0482 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34 39.676 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.301 27.876 Td [(,)-349(con)30(ten)24(u)-349(dans)]TJ /F8 1 Tf 6.7108 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.216 39.676 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.139 27.876 Td [(,)-349(est)-349(la)-343(r)30(\266)469(eunion)-349(disjoin)30(te)-349(des)]TJ /F8 1 Tf -26.807 -1.1988 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7155 37.961 Td [(p;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.2831 26.677 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.664 37.961 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7229 26.677 Td [(p)-30(our)]TJ /F8 1 Tf 2.3614 0 Td [(p)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.93374 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(i)]TJ /F6 1 Tf 0.62048 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.793 37.961 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.813 26.677 Td [(qui)-331(son)24(t)-331(des)-307(\266)469(el)24(\266)469(emen)30(ts)-331(de)]TJ /F6 1 Tf 10.976 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)-445(La)-331(propri)24(\266)469(et)24(\266)475(e)-337(d'additivit)30(\266)469(e)-337(de)]TJ /F8 1 Tf 13.145 0 Td [(m)]TJ /F2 1 Tf -38.283 -1.1988 Td [(assure)-349(donc)-349(que)]TJ /F8 1 Tf 7.3133 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.888 36.245 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.139 25.479 Td [(\))-355(est)-355(la)-349(somme)-349(des)]TJ /F8 1 Tf 8.5482 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.793 36.245 Td [(p;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.928 25.479 Td [(\\)]TJ /F8 1 Tf 0.89759 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.802 36.245 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.054 25.479 Td [(\))-355(p)-30(our)]TJ /F8 1 Tf 3.1205 0 Td [(p)]TJ /F11 1 Tf 0.80723 0 Td [(6)]TJ /F8 1 Tf 1.0843 0 Td [(n)]TJ /F2 1 Tf 0.95181 0 Td [(et)]TJ /F8 1 Tf 1.1807 0 Td [(i)]TJ /F6 1 Tf 0.6506 0 Td [(2)]TJ /F8 1 Tf 0.96988 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.75 36.245 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.723 25.479 Td [(.)-493(Cep)-24(endan)24(t,)]TJ -33.392 -1.2048 Td [(les)]TJ /F8 1 Tf 1.5181 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.888 34.521 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0723 24.274 Td [(p)-30(our)]TJ /F8 1 Tf 2.4277 0 Td [(i)]TJ /F6 1 Tf 0.73494 0 Td [(2)]TJ /F8 1 Tf 1.0542 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.647 34.521 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.675 24.274 Td [(son)24(t)-403(con)30(ten)24(us)-409(dans)]TJ /F8 1 Tf 8.8012 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.569 34.521 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.102 24.274 Td [(et)-403(donc)-403(la)-397(somme)-403(pr)30(\266)469(ec)24(\266)469(eden)30(te)-403(fait)-403(in)30(terv)24(enir)]TJ -19.771 -1.4337 Td [(tous)-307(les)]TJ /F8 1 Tf 3.5723 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.638 32.469 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.994 22.84 Td [(\))-313(p)-30(our)]TJ /F8 1 Tf 3.0422 0 Td [(i)]TJ /F6 1 Tf 0.62048 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.388 32.469 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.584 22.84 Td [(.)-439(On)-307(a)-313(donc)]TJ /F8 1 Tf 5.5301 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.871 32.469 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.614 22.84 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(>)]TJ /F9 1 Tf 1.1687 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.121 30.969 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 57.024 43.09 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.169 22.84 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.638 32.469 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.259 22.84 Td [(\).)-439(On)-313(en)-307(d)24(\266)469(eduit)-307(que)-313(la)-307(taille)]TJ -26.928 -2.5482 Td [(du)-337(recouvremen)24(t)]TJ /F9 1 Tf 7.6747 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.707 28.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.114 20.292 Td [(\))-337(est)-337(minor)24(\266)469(ee)-337(par)-337(la)-331(somme)-337(de)-331(tous)-337(les)]TJ /F8 1 Tf 17.259 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.086 28.823 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.464 20.292 Td [(\).)]TJ -32.133 -2.1265 Td [(Mais)-445(alors)-445(les)]TJ /F8 1 Tf 6.5904 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.147 25.78 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.187 18.165 Td [(on)24(t)-439(la)-445(m)24(^)475(eme)-439(r)24(\266)475(eunion)-445(que)-445(les)]TJ /F8 1 Tf 13.325 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.207 25.78 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.639 18.165 Td [(,)-469(donc)-439(que)-445(les)]TJ /F8 1 Tf 6.7651 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.647 25.78 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.633 18.165 Td [(,)-469(donc)-439(recouvren)18(t)]TJ /F8 1 Tf -31.301 -1.1988 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(.)-668(Ainsi)]TJ /F8 1 Tf 3.6084 0 Td [(A)]TJ /F2 1 Tf 1.1566 0 Td [(est)-415(la)-409(r)30(\266)469(eunion)-409(disjoin)24(te)-409(des)]TJ /F8 1 Tf 12.398 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.345 24.064 Td [(n;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.337 16.967 Td [(\\)]TJ /F8 1 Tf 0.93373 0 Td [(A)]TJ /F2 1 Tf 1.1627 0 Td [(qui)-403(appartiennen)24(t)-415(\265)500(a)]TJ /F6 1 Tf 9.1687 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(,)-427(donc)]TJ /F8 1 Tf 3.1687 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-409(est)]TJ -37.139 -1.2048 Td [(la)-409(somme)-409(des)]TJ /F8 1 Tf 6.4096 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.698 22.34 Td [(n;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.108 15.762 Td [(\\)]TJ /F8 1 Tf 0.93976 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\).)-680(La)-415(remarque)-415(faite)-409(\265)500(a)-409(la)-415(\257n)-409(de)-409(la)-415(preuv)30(e)-415(du)-415(p)-30(oin)24(t)]TJ /F1 1 Tf 23.446 0 Td [(1.)]TJ /F2 1 Tf 1.5723 0 Td [(assure)]TJ -36.482 -1.1988 Td [(alors)]TJ /F8 1 Tf 2.4578 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.043 20.625 Td [(n;i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.139 14.563 Td [(\\)]TJ /F8 1 Tf 0.93373 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F11 1 Tf 0.76506 0 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24 20.625 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.831 14.563 Td [(\).)-632(En)-391(somman)24(t)-397(toutes)-397(ces)-391(relations)-397(on)-397(obtien)30(t)-397(les)-397(in)30(\266)469(egalit)24(\266)469(es)]TJ /F8 1 Tf -12.5 -1.4337 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F11 1 Tf 0.68072 0 Td [(6)]TJ /F9 1 Tf 1.0723 0.94578 Td [(X)]TJ /F8 1 Tf 1.6084 -0.94578 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.233 18.573 Td [(n;i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.807 13.129 Td [(\))]TJ /F11 1 Tf 0.68072 0 Td [(6)]TJ /F9 1 Tf 1.0723 0.94578 Td [(X)]TJ /F8 1 Tf 1.6084 -0.94578 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.362 18.573 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.669 13.129 Td [(\).)-475(La)-349(taille)-343(du)-349(recouvremen)24(t)-349(est)-343(bien)-349(minor)24(\266)469(ee)-343(par)]TJ /F8 1 Tf 22.139 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74698 0 Td [(\),)]TJ -38.494 -1.3494 Td [(ce)-337(qui)-331(ac)24(h)24(\265)469(ev)24(e)-337(la)-331(d)24(\266)475(emonstration.)]TJ ET 512.64 139.64 m 519.84 139.64 l 519.84 148.06 l 512.64 148.06 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 9.4726 Td [(Nous)-259(p)-30(ouv)18(ons)-265(alors)-265(d)30(\266)469(emon)24(trer)-265(le)-265(th)24(\266)469(eor)24(\265)469(eme)-259(de)-259(Carath)18(\266)475(eo)-36(dory)-259(don)24(t)-265(nous)-259(rapp)-30(elons)-265(l')24(\266)469(enonc)30(\266)469(e)-265(:)]TJ /F15 1 Tf 0 -1.9819 Td [(Th)]TJ 1.4036 0.078313 Td [(\266)]TJ -0.0060239 -0.078313 Td [(eor)]TJ 1.8012 0.078313 Td [(\265)]TJ -0.0060244 -0.078313 Td [(eme.)]TJ /F12 1 Tf 2.7831 0 Td [(\()]TJ /F1 1 Tf 0.38554 0 Td [(de)-427(Carath)24(\266)548(eo)-36(dory)]TJ /F12 1 Tf 8.5301 0 Td [(\))-367(Soit)]TJ /F8 1 Tf 2.8434 0 Td [(m)]TJ /F12 1 Tf 1.241 0 Td [(une)-367(pr)24(\266)469(e-mesure)-361(d)24(\266)469(e\257nie)-367(sur)-367(un)-367(semi-anneau)]TJ /F6 1 Tf 19.229 0 Td [(S)]TJ /F12 1 Tf 0.68072 0 Td [(.)]TJ -38.886 -1.2048 Td [(Alors)]TJ /F8 1 Tf 2.6446 0 Td [(m)]TJ /F12 1 Tf 1.2169 0 Td [(p)-24(eut)-337(se)-337(prolonger)-337(en)-337(une)-331(mesure)-343(d)30(\266)469(e\257nie)-337(sur)-337(la)-331(tribu)-337(engendr)24(\266)475(ee)-337(par)]TJ /F6 1 Tf 29.464 0 Td [(S)]TJ /F12 1 Tf 0.68073 0 Td [(.)]TJ /F2 1 Tf -15.09 -2.3976 Td [(97)]TJ /F5 1 Tf 6.259 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 327 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 326 0 R /ProcSet 2 0 R >> /Contents 325 0 R >> endobj 326 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 328 0 obj << /Length 18417 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F12 1 Tf 0 -2.4036 Td [(En)-234(outre)-240(il)-234(n'y)-240(a)-234(qu'un)-234(tel)-240(prolongemen)24(t)-234(lorsque)]TJ /F8 1 Tf 20.657 0 Td [(X)]TJ /F12 1 Tf 1.1446 0 Td [(est)-240(la)-234(r)30(\266)469(eunion)-240(d'une)-240(famille)-234(d)24(\266)469(enom)24(brable)]TJ -21.801 -1.1988 Td [(d')24(\266)475(el)24(\266)469(emen)24(ts)-331(de)]TJ /F6 1 Tf 6.259 0 Td [(S)]TJ /F12 1 Tf 1.012 0 Td [(de)-337(pr)30(\266)469(e-mesure)-331(\257nie.)]TJ /F1 1 Tf -7.2711 -1.8614 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Appliquons)-349(la)-355(construction)-355(pr)24(\266)475(ec)24(\266)469(eden)24(te.)-500(Nous)-355(obtenons)-361(une)-355(mesure)]TJ /F8 1 Tf 29.741 0 Td [(\271)]TJ /F2 1 Tf -38.56 -1.2048 Td [(sur)-283(une)-277(tribu)]TJ /F6 1 Tf 5.9036 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(.)-427(Comme)]TJ /F6 1 Tf 4.3193 0 Td [(S)]TJ /F2 1 Tf 0.95783 0 Td [(est)-283(un)-283(semi-anneau)-283(et)]TJ /F8 1 Tf 9.6807 0 Td [(m)]TJ /F2 1 Tf 1.1566 0 Td [(est)-283(d)24(\266)475(enom)24(brablemen)24(t)-283(additiv)30(e)-283(on)-277(v)24(oit)]TJ -22.813 -1.1988 Td [(que)]TJ /F6 1 Tf 1.8855 0 Td [(T)]TJ /F2 1 Tf 1.1566 0 Td [(con)30(tien)24(t)]TJ /F6 1 Tf 3.8554 0 Td [(S)]TJ /F2 1 Tf 1.0361 0 Td [(donc)-361(la)-355(tribu)-361(engendr)24(\266)475(ee)-361(par)]TJ /F6 1 Tf 12.578 0 Td [(S)]TJ /F2 1 Tf 1.0361 0 Td [(et)-361(co)108(\304)385(\263ncide)-355(a)24(v)24(ec)]TJ /F8 1 Tf 7.2711 0 Td [(m)]TJ /F2 1 Tf 1.2349 0 Td [(sur)]TJ /F6 1 Tf 1.6988 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(.)-512(La)-361(restriction)]TJ -32.434 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(\271)]TJ /F2 1 Tf 0.93976 0 Td [(\265)500(a)-331(la)-337(tribu)-331(engendr)24(\266)469(ee)-337(par)]TJ /F6 1 Tf 10.898 0 Td [(S)]TJ /F2 1 Tf 1.0181 0 Td [(est)-337(donc)-337(une)-337(mesure)-337(qui)-331(prolonge)]TJ /F8 1 Tf 14.994 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ -30.06 -1.5843 Td [(Rapp)-30(elons)-325(que)-319(l'unicit)24(\266)469(e)-319(lorsque)]TJ /F8 1 Tf 13.94 0 Td [(X)]TJ /F2 1 Tf 1.2229 0 Td [(est)-325(la)-319(r)30(\266)469(eunion)-319(d'une)-325(famille)-319(d)24(\266)475(enom)24(brable)-319(d')24(\266)469(el)24(\266)469(emen)30(ts)-319(de)]TJ /F6 1 Tf -15.163 -1.1988 Td [(S)]TJ /F2 1 Tf 1.012 0 Td [(de)-337(pr)30(\266)469(e-mesure)-331(\257nie)-337(r)24(\266)475(esulte)-337(de)-331(la)-337(prop)-30(osition)]TJ /F1 1 Tf 19.566 0 Td [(I)-36(I)-30(I.4.5.)]TJ /F2 1 Tf 3.4819 0 Td [(.)]TJ ET 512.64 629.53 m 519.84 629.53 l 519.84 637.95 l 512.64 637.95 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 40.344 Td [(VI)-35(I.2.)-236(Application)-371(\265)552(a)-376(la)-376(mesure)-376(de)-381(Leb)-35(esgue.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 46.78 Td [(Le)-265(th)24(\266)469(eor)24(\265)475(eme)-265(de)-259(Carath)18(\266)475(eo)-36(dory)-259(p)-30(eut-)30(^)469(etre)-265(utilis)24(\266)475(e)-265(en)-265(prenan)24(t)-265(p)-30(our)]TJ /F6 1 Tf 27.428 0 Td [(S)]TJ /F2 1 Tf 0.93976 0 Td [(l'ensem)30(ble)-265(des)-265(in)24(terv)54(alles)]TJ -28.367 -1.1988 Td [(de)]TJ /F3 1 Tf 1.4578 0 Td [(R)]TJ /F2 1 Tf 1.1867 0 Td [(et)-457(p)-30(our)]TJ /F8 1 Tf 3.7831 0 Td [(m)]TJ /F2 1 Tf 1.3373 0 Td [(la)-463(longueur)-463(p)-30(ourvu)-463(que)-463(l'on)-463(prouv)24(e)-463(que)-463(celle-ci)-463(est)-463(d)24(\266)475(enom)24(brablemen)24(t)]TJ -7.7651 -1.1988 Td [(additiv)30(e)-355(\(ce)-343(p)-30(oin)24(t)-349(est)-355(loin)-349(d')30(^)469(etre)-319(\266)469(eviden)24(t,)-355(nous)-349(in)30(vitons)-349(d'ailleurs)-349(le)-349(lecteur)-349(\265)499(a)-349(essa)24(y)24(er)-355(d'en)]TJ 0 -1.2048 Td [(fournir)-301(une)-307(preuv)30(e\).)-439(On)-301(en)-301(d)24(\266)469(eduit)-301(l'existence)-307(et)-301(l'unicit)24(\266)475(e)-307(d'une)-307(mesure)-307(sur)-307(les)-301(b)-30(or)24(\266)475(eliens)-307(de)]TJ /F3 1 Tf 0 -1.1988 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(qui)-331(prolonge)-337(la)-337(longueur,)-331(c'est)-343(la)-331(mesure)-337(de)-337(Borel.)]TJ -1.0542 -1.5843 Td [(En)-240(fait,)-259(en)-240(reprenan)24(t)-240(la)-240(d)30(\266)469(emonstration)-240(de)-240(Carath)24(\266)469(eo)-30(dory)-240(sur)-240(cet)-240(exemple)-246(nous)-234(allons)-240(obtenir)]TJ 0 -1.1988 Td [(une)-331(description)-343(concr)24(\265)469(ete)-331(de)-337(la)-331(mesure)-337(de)-337(Leb)-30(esgue.)]TJ 0 -1.8614 Td [(Appliquons)-385(l'exemple)-385(de)-391(Carath)24(\266)469(eo)-30(dory)-385(\(cf.)]TJ /F1 1 Tf 19.614 0 Td [(VI)-36(I.1.3.)]TJ /F2 1 Tf 3.8855 0 Td [(\))-391(en)-385(prenan)24(t)-391(p)-30(our)]TJ /F6 1 Tf 8.3313 0 Td [(S)]TJ /F2 1 Tf 1.0723 0 Td [(l'ensem)30(ble)-391(des)]TJ -32.904 -1.2048 Td [(in)24(terv)60(alles)-379(ouv)18(erts)-379(b)-30(orn)24(\266)475(es)-379(de)]TJ /F3 1 Tf 12.952 0 Td [(R)]TJ /F2 1 Tf 1.0964 0 Td [(et)-379(p)-30(our)]TJ /F8 1 Tf 3.6145 0 Td [(m)]TJ /F2 1 Tf 1.259 0 Td [(la)-373(fonction)-379(longueur)]TJ /F8 1 Tf 9.2169 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(.)-572(On)-379(a)-373(donc)]TJ /F8 1 Tf 5.8675 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\(])]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [([\))-355(=)]TJ /F8 1 Tf -37.367 -1.1988 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\(])]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [([\))-343(=)]TJ /F8 1 Tf 2.1325 0 Td [(b)]TJ /F6 1 Tf 0.6747 0 Td [(\241)]TJ /F8 1 Tf 1.0301 0 Td [(a)]TJ /F2 1 Tf 0.89759 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.4096 0 Td [(a)-349(<)-343(b)]TJ /F2 1 Tf 2.4217 0 Td [(.)-566(Cette)-379(donn)24(\266)475(ee)-379(p)-30(ermet)-379(de)-373(construire)-373(une)-373(pseudo-mesure)]TJ /F8 1 Tf 25.301 0 Td [(\271)]TJ /F2 1 Tf 0.97591 0 Td [(et)]TJ -38.331 -1.1988 Td [(donc)-427(une)-427(tribu)-427(sur)-427(laquelle)]TJ /F8 1 Tf 12.56 0 Td [(\271)]TJ /F2 1 Tf 1.0241 0 Td [(est)-433(une)-427(mesure)-427(compl)24(\265)469(ete.)-722(Cette)-427(tribu)-427(se)-427(note)]TJ /F6 1 Tf 21.006 0 Td [(L)]TJ /F2 1 Tf 0.69277 0 Td [(,)-451(c'est)-427(la)]TJ /F1 1 Tf -35.283 -1.1988 Td [(tribu)-385(de)-385(Leb)-36(esgue)]TJ /F2 1 Tf 9.3976 0 Td [(et)-337(la)-331(mesure)-343(d)30(\266)469(e\257nie)-337(sur)]TJ /F6 1 Tf 10.524 0 Td [(L)]TJ /F2 1 Tf 1.0241 0 Td [(est)-337(la)]TJ /F1 1 Tf 2.6687 0 Td [(mesure)-385(de)-391(Leb)-36(esgue)]TJ /F2 1 Tf 10.133 0 Td [(.)]TJ -33.747 -1.5843 Td [(Nous)-403(allons)-403(prouv)30(er)-409(que)-403(la)-403(tribu)]TJ /F6 1 Tf 14.934 0 Td [(L)]TJ /F2 1 Tf 1.0904 0 Td [(p)-30(oss)24(\265)475(ede)-409(tous)-403(les)-403(in)24(terv)54(alles)-403(])]TJ /F8 1 Tf 12.56 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)-403(\(p)-30(our)]TJ /F8 1 Tf 3.5 0 Td [(a)]TJ /F6 1 Tf 0.92169 0 Td [(2)]TJ /F3 1 Tf 1.0602 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))-403(et)]TJ -37.542 -1.1988 Td [(donc)-355(con)30(tien)24(t)-361(la)-355(tribu)-355(qu'ils)-355(engendren)24(t,)-361(c'est-)0(\265)500(a-dire)-361(celle)-355(des)-355(b)-30(or)24(\266)475(eliens.)-512(Nous)-355(prouv)24(erons)]TJ 0 -1.2048 Td [(ensuite)]TJ /F8 1 Tf 3.3614 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(b)]TJ /F6 1 Tf 0.59036 0 Td [(\241)]TJ /F8 1 Tf 0.93976 0 Td [(a)]TJ /F2 1 Tf 0.83132 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3313 0 Td [(I)]TJ /F2 1 Tf 0.8253 0 Td [(in)30(terv)54(alle)-307(d'extr)24(\266)475(emit)24(\266)469(es)]TJ /F8 1 Tf 9.9819 0 Td [(a)]TJ /F2 1 Tf 0.83133 0 Td [(et)]TJ /F8 1 Tf 1.1325 0 Td [(b)]TJ /F2 1 Tf 0.73494 0 Td [(ce)-307(qui)-301(prouv)24(era)-307(que)-307(la)-301(restriction)]TJ -24.795 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(\271)]TJ /F2 1 Tf 0.93976 0 Td [(\265)500(a)-331(la)-337(tribu)-331(des)-337(b)-30(or)24(\266)475(eliens)-337(est)-337(la)-337(mesure)-337(de)-337(Borel.)]TJ /F1 1 Tf -1.4578 -1.5783 Td [(1.)-668(P)30(our)]TJ /F6 1 Tf 4.3855 0 Td [(\2411)]TJ /F8 1 Tf 2.0602 0 Td [(<)-277(a)]TJ /F11 1 Tf 1.8614 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(b)-283(<)]TJ /F2 1 Tf 1.7651 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F1 1 Tf 1 0 Td [(,)]TJ /F8 1 Tf 0.70482 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))]TJ /F11 1 Tf 0.94578 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F1 1 Tf 0.53012 0 Td [(.)]TJ /F2 1 Tf -21.277 -1.5843 Td [(P)30(our)]TJ /F8 1 Tf 2.4699 0 Td [(")-337(>)]TJ /F2 1 Tf 1.9157 0 Td [(0,)-379(la)-367(famille)-373(r)30(\266)469(eduite)-373(au)-367(seul)-373(in)24(terv)54(alle)-367(ouv)24(ert)-373(])]TJ /F8 1 Tf 20.283 0 Td [(a)]TJ /F6 1 Tf 0.77711 0 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(")]TJ /F2 1 Tf 0.46988 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.6747 0 Td [(+)]TJ /F8 1 Tf 1.0241 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [([)-367(est)-379(un)-367(recouvremen)18(t)]TJ -29.542 -1.1988 Td [(d)30(\266)469(enom)24(brable)-289(de)-283([)]TJ /F8 1 Tf 7.4398 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(],)-295(de)-283(taille)]TJ /F8 1 Tf 4.5904 0 Td [(b)]TJ /F6 1 Tf 0.55422 0 Td [(\241)]TJ /F8 1 Tf 0.90361 0 Td [(a)]TJ /F2 1 Tf 0.65663 0 Td [(+)-126(2)]TJ /F8 1 Tf 1.4096 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [(.)-427(P)24(ar)-283(suite)]TJ /F8 1 Tf 4.8855 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(]\))]TJ /F11 1 Tf 0.94579 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F6 1 Tf 0.55421 0 Td [(\241)]TJ /F8 1 Tf 0.90361 0 Td [(a)]TJ /F2 1 Tf 0.65663 0 Td [(+)-132(2)]TJ /F8 1 Tf 1.4096 0 Td [(")]TJ /F2 1 Tf 0.46386 0 Td [(.)-427(En)-289(faisan)24(t)-289(tendre)]TJ /F8 1 Tf -30.97 -1.2048 Td [(")]TJ /F2 1 Tf 0.8012 0 Td [(v)24(ers)-331(0)-337(on)-331(obtien)24(t)-337(la)-331(ma)-60(joration)-337(annonc)24(\266)469(ee.)]TJ /F1 1 Tf 0.012049 -1.5783 Td [(2.)-668(P)30(our)]TJ /F8 1 Tf 4.3855 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(R)]TJ /F1 1 Tf 1.1084 0 Td [(on)-385(a)]TJ /F2 1 Tf 2.5361 0 Td [(])]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)]TJ /F6 1 Tf 0.27711 0 Td [(2)-283(L)]TJ /F1 1 Tf 1.6386 0 Td [(.)]TJ /F2 1 Tf -15.542 -1.5843 Td [(D'apr)30(\265)469(es)-337(la)-331(prop)-36(osition)]TJ /F1 1 Tf 10.036 0 Td [(VI)-36(I.1.4.)]TJ /F2 1 Tf 4.2169 0 Td [(il)-331(su\261t)-337(de)-337(prouv)24(er,)-337(p)-30(our)]TJ /F8 1 Tf 11.048 0 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=])]TJ /F8 1 Tf 1.0542 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(v)]TJ /F2 1 Tf 0.51807 0 Td [([,)-337(la)-337(ma)-54(joration)]TJ /F8 1 Tf -18.084 -1.1988 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\(])]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)]TJ /F6 1 Tf 0.44578 0 Td [(\\)]TJ /F8 1 Tf 0.66265 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\(])]TJ /F6 1 Tf 0.89157 0 Td [(\241)-222(1)]TJ /F2 1 Tf 2 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(])]TJ /F6 1 Tf 0.5 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(v)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(.)]TJ -28.283 -1.5843 Td [(P)30(our)]TJ /F8 1 Tf 2.4458 0 Td [(a)]TJ /F11 1 Tf 0.81928 0 Td [(6)]TJ /F8 1 Tf 1.0723 0 Td [(u)]TJ /F2 1 Tf 0.91566 0 Td [(la)-349(relation)-343(s')24(\266)475(ecrit)]TJ /F8 1 Tf 7.7711 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-234(+)]TJ /F8 1 Tf 1.6265 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(\))]TJ /F11 1 Tf 0.68072 0 Td [(6)]TJ /F8 1 Tf 1.0723 0 Td [(v)]TJ /F6 1 Tf 0.75301 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(,)-349(de)-343(m)24(^)475(eme)-343(p)-30(our)]TJ /F8 1 Tf 7.247 0 Td [(a)]TJ /F11 1 Tf 0.81928 0 Td [(>)]TJ /F8 1 Tf 1.0723 0 Td [(v)]TJ /F2 1 Tf 0.86747 0 Td [(la)-343(relation)-343(s')24(\266)469(ecrit)]TJ /F8 1 Tf -31.741 -1.1988 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(v)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(,)-337(elles)-337(r)30(\266)469(esulten)30(t)-337(de)]TJ /F8 1 Tf 8.2711 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(\))-283(=)-277(0)-337(et)]TJ /F8 1 Tf 3.7289 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(`)]TJ /F2 1 Tf 0.42169 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(v)]TJ /F6 1 Tf 0.74699 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(.)]TJ -30.735 -1.5843 Td [(P)30(our)]TJ /F8 1 Tf 2.5663 0 Td [(u)-506(<)-506(a)-512(<)-500(v)]TJ /F2 1 Tf 5.6687 0 Td [(la)-469(relation)-469(s')24(\266)475(ecrit)]TJ /F8 1 Tf 8.1446 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\(])]TJ /F8 1 Tf 0.66868 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(v)]TJ /F2 1 Tf 0.5241 0 Td [([\))-313(+)]TJ /F8 1 Tf 2.0663 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\(])]TJ /F8 1 Tf 0.66867 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(]\))]TJ /F11 1 Tf 1.1687 0 Td [(6)]TJ /F8 1 Tf 1.2831 0 Td [(v)]TJ /F6 1 Tf 0.83735 0 Td [(\241)]TJ /F8 1 Tf 1.0904 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(.)-849(Elle)-469(r)24(\266)475(esulte)-469(alors)-475(de)]TJ /F8 1 Tf -28.982 -1.1988 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\(])]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(v)]TJ /F2 1 Tf 0.5241 0 Td [([\))]TJ /F11 1 Tf 0.93976 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\(])]TJ /F8 1 Tf 0.66868 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(v)]TJ /F2 1 Tf 0.5241 0 Td [([\))-283(=)]TJ /F8 1 Tf 2 0 Td [(v)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(et)-337(de)]TJ /F8 1 Tf 2.506 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\(])]TJ /F8 1 Tf 0.66265 0 Td [(u)]TJ /F2 1 Tf 0.57831 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(]\))]TJ /F11 1 Tf 0.94579 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(]\))]TJ /F11 1 Tf 0.94578 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(a)]TJ /F6 1 Tf 0.75301 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(.)]TJ /F1 1 Tf -25.596 -1.5843 Td [(3.)-668(P)30(our)]TJ /F6 1 Tf 4.3855 0 Td [(\2411)]TJ /F8 1 Tf 2.0602 0 Td [(<)-277(a)]TJ /F11 1 Tf 1.8614 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(b)-283(<)]TJ /F2 1 Tf 1.7651 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F1 1 Tf 1 0 Td [(,)]TJ /F8 1 Tf 0.70482 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))-283(=)]TJ /F8 1 Tf 2.006 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F1 1 Tf 0.53012 0 Td [(.)]TJ /F2 1 Tf -21.277 -1.5783 Td [(On)-463(a)-469(d)30(\266)469(ej)-6(\265)500(a)-463(vu)]TJ /F8 1 Tf 6.5843 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66868 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))]TJ /F11 1 Tf 1.1627 0 Td [(6)]TJ /F8 1 Tf 1.2771 0 Td [(b)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1.0904 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(,)-499(il)-469(s'agit)-469(de)-463(prouv)24(er)-469(l'in)24(\266)475(egalit)18(\266)475(e)-469(in)24(v)24(erse,)-500(c'est-)0(\265)500(a-dire)-469(que)]TJ -14.054 -1.1988 Td [(tout)-433(recouvremen)24(t)-433(d)24(\266)475(enom)24(brable)-433(\()]TJ /F8 1 Tf 15 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.069 10.504 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.349 7.4906 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.103 10.504 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.446 7.4906 Td [(de)-433([)]TJ /F8 1 Tf 1.7108 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])-433(par)-427(des)-433(in)24(terv)54(alles)-433(ouv)18(erts)-433(a)-433(une)-427(taille)]TJ -21.229 -1.2048 Td [(sup)0(\266)475(erieure)-343(ou)-307(\266)475(egale)-343(\265)500(a)]TJ /F8 1 Tf 9.5422 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)]TJ /F5 1 Tf -11.723 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19.247 0 Td [(98)]TJ ET endstream endobj 330 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 329 0 R /ProcSet 2 0 R >> /Contents 328 0 R >> endobj 329 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R >> endobj 331 0 obj << /Length 17960 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4036 Td [(Remarquons)-240(que)-246(la)-246(compacit)30(\266)469(e)-246(de)-240([)]TJ /F8 1 Tf 14.627 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])-246(assure)-240(l'existence)-246(d'un)-246(sous-recouvremen)24(t)-246(\257ni,)-265(lequel)]TJ -16.03 -1.1988 Td [(a)-313(\266)469(evidemmen)24(t)-349(une)-343(taille)-355(inf)30(\266)469(erieure.)-475(Cela)-349(p)-30(ermet)-349(de)-349(se)-343(ramener)-349(au)-343(cas)-349(d'un)-349(recouvremen)18(t)]TJ 0 -1.1988 Td [(\257ni)-337(\()]TJ /F8 1 Tf 2.1084 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(\257ni\))-331(et)-337(de)-337(raisonner)-331(par)-337(r)30(\266)469(ecurrence)-337(sur)-337(le)-337(nom)18(bre)-331(d')24(\266)475(el)18(\266)475(emen)24(ts)-337(de)]TJ /F8 1 Tf 28.163 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)]TJ -31.645 -1.8374 Td [(Notons)]TJ /F8 1 Tf 3.4518 0 Td [(T)]TJ /F2 1 Tf 1.0482 0 Td [(=)]TJ /F9 1 Tf 1.1024 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.802 81.271 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.548 57.985 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.086 82.771 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.367 57.985 Td [(\))-367(la)-361(taille)-367(du)-367(recouvremen)24(t.)-536(Il)-361(existe)]TJ /F8 1 Tf 16.44 0 Td [(j)]TJ /F6 1 Tf 0.79518 0 Td [(2)]TJ /F8 1 Tf 0.99398 0 Td [(I)]TJ /F2 1 Tf 0.87952 0 Td [(tel)-367(que)]TJ /F8 1 Tf 3.3675 0 Td [(a)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.99398 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.336 82.771 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.789 57.985 Td [(,)-373(notons)]TJ /F8 1 Tf 3.9096 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 83.573 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 55.491 Td [(le)-373(compl)24(\266)469(emen)30(taire)-373(de)]TJ /F6 1 Tf 9.6928 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(j)]TJ /F6 1 Tf 0.46988 0 Td [(g)]TJ /F2 1 Tf 0.87349 0 Td [(dans)]TJ /F8 1 Tf 2.3735 0 Td [(I)]TJ /F2 1 Tf 0.88554 0 Td [(et)-373(p)-30(osons)]TJ /F8 1 Tf 4.5 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.276 79.995 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.97 55.491 Td [(=)]TJ /F9 1 Tf 1.1446 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.759 77.693 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 56.627 108.98 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.753 55.491 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.276 79.193 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.572 55.491 Td [(\).)-554(Si)]TJ /F8 1 Tf 2.4217 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.19 79.193 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.434 55.491 Td [(=])]TJ /F8 1 Tf 1.0602 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(;)]TJ /F8 1 Tf 0.43976 0 Td [(v)]TJ /F2 1 Tf 0.5241 0 Td [([)-367(on)-373(a)]TJ /F8 1 Tf 2.9398 0 Td [(u)-337(<)-337(a)-343(<)-337(v)]TJ /F2 1 Tf -34.639 -2.253 Td [(\(puisque)]TJ /F8 1 Tf 4.1024 0 Td [(a)]TJ /F6 1 Tf 0.92169 0 Td [(2)]TJ /F8 1 Tf 1.0602 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.319 75.969 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.524 53.238 Td [(\))-403(et)]TJ /F8 1 Tf 2.0241 0 Td [(T)]TJ /F2 1 Tf 1.1205 0 Td [(=)]TJ /F8 1 Tf 1.1687 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.129 76.771 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.108 53.238 Td [(+)-271(\()]TJ /F8 1 Tf 1.4398 0 Td [(v)]TJ /F6 1 Tf 0.78916 0 Td [(\241)]TJ /F8 1 Tf 1.0422 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\).)-656(Alors,)-427(si)]TJ /F8 1 Tf 5.4096 0 Td [(v)-433(>)-391(b)]TJ /F2 1 Tf 2.5181 0 Td [(,)-415(la)-403(taille)]TJ /F8 1 Tf 4.4458 0 Td [(T)]TJ /F2 1 Tf 1.1265 0 Td [(du)-403(recouvremen)18(t)-403(est)]TJ -30.12 -1.1988 Td [(sup)0(\266)475(erieure)-234(\265)500(a)]TJ /F8 1 Tf 5.4337 0 Td [(v)]TJ /F6 1 Tf 0.5241 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(u)]TJ /F2 1 Tf 0.79518 0 Td [(donc)-222(\265)500(a)]TJ /F8 1 Tf 3.006 0 Td [(b)]TJ /F6 1 Tf 0.42771 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(a)]TJ /F2 1 Tf 0.75301 0 Td [(;)-259(si)]TJ /F8 1 Tf 1.4337 0 Td [(v)]TJ /F11 1 Tf 0.80121 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F2 1 Tf 0.65663 0 Td [(la)-222(sous-famille)-222(des)]TJ /F8 1 Tf 7.9398 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.509 74.254 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.964 52.039 Td [(p)-30(our)]TJ /F8 1 Tf 2.253 0 Td [(i)]TJ /F6 1 Tf 0.62651 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.526 75.056 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.807 52.039 Td [(est)-228(un)-228(recouvremen)18(t)]TJ -30.476 -1.2048 Td [(\257ni)-301(de)-295([)]TJ /F8 1 Tf 3.259 0 Td [(v)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(])-301(de)-295(taille)]TJ /F8 1 Tf 4.3373 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.543 73.34 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.633 50.834 Td [(et)-295(l'h)24(yp)-36(oth)24(\265)469(ese)-295(de)-301(r)30(\266)469(ecurrence)-301(assure)]TJ /F8 1 Tf 15.44 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.5 73.34 Td [(0)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 32.349 50.834 Td [(>)]TJ /F8 1 Tf 1.0602 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(v)]TJ /F2 1 Tf 0.51807 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))]TJ /F11 1 Tf 0.94579 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F6 1 Tf 0.58434 0 Td [(\241)]TJ /F8 1 Tf 0.92772 0 Td [(v)]TJ /F2 1 Tf 0.51807 0 Td [(,)-307(par)-301(suite)]TJ /F8 1 Tf -34.771 -1.1988 Td [(T)]TJ /F11 1 Tf 1 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(v)]TJ /F2 1 Tf 0.5241 0 Td [(\))-222(+)-222(\()]TJ /F8 1 Tf 2 0 Td [(v)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(u)]TJ /F11 1 Tf 0.8494 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(.)]TJ /F1 1 Tf -16.398 -2.5783 Td [(Remarque.)]TJ /F2 1 Tf 6.247 0 Td [(On)-331(p)-30(eut)-331(s')24(\266)469(etonner)-331(que)-337(la)-331(preuv)30(e)-337(de)]TJ /F8 1 Tf 15.765 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))]TJ /F11 1 Tf 0.94578 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(b)]TJ /F6 1 Tf 0.64458 0 Td [(\241)]TJ /F8 1 Tf 0.99398 0 Td [(a)]TJ /F2 1 Tf 0.85542 0 Td [(passe)-331(par)-331(un)-331(argumen)18(t)]TJ -29.181 -1.1988 Td [(aussi)-307(sophistiqu)24(\266)475(e)-313(que)-313(la)-307(compacit)30(\266)469(e.)-439(On)-307(remarquera)-307(toutefois)-313(que)-307(tout)-313(ce)-307(qui)-307(pr)24(\266)475(ec)24(\265)469(ede)-313(cette)]TJ 0 -1.2048 Td [(preuv)30(e)-367(p)-24(eut)-337(^)469(etre)-355(fait)-361(en)-361(rempla)-6(\273)439(can)30(t)]TJ /F3 1 Tf 16.024 0 Td [(R)]TJ /F2 1 Tf 1.0843 0 Td [(par)-361(l'ensem)30(ble)]TJ /F3 1 Tf 6.6446 0 Td [(Q)]TJ /F2 1 Tf 1.1386 0 Td [(des)-361(rationnels.)-524(Dans)-361(ce)-361(cas)-361(on)-361(a)]TJ /F8 1 Tf -24.892 -1.1988 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(]\))]TJ /F11 1 Tf 1.012 0 Td [(6)]TJ /F8 1 Tf 1.1205 0 Td [(a)]TJ /F6 1 Tf 0.77711 0 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(a)]TJ /F2 1 Tf 0.87349 0 Td [(=)-349(0,)-379(autremen)24(t)-373(dit)-379(les)-379(singletons)-373(son)24(t)-379(n)30(\266)469(egligeables.)-566(Cep)-30(endan)24(t)]TJ /F3 1 Tf 28.181 0 Td [(Q)]TJ /F2 1 Tf 1.1265 0 Td [(\266)475(etan)18(t)]TJ -36.886 -1.1988 Td [(d)30(\266)469(enom)24(brable)-385(on)-385(conclut)-391(qu'alors)-385(toutes)-385(les)-385(parties)-391(son)24(t)-385(n)24(\266)475(egligeables)-391(et,)-397(en)-379(particulier,)-403(si)]TJ 0 -1.2048 Td [(l'on)-337(se)-331(place)-337(sur)]TJ /F3 1 Tf 7.3494 0 Td [(Q)]TJ /F2 1 Tf 0.77711 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66265 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))-337(est)-343(n)30(ul)-337(donc)-337(en)-331(g)24(\266)475(en)24(\266)475(eral)-337(strictemen)24(t)-337(inf)24(\266)475(erieur)-337(\265)500(a)]TJ /F8 1 Tf 21.506 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F1 1 Tf -33.753 -1.6325 Td [(4.)-668(Conclusion.)]TJ /F2 1 Tf -0.81325 -1.6265 Td [(Ainsi,)]TJ /F8 1 Tf 3 0 Td [(\271)]TJ /F2 1 Tf 1.0361 0 Td [(d)24(\266)469(e\257nit)-439(une)-439(mesure)-439(sur)-445(la)-439(tribu)]TJ /F6 1 Tf 14.325 0 Td [(L)]TJ /F2 1 Tf 1.1265 0 Td [(qui)-439(con)24(tien)24(t)-439(les)-439(])]TJ /F8 1 Tf 7.5723 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-168(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)-439(donc)-439(la)-439(tribu)-439(qu'ils)]TJ -29.813 -1.2048 Td [(engendren)24(t,)-457(c'est-)0(\265)500(a-dire)-433(celle)-427(des)-433(b)-30(or)30(\266)469(eliens.)-734(P)30(our)-433(un)-433(segmen)24(t)-433(nous)-427(a)30(v)24(ons)-433(vu)]TJ /F8 1 Tf 34.608 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44579 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))-439(=)]TJ /F8 1 Tf -37.283 -1.1988 Td [(b)]TJ /F6 1 Tf 0.72289 0 Td [(\241)]TJ /F8 1 Tf 1.0783 0 Td [(a)]TJ /F2 1 Tf 0.98795 0 Td [(=)]TJ /F8 1 Tf 1.247 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\([)]TJ /F8 1 Tf 0.66867 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\),)-481(en)-445(particulier,)-475(p)-30(our)-445(un)-445(singleton,)]TJ /F8 1 Tf 16.807 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(\))-469(=)-463(0.)-783(Plus)-445(g)24(\266)475(en)24(\266)475(eralemen)24(t)-451(un)]TJ -25.843 -1.1988 Td [(in)24(terv)60(alle)]TJ /F8 1 Tf 4.2349 0 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)-283(\()]TJ /F8 1 Tf 1.4458 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))-234(d'extr)24(\266)469(emit)24(\266)475(es)]TJ /F8 1 Tf 6.2229 0 Td [(a)]TJ /F2 1 Tf 0.76506 0 Td [(et)]TJ /F8 1 Tf 1.0602 0 Td [(b)]TJ /F2 1 Tf 0.63856 0 Td [(\266)475(etan)24(t)-210(\266)475(egal)-234(\265)500(a)-234([)]TJ /F8 1 Tf 5.4759 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(])-228(priv)54(\266)469(e)-234(d'un)-234(nom)24(bre)-234(\257ni)-234(de)-234(singletons)]TJ -23.446 -1.2048 Td [(on)-331(conclut)]TJ /F8 1 Tf 4.8916 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\([)]TJ /F8 1 Tf 0.66868 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(]\))-283(=)]TJ /F8 1 Tf 2 0 Td [(b)]TJ /F6 1 Tf 0.6506 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(a)]TJ /F2 1 Tf 0.8012 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(`)]TJ /F2 1 Tf 0.41566 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(\).)]TJ -17.639 -1.6325 Td [(La)-337(restriction)-337(de)]TJ /F8 1 Tf 7.5843 0 Td [(\271)]TJ /F2 1 Tf 0.93373 0 Td [(aux)-337(b)-30(or)24(\266)475(eliens)-337(est)-343(par)-331(cons)24(\266)469(equen)24(t)-337(la)-337(mesure)-337(de)-331(Borel.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 23.595 Td [(VI)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8193 28.286 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)18(oposition.)]TJ /F12 1 Tf 6.9157 0 Td [(P)30(our)-349(la)-349(mesure)-349(de)-349(Leb)-30(esgue)-349(sur)]TJ /F3 1 Tf 14.355 0 Td [(R)]TJ /F12 1 Tf 1.0723 0 Td [(la)-349(mesure)-349(ext)30(\266)469(erieure)-349(d'une)]TJ -27.578 -1.2048 Td [(partie)-337(est)-337(la)-337(b)-30(orne)-331(inf)24(\266)475(erieure)-343(des)-337(mesures)-331(des)-337(ouv)18(erts)-337(qui)-331(la)-337(con)30(tiennen)24(t.)]TJ /F1 1 Tf 0 -1.9458 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Av)30(ec)-343(les)-343(notations)-337(pr)24(\266)469(ec)24(\266)475(eden)24(tes)-343(les)-313(\266)475(el)18(\266)475(emen)24(ts)-337(de)]TJ /F6 1 Tf 20.596 0 Td [(S)]TJ /F2 1 Tf 1.0181 0 Td [(son)18(t)-337(mesurables)-343(\(au)]TJ -30.434 -1.1988 Td [(sens)-361(de)-367(Leb)-30(esgue\),)-373(par)-361(suite)-367(la)-367(prop)-30(osition)]TJ /F1 1 Tf 19.217 0 Td [(VI)-36(I.1.4.)]TJ /F2 1 Tf 4.247 0 Td [(a\261rme)-367(que,)-373(p)-30(our)]TJ /F8 1 Tf 7.9398 0 Td [(A)]TJ /F6 1 Tf 1.0783 0 Td [(\275)]TJ /F3 1 Tf 1.1024 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-373(la)-361(mesure)]TJ -34.307 -1.2048 Td 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0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 333 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 332 0 R /ProcSet 2 0 R >> /Contents 331 0 R >> endobj 332 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F18 107 0 R >> endobj 334 0 obj << /Length 19016 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ 0.81325 -2.4036 Td [(2.)]TJ /F12 1 Tf 1.5181 0 Td [(Les)-313(parties)-313(mesurables)-319(au)-313(sens)-313(de)-313(Leb)-30(esgue)-319(son)24(t)-313(les)-313(r)24(\266)469(eunions)-313(d'une)-313(partie)-319(b)-30(or)30(\266)469(elienne)]TJ -2.3313 -1.1988 Td [(et)-337(d'une)-337(partie)-337(n)24(\266)475(egligeable.)]TJ /F1 1 Tf 0.81325 -1.6325 Td [(3.)]TJ /F12 1 Tf 1.3795 0 Td 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[(\))-415(d'apr)24(\265)469(es)-409(l')24(\266)469(enonc)30(\266)469(e)-415(pr)30(\266)469(ec)24(\266)469(eden)30(t.)-680(On)-409(en)-409(d)24(\266)469(eduit)-409(l')30(\266)469(egalit)24(\266)469(e)-415(des)-409(mesures)-415(ext)30(\266)469(erieures)]TJ -5.241 -1.1988 Td [(d)30(\266)469(e\257nies)-337(par)-337(la)-331(mesure)-343(de)-331(Borel)-337(et)-337(par)-337(celle)-337(de)-331(Leb)-36(esgue)-331(et)-337(l')24(\266)475(enonc)24(\266)469(e)]TJ /F1 1 Tf 29.681 0 Td [(1.)]TJ /F2 1 Tf 1.3373 0 Td [(en)-337(d)30(\266)469(ecoule.)]TJ -31.018 -1.6325 Td [(Cela)-379(prouv)24(e)-379(aussi)-379(que,)-391(sur)]TJ /F6 1 Tf 12.06 0 Td [(L)]TJ /F2 1 Tf 0.68675 0 Td [(,)]TJ /F8 1 Tf 0.66867 0 Td [(\271)]TJ /F2 1 Tf 0.9759 0 Td [(co)102(\304)391(\263ncide)-379(a)24(v)24(ec)]TJ /F8 1 Tf 6.1205 0 Td [(\257)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.871 65.4 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.922 45.286 Td 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[(appartenan)24(t)-295(\265)499(a)]TJ /F6 1 Tf 6.3916 0 Td [(L)]TJ /F2 1 Tf 0.9759 0 Td [(et)-289(prouv)24(ons)-295(que)-295(l'on)-289(p)-30(eut)-295(l')30(\266)469(ecrire)]TJ /F8 1 Tf 14.584 0 Td [(B)]TJ /F6 1 Tf 0.94578 0 Td [([)]TJ /F8 1 Tf 0.8012 0 Td [(N)]TJ /F2 1 Tf 1.1988 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf -37.301 -1.1988 Td [(B)]TJ /F2 1 Tf 1.2229 0 Td [(b)-30(or)30(\266)469(elienne)-415(et)]TJ /F8 1 Tf 6.1386 0 Td [(N)]TJ /F2 1 Tf 1.3313 0 Td [(n)30(\266)469(egligeable.)-692(Supp)-24(osons)-415(tout)-421(d'ab)-30(ord)-415(que)-421(le)-421(compl)24(\266)475(emen)24(taire)]TJ /F8 1 Tf 27.145 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.991 55.581 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.747 38.424 Td [(\(qui)]TJ -37.416 -1.1988 Td [(appartien)24(t)-265(encore)-265(\265)500(a)]TJ /F6 1 Tf 8.6024 0 Td [(L)]TJ /F2 1 Tf 0.68675 0 Td [(\))-265(soit)-265(de)-265(mesure)-265(\257nie.)-421(Alors)]TJ /F8 1 Tf 12.175 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.845 53.857 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.946 37.226 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\257)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.776 53.857 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.747 37.226 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.06 53.857 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.295 37.226 Td [(\),)-283(il)-259(existe)-271(donc)-259(une)-265(partie)]TJ /F8 1 Tf -27.964 -1.1988 Td [(C)]TJ /F2 1 Tf 1.1084 0 Td [(b)-30(or)30(\266)469(elienne)-325(con)24(tenan)24(t)]TJ /F8 1 Tf 9.4036 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.75 52.142 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.325 36.027 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1867 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.414 52.142 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.663 36.027 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\257)]TJ /F2 1 Tf 0.62048 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(C)]TJ /F2 1 Tf 0.78915 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F2 1 Tf 0.78313 0 Td [(\).)-445(Comme)]TJ /F8 1 Tf 4.7711 0 Td [(C)]TJ /F2 1 Tf 1.1084 0 Td [(est)-331(la)-325(r)30(\266)469(eunion)-325(disjoin)24(te)]TJ -29.229 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.612 50.426 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1566 34.822 Td [(et)-337(de)]TJ /F8 1 Tf 2.506 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(,)-331(on)-337(conclut)]TJ /F8 1 Tf 5.506 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(C)]TJ /F2 1 Tf 0.78313 0 Td [(\))-283(=)-283(0,)-337(par)-337(suite)]TJ /F8 1 Tf 7.0181 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(est)-337(n)24(\266)469(egligeable.)-445(Alors)]TJ /F8 1 Tf 9.7169 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(est)]TJ -37.934 -1.1988 Td [(la)-331(r)24(\266)475(eunion)-337(du)-337(compl)24(\266)469(emen)24(taire)-337(de)]TJ /F8 1 Tf 14.705 0 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(qui)-331(est)-343(b)-30(or)30(\266)469(elien)-331(et)]TJ /F8 1 Tf 8.2349 0 Td [(A)]TJ /F6 1 Tf 0.96988 0 Td [(\\)]TJ /F8 1 Tf 0.89157 0 Td [(C)]TJ /F2 1 Tf 1.1205 0 Td [(qui)-331(est)-343(n)30(\266)469(egligeable.)]TJ -27.042 -1.6325 Td [(Dans)-439(le)-439(cas)-439(g)30(\266)469(en)30(\266)469(eral)-433(notons)]TJ /F8 1 Tf 12.464 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.655 45.564 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.602 31.991 Td [(le)-445(compl)24(\266)475(emen)24(taire)-439(du)-439(segmen)24(t)-439([)]TJ /F6 1 Tf 14.259 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(])-439(et)]TJ /F8 1 Tf 1.9879 0 Td 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[(est)-493(une)-493(partie)-493(b)-30(or)30(\266)469(elienne)-487(con)24(ten)24(ue)-487(dans)]TJ /F9 1 Tf 18.596 0.94578 Td [(\\)]TJ /F8 1 Tf 1.2771 -0.94578 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.983 40.211 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.904 28.25 Td [(=)]TJ /F8 1 Tf 1.3133 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(,)-530(le)]TJ -37.633 -1.3494 Td [(compl)24(\266)475(emen)24(taire)]TJ /F8 1 Tf 7.2229 0 Td [(N)]TJ /F2 1 Tf 1.2651 0 Td [(de)]TJ /F8 1 Tf 1.3554 0 Td [(B)]TJ /F2 1 Tf 1.1627 0 Td [(dans)]TJ /F8 1 Tf 2.3614 0 Td [(A)]TJ /F2 1 Tf 1.1084 0 Td [(est)-361(con)30(ten)24(u)-355(dans)-355(la)-361(r)30(\266)469(eunion)-355(des)]TJ /F8 1 Tf 14.211 0 Td [(N)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.828 38.28 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.723 26.9 Td [(donc)-355(est)-361(n)30(\266)469(egligeable)]TJ -30.392 -1.1988 Td [(et)-337(l'on)-331(a)]TJ /F8 1 Tf 3.9458 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(B)]TJ /F6 1 Tf 1.0301 0 Td [([)]TJ /F8 1 Tf 0.89157 0 Td [(N)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ ET 512.64 305.96 m 519.84 305.96 l 519.84 314.38 l 512.64 314.38 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 17.691 Td [(VI)-35(I.3.)-236(L'analogue)-381(fonctionnel.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 19.581 Td [(Nous)-343(v)24(enons)-349(de)-343(v)24(oir)-343(commen)24(t)-349(une)-343(fonction)-349(n)30(um)24(\266)469(erique)-343(d)24(\266)469(e\257nie)-343(p)-30(our)-343(certaines)-343(parties)-343(d'un)]TJ 0 -1.1988 Td [(ensem)24(ble)]TJ /F8 1 Tf 4.3253 0 Td [(X)]TJ /F2 1 Tf 1.3133 0 Td [(p)-30(ermet)-409(de)-403(construire)-409(une)-403(mesure)-415(sur)]TJ /F8 1 Tf 16.831 0 Td [(X)]TJ /F2 1 Tf 1.3072 0 Td [(qui,)-427(dans)-403(les)-409(b)-24(ons)-415(cas,)-421(prolonge)-409(la)]TJ -23.777 -1.2048 Td 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[(croissan)30(te,)-337(p)-30(ositiv)30(emen)24(t)-337(homog)24(\265)469(ene)-337(et)-337(d)24(\266)475(enom)24(brablemen)24(t)-331(sous-additiv)18(e.)]TJ -6.3614 -1.6265 Td [(On)-349(p)-30(eut)-349(alors)-355(d)30(\266)469(e\257nir)]TJ /F8 1 Tf 9.5422 0 Td [(\271)]TJ /F2 1 Tf 0.95181 0 Td [(de)]TJ /F6 1 Tf 1.3494 0 Td [(P)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-355(dans)]TJ ET 266.98 113.76 m 275.62 113.76 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 22.337 8.6894 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 33 12.219 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.072 8.6894 Td [(par)]TJ /F8 1 Tf 1.7952 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-301(=)]TJ /F8 1 Tf 1.7711 0 Td [(\301)]TJ /F2 1 Tf 0.59036 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.259 12.219 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.578 8.6894 Td [(\))-355(et)-349(il)-349(est)-355(imm)24(\266)475(ediat)-355(que)]TJ /F8 1 Tf 10.735 0 Td [(\271)]TJ /F2 1 Tf 0.95181 0 Td [(est)]TJ -37.934 -1.1988 Td [(une)-415(pseudo-mesure)-415(sur)]TJ /F8 1 Tf 10.548 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)-686(L')24(\266)469(etude)-415(pr)24(\266)469(ec)24(\266)475(eden)24(te)-415(p)-30(ermet)-421(d'asso)-30(cier)-415(\265)500(a)-415(cette)-421(pseudo-mesure)]TJ -11.458 -1.2048 Td [(une)-331(tribu)]TJ /F6 1 Tf 4.3916 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(et)-337(une)-337(mesure)]TJ /F8 1 Tf 6.4578 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(qui)-331(est)-343(la)-331(restriction)-343(de)]TJ /F8 1 Tf 10.494 0 Td [(\271)]TJ /F2 1 Tf 0.93374 0 Td [(\265)499(a)]TJ /F6 1 Tf 0.83735 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)]TJ /F5 1 Tf -26.259 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(100)]TJ ET endstream endobj 336 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 335 0 R /ProcSet 2 0 R >> /Contents 334 0 R >> endobj 335 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R >> endobj 337 0 obj << /Length 21491 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(VI)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8193 62.22 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)18(oposition.)]TJ /F12 1 Tf 6.8494 0 Td [(Av)24(ec)-283(les)-277(notations)-283(et)-277(h)24(yp)-30(oth)24(\265)469(eses)-283(pr)30(\266)469(ec)24(\266)469(eden)30(tes)-283(la)-277(fonctionnelle)]TJ /F8 1 Tf 26.482 0 Td [(\301)]TJ /F12 1 Tf -38.566 -1.1988 Td [(co)108(\304)385(\263ncide)-295(a)30(v)24(ec)-295(l'in)24(t)24(\266)469(egrale)-295(d)24(\266)469(e\257nie)-295(par)]TJ /F8 1 Tf 15.422 0 Td [(m)]TJ /F12 1 Tf 1.1687 0 Td [(sur)-295(l'ensem)24(ble)-295(des)-295(fonctions)]TJ /F6 1 Tf 12.313 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurables)-289(p)-30(ositiv)24(es.)]TJ /F1 1 Tf -29.705 -2.3012 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Il)-427(s'agit)-433(donc)-433(de)-433(prouv)24(er)-433(l')24(\266)475(egalit)24(\266)469(e)]TJ /F8 1 Tf 15.349 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-445(=)]TJ /F9 1 Tf 2.0482 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5964 0 Td [(lorsque)]TJ /F8 1 Tf 3.5241 0 Td [(f)]TJ /F2 1 Tf 1.0241 0 Td [(est)-439(une)]TJ -35.946 -1.6566 Td [(fonction)-337(mesurable)-337(p)-30(ositiv)30(e.)]TJ 0 -1.5723 Td [(Remarquons)-409(tout)-415(d'ab)-30(ord)-415(que,)-433(si)]TJ /F8 1 Tf 15.187 0 Td [(f)]TJ /F2 1 Tf 1.006 0 Td [(=)-409(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.302 79.193 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.331 55.491 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.2711 0 Td [(A)]TJ /F2 1 Tf 1.1627 0 Td [(mesurable,)-433(on)-409(a,)-433(par)-409(d)24(\266)475(e\257nition)-415(de)]TJ /F8 1 Tf 15.572 0 Td [(m)]TJ /F2 1 Tf 0.87951 0 Td [(,)]TJ /F8 1 Tf -38.886 -1.6566 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.569 76.823 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.831 53.834 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.517 76.823 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.145 53.834 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)-445(L')24(\266)469(egalit)24(\266)469(e)-337(est)-337(assur)24(\266)469(ee)-337(dans)-331(ce)-337(cas)-337(l)0(\265)500(a.)]TJ -18.211 -2.3855 Td [(Dans)-403(le)-409(cas)-409(g)30(\266)469(en)30(\266)469(eral)]TJ /F8 1 Tf 9.006 0 Td [(f)]TJ /F2 1 Tf 1 0 Td [(p)-30(eut)-403(s')24(\266)469(ecrire)]TJ /F9 1 Tf 5.8494 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.466 71.918 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.795 51.448 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.543 73.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.771 51.448 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.819 73.418 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 49.867 102.4 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.705 51.448 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.2651 0 Td [(I)]TJ /F2 1 Tf 0.92169 0 Td [(d)24(\266)475(enom)24(brable,)-427(des)]TJ /F8 1 Tf 8.0843 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.991 73.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.464 51.448 Td [(mesurables)-409(et)]TJ -33.133 -2.1386 Td [(des)]TJ /F8 1 Tf 1.7892 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.103 70.349 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.488 49.31 Td [(r)30(\266)469(eels)-397(p)-30(ositifs)-397(que)-397(l'on)-403(p)-24(eut)-403(toujours)-397(supp)-30(oser)-397(non)-397(n)30(uls.)-632(Il)-397(s'agit)-397(alors)-397(de)-397(prouv)30(er)]TJ /F8 1 Tf -3.1566 -1.3494 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.578 68.418 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.223 47.961 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.948 68.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.572 47.961 Td [(\).)]TJ -8.241 -2.0723 Td [(La)-487(sous-additivit)24(\266)469(e)-487(de)]TJ /F8 1 Tf 9.9518 0 Td [(\301)]TJ /F2 1 Tf 1.0843 0 Td [(assure)]TJ /F8 1 Tf 3.1627 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.92169 0 Td [(6)]TJ /F9 1 Tf 1.3072 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.56 63.961 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.952 45.888 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.629 65.452 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.928 45.888 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.319 65.452 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 56.157 91.282 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.446 45.888 Td [(\))-530(=)]TJ /F9 1 Tf 2.2229 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.112 65.452 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.253 45.888 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.474 65.452 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.608 45.888 Td [(\).)-897(Il)-487(su\261t)-487(donc)-487(de)]TJ -30.277 -2.1988 Td [(prouv)30(er)-421(l'in)24(\266)469(egalit)24(\266)475(e)-421(in)30(v)18(erse.)-680(P)24(our)-415(cela)-415(on)-415(p)-30(eut)-385(\266)469(evidemmen)24(t)-415(supp)-30(oser)]TJ /F8 1 Tf 30.711 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-415(\257ni.)-686(Puisque,)]TJ -32.295 -1.1988 Td [(p)-30(our)-391(tout)]TJ /F8 1 Tf 4.6506 0 Td [(i)]TJ /F2 1 Tf 0.34337 0 Td [(,)-409(on)-397(a)]TJ /F8 1 Tf 3.0301 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.026 60.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.325 42.491 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.31 60.59 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 30.988 84.475 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 16.235 42.491 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)-409(cela)-397(implique)]TJ /F8 1 Tf 6.8916 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.931 60.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.837 42.491 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.207 60.59 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 56 84.475 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.367 42.491 Td [(\))-379(=)]TJ /F8 1 Tf 1.9217 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.259 60.59 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.265 42.491 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.629 60.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.614 42.491 Td [(\))]TJ /F11 1 Tf 0.76506 0 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ /F8 1 Tf 0.76506 0 Td [(<)]TJ /F2 1 Tf 1.1566 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.3916 0 Td [(et)-397(donc)]TJ /F8 1 Tf -35.88 -1.1988 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.517 58.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6807 41.292 Td [(\))-337(\257ni.)]TJ -2.3494 -1.5723 Td [(Consid)30(\266)469(erons)-361(tout)-361(d'ab)-36(ord)-361(le)-361(cas)-361(o)-30(\265)530(u)]TJ /F8 1 Tf 15.765 0 Td [(I)]TJ /F2 1 Tf 0.87952 0 Td [(est)-361(\257ni.)-530(Notons)]TJ /F8 1 Tf 7.2229 0 Td [(A)]TJ /F2 1 Tf 1.1145 0 Td [(la)-355(r)24(\266)469(eunion)-361(des)]TJ /F8 1 Tf 6.5301 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.802 56.625 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.928 39.72 Td [(,)]TJ /F8 1 Tf 0.64458 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.284 56.625 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.024 39.72 Td [(le)-361(compl)24(\266)469(e-)]TJ -34.693 -1.3494 Td [(men)24(taire)-337(de)]TJ /F8 1 Tf 5.4759 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.543 54.702 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.223 38.37 Td [(dans)]TJ /F8 1 Tf 2.3434 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(et)-331(p)-30(osons)]TJ /F8 1 Tf 4.4277 0 Td [(S)]TJ /F2 1 Tf 0.95181 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.828 54.702 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25 38.37 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(g)]TJ /F2 1 Tf 0.78916 0 Td [(=)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.319 54.702 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.602 38.37 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.595 54.702 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 63.53 76.246 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.133 38.37 Td [(.)]TJ -26.801 -1.7711 Td [(On)-337(a)-331(alors)]TJ /F8 1 Tf 10.578 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.655 52.159 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.259 36.599 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.336 52.159 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.235 36.599 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F8 1 Tf 2.3855 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(S)]TJ /F2 1 Tf 0.6747 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.603 52.159 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.151 34.93 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.638 49.771 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.042 34.93 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33 49.771 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.392 34.93 Td [(\))]TJ /F8 1 Tf 2.7289 0 Td [(\301)]TJ /F2 1 Tf 0.59036 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.207 49.771 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.922 34.93 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.586 49.771 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.283 34.93 Td [(\))]TJ -28.952 -1.7169 Td [(et)-337(donc)]TJ /F8 1 Tf 1.7289 -1.2048 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(S)]TJ /F2 1 Tf 0.66867 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.284 45.59 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.9337 32.009 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(S)-60(m)]TJ /F2 1 Tf 1.5482 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.81928 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.612 45.59 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 30.108 32.822 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.629 45.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.922 32.009 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.319 45.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.892 32.009 Td [(\))]TJ /F9 1 Tf 0.38554 0.81325 Td [(\242)]TJ /F2 1 Tf 0.74096 -0.81325 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(S)-60(m)]TJ /F2 1 Tf 1.5482 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-337(.)]TJ -36.428 -1.7711 Td [(Comme)-427(il)-421(s'agit)-427(de)-421(relations)-427(en)24(tre)-427(r)30(\266)469(eels)-421(\(\257nis\),)-451(l')24(\266)475(egalit)24(\266)469(e)-427(en)24(tre)-421(les)-427(termes)-427(extr)30(^)469(emes)-427(de)-421(la)]TJ 0 -1.1988 Td [(derni)24(\265)475(ere)-349(ligne)-349(mon)24(tre)-343(qu'aucune)-349(des)-343(in)24(\266)469(egalit)24(\266)469(es)-343(pr)24(\266)475(ec)24(\266)469(eden)24(tes)-349(ne)-343(p)-30(eut)-319(^)475(etre)-349(stricte.)-475(En)-349(parti-)]TJ 0 -1.3434 Td [(culier)-331(on)-337(a)-331(bien)]TJ /F8 1 Tf 7.1145 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.94578 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94578 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.767 39.418 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.337 27.695 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.129 39.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.687 27.695 Td [(\).)]TJ -15.355 -2.6205 Td [(Dans)-259(le)-265(cas)-259(g)30(\266)469(en)24(\266)475(eral,)-277(il)-259(su\261t)-259(de)-259(prouv)24(er,)-277(p)-30(our)-259(toute)]TJ /F8 1 Tf 21.934 0 Td [(J)]TJ /F2 1 Tf 0.90964 0 Td [(partie)-259(\257nie)-265(de)]TJ /F8 1 Tf 6.1687 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(,)]TJ /F8 1 Tf 0.54819 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(>)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.422 34.176 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.325 25.075 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.629 35.668 Td [(j)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.392 25.075 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.112 35.668 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.825 25.075 Td [(\).)]TJ -38.494 -2.6807 Td [(Mais,)-355(puisque)]TJ /F8 1 Tf 6.3855 0 Td [(J)]TJ /F2 1 Tf 1 0 Td [(est)-355(\257nie,)-355(le)-355(cas)-355(pr)30(\266)469(ec)24(\266)469(eden)30(t)-355(assure)]TJ /F9 1 Tf 14.386 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.802 30.34 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.717 22.394 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.983 31.831 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.157 22.394 Td [(\))-307(=)]TJ /F8 1 Tf 1.7771 0 Td [(\301)]TJ /F9 1 Tf 0.59639 1.1084 Td [(\263)]TJ 0.76506 -0.15662 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.095 30.34 Td [(j)]TJ /F10 1 Tf 0.53448 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(J)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.904 22.394 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.302 31.831 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.97 22.394 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.707 31.831 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 76.265 44.295 Td [(j)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 38.554 23.503 Td [(\264)]TJ /F2 1 Tf 0.59639 -1.1084 Td [(.)-493(L'in)24(\266)469(egalit)24(\266)475(e)]TJ 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[(2.)-289(Le)-385(th)24(\266)548(eor)18(\265)548(eme)-385(de)-385(Daniell-Stone.)]TJ /F15 1 Tf -3.488 -1.8434 Td [(D)]TJ 0.83133 0.084337 Td [(\266)]TJ -0.0060239 -0.084337 Td [(efinition.)]TJ /F12 1 Tf 5.2831 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0301 0 Td [(X)]TJ /F12 1 Tf 1.247 0 Td [(un)-337(ensem)24(ble)-337(et)]TJ /F6 1 Tf 6.8795 0 Td [(E)]TJ /F12 1 Tf 0.95181 0 Td [(un)-337(sous-espace)-343(v)24(ectoriel)-337(de)-337(l'espace)-343(des)-337(applica-)]TJ -18.217 -1.1988 Td [(tions)-337(de)]TJ /F8 1 Tf 3.7831 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(dans)]TJ /F3 1 Tf 2.3434 0 Td [(R)]TJ /F12 1 Tf 0.71687 0 Td [(.)-445(On)-337(dit)-343(que)]TJ /F6 1 Tf 5.8133 0 Td [(E)]TJ /F12 1 Tf 0.95181 0 Td [(est)]TJ /F1 1 Tf 1.5602 0 Td [(r)24(\266)548(eticul)18(\266)548(e)]TJ /F12 1 Tf 4.0964 0 Td [(lorsque)-337(:)]TJ /F6 1 Tf -5.2711 -1.2048 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(\256)]TJ /F6 1 Tf 0.92168 0 Td [(2)-283(E)-2096(j)]TJ /F8 1 Tf 3.8434 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td 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[(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F2 1 Tf 0.62048 0 Td [(=)-349(sup)0(\()]TJ /F8 1 Tf 3.0181 0 Td [(\256)-6(;)]TJ /F6 1 Tf 1.0904 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\))-379(mon)24(tren)24(t)-379(qu'il)-373(est)-355(\266)475(equiv)54(alen)24(t)-379(de)-373(dire)-373(que)]TJ /F6 1 Tf 19.151 0 Td [(E)]TJ /F2 1 Tf 0.99397 0 Td [(est)-379(r)30(\266)469(eticul)24(\266)469(e,)-385(qu'il)-379(est)-379(stable)]TJ -27.217 -1.2048 Td [(par)-331(inf)-415(ou)-337(qu'il)-331(est)-343(stable)-331(par)-337(sup)0(.)]TJ 18.663 -2.3976 Td [(101)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 339 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 338 0 R /ProcSet 2 0 R >> /Contents 337 0 R >> endobj 338 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R >> endobj 340 0 obj << /Length 19951 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4036 Td [(En)-415(particulier,)-439(raisonnan)24(t)-421(par)-421(r)30(\266)469(ecurrence,)-439(on)-421(v)24(oit)-421(que,)-445(si)]TJ /F6 1 Tf 25.867 0 Td [(E)]TJ /F2 1 Tf 1.0361 0 Td [(est)-421(r)24(\266)475(eticul)18(\266)475(e,)-445(le)-421(sup)-415(et)-421(le)-421(inf)]TJ -26.904 -1.1988 Td [(d'une)-337(famille)-337(\257nie)-331(d')24(\266)469(el)24(\266)469(emen)30(ts)-337(de)]TJ /F6 1 Tf 14.398 0 Td [(E)]TJ /F2 1 Tf 0.95181 0 Td [(son)24(t)-337(encore)-343(dans)]TJ /F6 1 Tf 7.6024 0 Td [(E)]TJ /F2 1 Tf 0.62048 0 Td [(.)]TJ -23.572 -2.0301 Td [(En)-331(application)-337(du)-337(r)24(\266)475(esultat)-343(de)-331(la)-337(section)-331(pr)24(\266)469(ec)24(\266)475(eden)24(te)-331(nous)-337(allons)-331(prouv)24(er)-337(:)]TJ /F15 1 Tf 0 -1.7892 Td [(Th)]TJ 1.4036 0.084335 Td [(\266)]TJ -0.0060239 -0.084335 Td [(eor)]TJ 1.8012 0.084335 Td [(\265)]TJ -0.0060244 -0.084335 Td [(eme.)]TJ /F12 1 Tf 2.7169 0 Td [(\()]TJ /F1 1 Tf 0.38554 0 Td [(de)-403(Daniell-Stone)]TJ /F12 1 Tf 8.3434 0 Td [(\))-1006(Soien)24(t)]TJ /F6 1 Tf 4.4337 0 Td [(E)]TJ /F12 1 Tf 0.95783 0 Td [(un)-349(espace)-349(v)24(ectoriel)-343(r)24(\266)469(eticul)24(\266)475(e)-349(de)-349(fonctions)-343(sur)]TJ -20.03 -1.1988 Td [(un)-259(ensem)18(ble)]TJ /F8 1 Tf 5.5542 0 Td [(X)]TJ /F12 1 Tf 1.1687 0 Td [(et)]TJ /F8 1 Tf 1.0964 0 Td [(\256)]TJ /F12 1 Tf 0.90361 0 Td [(une)-259(forme)-259(lin)24(\266)469(eaire)-265(sur)]TJ /F6 1 Tf 9.5843 0 Td [(E)]TJ /F12 1 Tf 0.62048 0 Td [(.)-421(On)-259(note)]TJ /F6 1 Tf 4.4398 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.828 79.926 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 30.151 56.003 Td [(l'ensem)24(ble)-259(des)-265(fonctions)-265(p)-24(ositiv)24(es)]TJ -24.819 -1.1988 Td [(qui)-331(appartiennen)24(t)-337(\265)500(a)]TJ /F6 1 Tf 8.9518 0 Td [(E)]TJ /F12 1 Tf 0.95181 0 Td [(et)-331(on)-337(fait)-337(les)-337(h)30(yp)-36(oth)24(\265)469(eses)-337(suiv)54(an)24(tes)-337(:)]TJ /F1 1 Tf -9.0904 -1.5422 Td [(1.)-668(propri)24(\266)548(et)18(\266)548(e)-385(de)-385(Stone)]TJ /F6 1 Tf 11.946 0 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-277(E)-427(8)]TJ /F8 1 Tf 2.4518 0 Td [(\270)]TJ /F6 1 Tf 0.86144 0 Td [(2)]TJ /F3 1 Tf 0.93976 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.06 76.004 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.331 53.262 Td [(inf)-78(\()]TJ /F8 1 Tf 1.6084 0 Td [(\270;)-168(f)]TJ /F2 1 Tf 1.6265 0 Td [(\))]TJ /F6 1 Tf 0.66868 0 Td [(2)-283(E)]TJ /F12 1 Tf 1.8916 0 Td [(;)]TJ /F1 1 Tf -25.982 -1.5482 Td [(2.)-668(p)-30(ositivit)18(\266)548(e)]TJ /F6 1 Tf 7.247 0 Td [(8)]TJ /F8 1 Tf 0.56024 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 23.319 73.788 Td [(+)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.964 51.714 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(0)]TJ /F12 1 Tf 0.83735 0 Td [(;)]TJ /F1 1 Tf -16.006 -1.5422 Td [(3.)-831(propri)24(\266)548(et)24(\266)548(e)-481(de)-475(Daniell)]TJ /F12 1 Tf 12.44 0 Td [(p)-30(our)-415(toute)-415(suite)-415(d)30(\266)469(ecroissan)24(te)-415(d')30(\266)469(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 18.295 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.483 71.581 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.331 50.171 Td [(de)]TJ /F6 1 Tf 1.4096 0 Td [(E)]TJ /F12 1 Tf 1.0361 0 Td [(qui)-409(tend)]TJ -35.446 -1.1988 Td [(simplemen)24(t)-337(v)24(ers)-337(0)-331(on)-337(a)]TJ /F8 1 Tf 11.434 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.172 69.866 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.831 48.973 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(!)]TJ /F2 1 Tf 1.2771 0 Td [(0)]TJ /F12 1 Tf 0.5 0 Td [(.)]TJ -15.946 -1.5422 Td [(Alors)-259(il)-259(existe)-259(une)-259(tribu)]TJ /F6 1 Tf 10.361 0 Td [(T)]TJ /F12 1 Tf 1.0602 0 Td [(et)-259(une)-259(mesure)]TJ /F8 1 Tf 6.2229 0 Td [(m)]TJ /F12 1 Tf 1.1386 0 Td [(sur)]TJ /F8 1 Tf 1.6024 0 Td [(X)]TJ /F12 1 Tf 1.1626 0 Td [(p)-30(our)-259(lesquelles)-265(toute)]TJ /F8 1 Tf 9.1265 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(E)]TJ /F12 1 Tf 1.8193 0 Td [(est)-265(in)30(t)24(\266)469(egrable)]TJ -33.367 -1.6566 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F12 1 Tf 2.1627 0 Td [(.)]TJ /F1 1 Tf -7.6807 -2.3012 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-361(m)24(\266)469(etho)-30(de)-361(v)54(a)-361(consister)-367(\265)500(a)-361(d)30(\266)469(e\257nir)-361(une)-367(fonctionnelle)]TJ /F8 1 Tf 22.892 0 Td [(\301)]TJ /F2 1 Tf 0.95783 0 Td [(sur)]TJ /F6 1 Tf 1.7048 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.845 61.995 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.09 43.473 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-361(qui)]TJ -37.054 -1.1988 Td [(prolonge)-397(l'application)]TJ /F8 1 Tf 9.9036 0 Td [(\256)]TJ /F2 1 Tf 1.0422 0 Td [(d)30(\266)469(e\257nie)-397(sur)]TJ /F6 1 Tf 4.9639 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 31.155 60.28 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.434 42.274 Td [(.)-626(On)-397(mon)24(trera)-397(que)]TJ /F8 1 Tf 8.9277 0 Td [(\301)]TJ /F2 1 Tf 0.99398 0 Td [(est)-397(croissan)24(te,)-415(p)-24(ositiv)24(emen)24(t)]TJ -27.024 -1.2048 Td [(homog)24(\265)469(ene)-337(et)-337(d)30(\266)469(enom)24(brablemen)30(t)-337(sous-additiv)24(e.)]TJ 0 -1.5422 Td [(L')24(\266)469(etude)-415(pr)24(\266)475(ec)24(\266)469(eden)30(te)-415(p)-30(ermettra)-415(alors)-415(de)-415(d)24(\266)475(e\257nir)-415(une)-415(tribu)]TJ /F6 1 Tf 25.181 0 Td [(T)]TJ /F2 1 Tf 1.2108 0 Td [(et)-415(une)-415(mesure)]TJ /F8 1 Tf 6.6988 0 Td [(m)]TJ /F2 1 Tf 1.2892 0 Td [(telle)-415(que)]TJ /F8 1 Tf 4.1867 0 Td [(\301)]TJ /F2 1 Tf -38.566 -1.1988 Td [(co)108(\304)385(\263ncide)-253(a)24(v)24(ec)-253(l'in)24(t)24(\266)469(egrale)-259(d)24(\266)475(e\257nie)-259(par)]TJ /F8 1 Tf 15.223 0 Td [(m)]TJ /F2 1 Tf 1.1325 0 Td [(sur)-259(les)-259(fonctions)]TJ /F6 1 Tf 7.1446 0 Td [(T)]TJ /F2 1 Tf 1.0542 0 Td [(mesurables)-259(p)-30(ositiv)30(es.)-421(Il)-253(ne)-253(restera)]TJ -24.554 -1.1988 Td [(plus)-331(qu')0(\265)500(a)-337(prouv)30(er)-343(que)-337(les)-337(fonctions)-331(de)]TJ /F6 1 Tf 16.892 0 Td [(E)]TJ /F2 1 Tf 0.95181 0 Td [(son)24(t)]TJ /F6 1 Tf 2.1446 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurables)-337(p)-30(our)-337(en)-331(d)24(\266)475(eduire)-337(le)-337(r)24(\266)475(esultat.)]TJ /F1 1 Tf -19.97 -1.5482 Td [(1.)-668(La)-385(construction)-391(de)]TJ /F8 1 Tf 11.361 0 Td [(\301)]TJ /F1 1 Tf 0.59639 0 Td [(.)]TJ /F2 1 Tf -12.771 -1.5422 Td [(Si)-367(\()]TJ /F8 1 Tf 1.5843 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.724 48.495 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8253 34.039 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.759 48.495 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.8614 34.039 Td [(est)-373(une)-367(famille)-367(d)30(\266)469(enom)24(brable)-367(d')24(\266)469(el)24(\266)475(emen)24(ts)-367(de)]TJ /F6 1 Tf 19.078 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 42.164 48.495 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.494 34.039 Td [(nous)-367(app)-30(ellerons)]TJ /F1 1 Tf 7.6627 0 Td [(taille)]TJ /F2 1 Tf 2.8614 0 Td [(de)-367(cette)]TJ -35.687 -1.3976 Td [(famille)-481(la)-481(somme)]TJ /F9 1 Tf 8.1506 0.94579 Td [(X)]TJ /F8 1 Tf 1.6084 -0.94579 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.897 46.495 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.036 32.641 Td [(\))-481(et)-487(nous)-481(dirons)-481(que)-487(c'est)-487(une)]TJ /F1 1 Tf 14.319 0 Td [(famille)-554(ma)-60(joran)30(te)]TJ /F2 1 Tf 9.512 0 Td [(p)-30(our)-487(un)]TJ -35.566 -1.7048 Td [(\266)469(el)24(\266)469(emen)30(t)]TJ /F8 1 Tf 3.7229 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(de)]TJ /F6 1 Tf 1.3313 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.19 44.056 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.675 30.936 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-337(si)-337(l'on)-337(a)]TJ /F8 1 Tf 4.506 0 Td [(f)]TJ /F11 1 Tf 0.87952 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.336 44.056 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.934 30.936 Td [(.)]TJ -17.602 -1.741 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.75 41.564 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.97 29.195 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-337(on)-337(note)-337(alors)]TJ /F8 1 Tf 6.7349 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))-337(la)-331(b)-30(orne)-337(inf)30(\266)469(erieure)-337(des)-337(tailles)-337(des)-337(familles)-337(ma)-60(joran)30(tes)]TJ -15.253 -1.1988 Td [(p)-30(our)]TJ /F8 1 Tf 2.3614 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(.)]TJ -2.9639 -1.5482 Td [(Si)]TJ /F8 1 Tf 1.2289 0 Td [(f)]TJ /F11 1 Tf 0.98795 0 Td [(6)]TJ /F8 1 Tf 1.1687 0 Td [(g)]TJ /F2 1 Tf 0.90964 0 Td [(il)-403(est)-403(clair)-403(que)-403(toute)-397(famille)-403(ma)-60(joran)30(te)-403(p)-30(our)]TJ /F8 1 Tf 20.084 0 Td [(g)]TJ /F2 1 Tf 0.90964 0 Td [(est)-403(aussi)-403(une)-403(famille)-403(ma)-54(joran)24(te)]TJ -25.289 -1.1988 Td [(p)-30(our)]TJ /F8 1 Tf 2.3976 0 Td [(f)]TJ /F2 1 Tf 0.95783 0 Td [(donc)-367(a)-367(une)-367(taille)-373(minor)24(\266)475(ee)-373(par)]TJ /F8 1 Tf 13.373 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)-548(Il)-361(s'ensuit)]TJ /F8 1 Tf 5.8675 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.72289 0 Td [(6)]TJ /F8 1 Tf 1.1084 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\).)-548(Autremen)24(t)-367(dit)]TJ /F8 1 Tf 7.8916 0 Td [(\301)]TJ /F2 1 Tf 0.95783 0 Td [(est)]TJ -37.934 -1.1988 Td [(croissan)30(te.)]TJ 0 -1.7892 Td [(Consid)30(\266)469(erons)-415(une)-415(suite)-415(d')24(\266)469(el)24(\266)475(emen)24(ts)]TJ /F8 1 Tf 15.157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.017 31.642 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.934 22.262 Td [(de)]TJ /F6 1 Tf 1.4157 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 34.44 31.642 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.735 22.262 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-415(telle)-415(que)-415(c)18(haque)]TJ /F8 1 Tf 8.4036 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.388 31.642 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.458 22.262 Td [(\))-415(soit)-415(\257ni.)-686(Fixons)]TJ /F8 1 Tf -31.127 -1.1988 Td [(")-277(>)]TJ /F2 1 Tf 1.8012 0 Td [(0)0(,)-415(p)-30(our)-403(c)24(haque)]TJ /F8 1 Tf 7.0301 0 Td [(n)]TJ /F2 1 Tf 1 0 Td [(il)-403(existe)-403(\()]TJ /F8 1 Tf 4.2289 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.569 29.926 Td [(i;n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.03 21.063 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.655 29.926 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 45.193 41.62 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.536 21.063 Td [(famille)-403(ma)-60(joran)24(te)-403(p)-24(our)]TJ /F8 1 Tf 10.602 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.56 29.926 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.578 21.063 Td [(don)24(t)-403(la)-403(taille)-403(est)-403(in-)]TJ -30.247 -1.1988 Td [(f)24(\266)475(erieure)-469(\265)500(a)]TJ /F8 1 Tf 4.6747 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.431 28.211 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.024 19.864 Td [(\))-307(+)]TJ /F8 1 Tf 1.7771 0 Td [("=)]TJ /F2 1 Tf 0.96988 0 Td [(2)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.853 29.012 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.813 19.864 Td [(.)-819(La)-463(famille)-463(de)-457(tous)-457(les)]TJ /F8 1 Tf 11.392 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.759 28.211 Td [(i;n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.307 19.864 Td [(est)-463(alors)-457(une)-463(famille)-457(d)24(\266)469(enom)24(brable)]TJ -23.976 -1.5422 Td [(d')24(\266)475(el)24(\266)469(emen)24(ts)-397(de)]TJ /F6 1 Tf 6.3855 0 Td [(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.793 26.012 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.096 18.322 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-397(telle)-397(que)]TJ /F9 1 Tf 4.9398 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.802 26.012 Td [(i;n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.976 18.322 Td [(=)]TJ /F9 1 Tf 1.2048 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.543 24.504 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 25.843 19.436 Td [(\263)]TJ 0.87349 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.078 24.512 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 55.566 34.054 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.44 18.322 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.526 26.012 Td [(i;n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 30.084 19.436 Td [(\264)]TJ /F11 1 Tf 0.98193 -1.1145 Td [(>)]TJ /F9 1 Tf 1.2048 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.112 24.504 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.934 18.322 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.259 26.012 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.964 18.322 Td [(.)-638(Il)-397(s'agit)-397(donc)-397(d'une)]TJ -29.633 -2.5422 Td [(famille)-337(ma)-54(joran)24(te)-331(p)-30(our)]TJ /F9 1 Tf 10.446 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.509 20.866 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.434 15.78 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.655 22.366 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.47 15.78 Td [(.)-445(La)-337(taille)-337(de)-337(cette)-337(famille)-337(est)]TJ /F9 1 Tf -8.9819 -1.6566 Td [(X)]TJ /F8 1 Tf 1.6084 -0.94578 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.181 18.642 Td [(i;n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.771 13.177 Td [(\))-283(=)]TJ /F9 1 Tf 1.7711 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.172 17.142 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 17.199 14.286 Td [(\263)]TJ 0.87952 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.707 17.142 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 38.289 23.764 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.801 13.177 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.638 18.642 Td [(i;n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.476 13.177 Td [(\))]TJ /F9 1 Tf 0.38554 1.1084 Td [(\264)]TJ /F11 1 Tf 0.87952 -1.1084 Td [(6)]TJ /F9 1 Tf 1.1024 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.483 17.142 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.337 13.177 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.353 18.642 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.741 13.177 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [("=)]TJ /F2 1 Tf 0.96385 0 Td [(2)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.543 19.443 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.367 13.177 Td [(\))-283(=)-277(2)]TJ /F8 1 Tf 2.2229 0 Td [(")]TJ /F2 1 Tf 0.68674 0 Td [(+)]TJ /F9 1 Tf 1.0482 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.914 17.142 Td [(n)]TJ /F10 1 Tf 0.71552 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.988 13.177 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.474 18.642 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.006 13.177 Td [(\).)]TJ -34.675 -2.994 Td [(On)-337(en)-331(d)24(\266)475(eduit)]TJ /F8 1 Tf 6.1145 0 Td [(\301)]TJ /F9 1 Tf 0.59639 1.1145 Td [(\263)]TJ 0.80723 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.328 12.857 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.512 10.183 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.466 14.357 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 15.542 11.298 Td [(\264)]TJ /F11 1 Tf 0.87952 -1.1145 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(2)]TJ /F8 1 Tf 0.5 0 Td [(")]TJ /F2 1 Tf 0.68675 0 Td [(+)]TJ /F9 1 Tf 1.0482 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.147 12.857 Td [(n)]TJ /F10 1 Tf 0.70689 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.373 10.183 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.698 14.357 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.392 10.183 Td [(\).)]TJ -18.06 -2.9458 Td [(Cela)-337(a)30(y)18(an)30(t)-337(lieu)-337(p)-30(our)-331(tout)]TJ /F8 1 Tf 11.422 0 Td [(")]TJ /F2 1 Tf 0.8012 0 Td [(on)-337(conclut)]TJ /F8 1 Tf 4.8916 0 Td [(\301)]TJ /F9 1 Tf 0.59638 1.1145 Td [(\263)]TJ 0.81325 -0.16265 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.069 8.6418 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.512 7.2376 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.207 10.142 Td [(n)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 26.548 8.3521 Td [(\264)]TJ /F11 1 Tf 0.8735 -1.1145 Td [(6)]TJ /F9 1 Tf 1.1024 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.75 8.6418 Td [(n)]TJ /F10 1 Tf 0.71552 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.187 7.2376 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.31 10.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.205 7.2376 Td [(\).)]TJ /F5 1 Tf -26.873 -3.3494 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(102)]TJ ET endstream endobj 342 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 341 0 R /ProcSet 2 0 R >> /Contents 340 0 R >> endobj 341 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F16 81 0 R >> endobj 343 0 obj << /Length 26367 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4036 Td [(En)-349(fait)-349(cette)-349(relation)-343(est)-325(\266)469(evidemmen)30(t)-355(encore)-349(v)54(alable)-349(si)-349(l'un)-355(au)-343(moins)-349(des)]TJ /F8 1 Tf 32.331 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.009 88.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.687 62.22 Td [(\))-349(est)-349(in\257ni.)]TJ -34.355 -1.1988 Td [(La)-337(fonctionnelle)]TJ /F8 1 Tf 7.3193 0 Td [(\301)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(donc)-337(d)30(\266)469(enom)24(brablemen)30(t)-337(sous-additiv)24(e.)]TJ -8.253 -2.2289 Td [(En\257n,)-283(p)-30(our)]TJ /F8 1 Tf 5.2048 0 Td [(\270)]TJ /F2 1 Tf 0.8494 0 Td [(r)24(\266)475(eel)-271(strictemen)18(t)-271(p)-30(ositif,)-283(il)-265(est)-277(clair)-265(que)-271(les)-271(familles)-277(ma)-54(joran)24(tes)-271(de)]TJ /F8 1 Tf 27.934 0 Td [(\270f)]TJ /F2 1 Tf 1.4458 0 Td [(s'obtien-)]TJ -35.434 -1.1988 Td [(nen)30(t)-289(en)-283(m)24(ultiplian)24(t)-289(par)]TJ /F8 1 Tf 10.331 0 Td [(\270)]TJ /F2 1 Tf 0.86747 0 Td [(les)-283(familles)-289(ma)-60(joran)30(tes)-289(p)-30(our)]TJ /F8 1 Tf 12.434 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(.)-433(On)-283(en)-283(tire)]TJ /F8 1 Tf 5.3916 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(\270f)]TJ /F2 1 Tf 1.1807 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\270\301)]TJ /F2 1 Tf 1.1807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\).)-427(Cette)]TJ -35.681 -1.1988 Td [(relation)-379(est)-391(encore)-385(v)54(alable)-385(p)-30(our)]TJ /F8 1 Tf 14.319 0 Td [(\270)]TJ /F2 1 Tf 0.93976 0 Td [(=)-361(0)-379(puisqu'alors)-385(les)-379(deux)-385(mem)24(bres)-385(de)-379(la)-385(relation)-379(son)18(t)]TJ -15.259 -1.2048 Td [(n)30(uls.)-415(Lorsque)]TJ /F8 1 Tf 6.1325 0 Td [(\270)]TJ /F2 1 Tf 0.86145 0 Td [(=)-283(+)]TJ /F6 1 Tf 1.8373 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-265(en)-246(consid)30(\266)469(eran)30(t)-253(que)-246(\(+)]TJ /F6 1 Tf 9.9578 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.84337 0 Td [(est)-253(la)-246(somme)-246(d'une)-246(famille)-246(d)24(\266)469(enom)24(brable)]TJ -22.024 -1.1988 Td [(in\257nie)-325(de)-325(termes)-301(\266)475(egaux)-331(\265)499(a)]TJ /F8 1 Tf 11.217 0 Td [(f)]TJ /F2 1 Tf 0.92169 0 Td [(on)-325(a)-325(\(d'apr)24(\265)475(es)-325(la)-331(sous-additivit)30(\266)469(e\))]TJ /F8 1 Tf 14.229 0 Td [(\301)]TJ /F9 1 Tf 0.59638 0.81325 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39156 0 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\242)]TJ /F11 1 Tf 0.73494 -0.81325 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\).)-445(P)30(ar)]TJ -36.512 -1.3012 Td [(ailleurs,)-246(p)-30(our)-228(tout)-228(en)24(tier)]TJ /F8 1 Tf 10.669 0 Td [(n)]TJ /F2 1 Tf 0.83133 0 Td [(on)-228(a)]TJ /F8 1 Tf 2.012 0 Td [(\301)]TJ /F9 1 Tf 0.59638 0.81325 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(\(+)]TJ /F6 1 Tf 1.1627 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F9 1 Tf 0.59638 0.81325 Td [(\242)]TJ /F11 1 Tf 0.74096 -0.81325 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(nf)]TJ /F2 1 Tf 1.1988 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(n\301)]TJ /F2 1 Tf 1.1988 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\),)-253(d'o)-30(\265)530(u)]TJ /F8 1 Tf 3.0301 0 Td [(\301)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\242)]TJ /F11 1 Tf 0.73494 -0.81325 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(\(+)]TJ /F6 1 Tf 1.1687 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39156 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ -38.777 -1.1988 Td [(et)-337(donc)-331(l')24(\266)469(egalit)24(\266)475(e.)-451(Ainsi)]TJ /F8 1 Tf 10.259 0 Td [(\301)]TJ /F2 1 Tf 0.92771 0 Td [(est)-343(p)-24(ositiv)24(emen)24(t)-337(homog)24(\265)475(ene.)]TJ -11.187 -2.6386 Td [(Le)-391(pro)-30(c)24(\266)469(ed)30(\266)469(e)-397(d)30(\266)469(etaill)24(\266)469(e)-391(pr)24(\266)469(ec)24(\266)475(edemmen)24(t)-397(p)-30(ermet)-391(donc)-391(de)-391(d)24(\266)475(e\257nir)-397(une)-391(tribu)]TJ /F6 1 Tf 30.47 0 Td [(T)]TJ /F2 1 Tf 1.1867 0 Td [(et)-397(une)-391(mesure)]TJ /F8 1 Tf 6.6265 0 Td [(m)]TJ /F2 1 Tf -38.283 -1.6566 Td [(sur)]TJ /F6 1 Tf 1.6747 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(telle)-337(que)]TJ /F8 1 Tf 4.0301 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.494 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(f)]TJ /F6 1 Tf 0.92771 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable)-337(p)-30(ositiv)30(e.)]TJ /F1 1 Tf -16.922 -2.2711 Td [(2.)]TJ /F8 1 Tf 1.5602 0 Td [(\301)]TJ /F1 1 Tf 0.98193 0 Td [(co)126(\304)445(\263ncide)-385(a)24(v)30(ec)]TJ /F8 1 Tf 6.9518 0 Td [(\256)]TJ /F1 1 Tf 1.0241 0 Td [(sur)]TJ /F6 1 Tf 1.9518 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.388 64.073 Td [(+)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 19.807 44.924 Td [(.)]TJ /F2 1 Tf -14.476 -1.8193 Td [(P)30(our)]TJ /F8 1 Tf 2.4639 0 Td [(f)]TJ /F6 1 Tf 0.92771 0 Td [(2)-331(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.664 61.469 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.91 43.105 Td [(,)-373(la)-367(famille)-367(r)30(\266)469(eduite)-367(\265)500(a)-361(l')24(\266)469(el)24(\266)469(emen)30(t)]TJ /F8 1 Tf 13.645 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(est)-367(ma)-60(joran)24(te)-367(p)-24(our)]TJ /F8 1 Tf 8.8012 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(et)-367(de)-361(taille)]TJ /F8 1 Tf 5.0964 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\),)-379(par)]TJ -36.675 -1.1988 Td [(suite)]TJ /F8 1 Tf 2.3916 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)]TJ -7.3313 -2.0181 Td [(Prouv)30(ons)-397(l'in)24(\266)469(egalit)24(\266)469(e)-397(in)30(v)24(erse)]TJ /F8 1 Tf 12.464 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.76506 0 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)-626(Cela)-397(revien)24(t)-397(\265)499(a)-391(prouv)24(er)]TJ /F8 1 Tf 11.651 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F11 1 Tf 0.76506 0 Td [(6)]TJ /F9 1 Tf 1.1566 0.94579 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94579 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.172 56.866 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.681 39.888 Td [(\))-397(p)-30(our)]TJ -36.349 -1.4036 Td [(toute)-295(famille)-289(\()]TJ /F8 1 Tf 6.1687 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.284 54.857 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.41 38.485 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.319 54.857 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.373 38.485 Td [(ma)-54(joran)24(te)-295(p)-30(our)]TJ /F8 1 Tf 7.0602 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(.)-427(P)24(our)-295(une)-289(telle)-295(famille)-295(on)-295(p)-30(eut)-295(consid)30(\266)469(erer)]TJ /F8 1 Tf 18.59 0 Td [(I)]TJ /F2 1 Tf 0.81325 0 Td [(comme)]TJ -36.108 -1.3976 Td [(la)-301(r)30(\266)469(eunion)-301(d'une)-307(suite)-301(croissan)24(te)-301(de)-301(parties)-301(\257nies)]TJ /F8 1 Tf 21.361 0 Td [(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.828 52.857 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.976 37.087 Td [(et)-307(in)30(tro)-36(duire)-301(les)-301(fonctions)]TJ /F8 1 Tf 11.422 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.06 52.857 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.693 37.087 Td [(=)]TJ /F9 1 Tf 1.1687 0.94578 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.75 51.357 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76725 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 85.855 71.584 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.584 37.087 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 52.857 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 34.84 Td [(et)]TJ /F8 1 Tf 1.2169 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.069 49.642 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.9398 34.84 Td [(=)-361(inf)-78(\()]TJ /F8 1 Tf 2.747 0 Td [(f)-54(;)-162(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.388 49.642 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.693 34.84 Td [(\).)-596(Remarquons)-385(alors)-385(que)]TJ /F8 1 Tf 11.446 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.224 49.642 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.542 34.84 Td [(somme)-379(\257nie)-385(d')24(\266)469(el)24(\266)469(emen)30(ts)-385(de)]TJ /F6 1 Tf 11.964 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.422 49.642 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.078 34.84 Td [(appartien)24(t)-385(\265)500(a)]TJ /F6 1 Tf -33.747 -1.1988 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3793 47.926 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8554 33.641 Td [(puisque)]TJ /F6 1 Tf 3.6446 0 Td [(E)]TJ /F2 1 Tf 0.95181 0 Td [(est)-337(un)-337(sous-espace)-343(v)24(ectoriel,)-337(de)-331(m)24(^)475(eme)]TJ /F8 1 Tf 16.813 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.147 47.926 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.578 33.641 Td [(2)-277(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.431 47.926 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.048 33.641 Td [(puisque)]TJ /F6 1 Tf 3.6446 0 Td [(E)]TJ /F2 1 Tf 0.95181 0 Td [(est)-337(r)24(\266)475(eticul)24(\266)469(e.)]TJ -31.313 -2.006 Td [(Il)-397(est)-397(clair)-397(que)-397(la)-397(suite)-397(des)]TJ /F8 1 Tf 12.295 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.905 45.056 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.042 31.635 Td [(est)-397(croissan)24(te)-397(et)-397(tend)-403(v)24(ers)]TJ /F9 1 Tf 11.982 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.526 45.056 Td [(i)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 33.928 31.635 Td [(>)]TJ /F8 1 Tf 1.1627 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)-632(P)30(ar)-397(cons)24(\266)469(equen)24(t)-397(la)]TJ -30.355 -1.6988 Td [(suite)-295(des)]TJ /F8 1 Tf 4.0482 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.121 42.625 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.705 29.936 Td [(est)-301(croissan)24(te)-295(et)-301(tend)-301(v)24(ers)-295(inf)-78(\()]TJ /F8 1 Tf 13.09 0 Td [(f)-54(;)]TJ /F9 1 Tf 0.98193 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.595 42.625 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.301 29.936 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)-433(Il)-295(s'ensuit)-295(que)-301(la)-295(suite)-295(des)]TJ /F8 1 Tf 12.175 0 Td [(f)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 0.92169 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 42.625 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 28.593 Td [(est)-343(une)-337(suite)-337(d)24(\266)475(ecroissan)24(te)-337(d')24(\266)475(el)18(\266)475(emen)24(ts)-337(de)]TJ /F6 1 Tf 17.723 0 Td [(E)]TJ /F2 1 Tf 0.95181 0 Td [(qui)-337(tend)-343(v)24(ers)-343(0.)-451(La)-343(propri)24(\266)469(et)30(\266)469(e)-343(de)-337(Daniell)-337(assure)]TJ -18.675 -1.2048 Td [(alors)-337(que)]TJ /F8 1 Tf 4.259 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F6 1 Tf 0.8253 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.509 38.978 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.476 27.388 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.569 38.978 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.512 27.388 Td [(\))-337(tend)-337(v)24(ers)-337(0,)-331(c'est-)0(\265)500(a-dire)-337(que)]TJ /F8 1 Tf 13.163 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.362 38.978 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.741 27.388 Td [(\))-337(tend)-337(v)24(ers)]TJ /F8 1 Tf 5.0663 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\).)]TJ -37.102 -2.0181 Td [(Cep)-24(endan)24(t)]TJ /F9 1 Tf 5.1145 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.103 34.599 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.06 25.37 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.56 36.09 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14 25.37 Td [(\))-397(est)-397(la)-391(limite)-397(des)-391(sommes)-397(partielles)]TJ /F9 1 Tf 16.506 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.5 34.599 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 63.145 48.15 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.229 25.37 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.422 36.09 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.175 25.37 Td [(\))-379(=)]TJ /F8 1 Tf 1.9157 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.81 36.09 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.151 25.37 Td [(\).)-620(P)24(ar)-391(ailleurs)]TJ /F8 1 Tf -32.819 -2.247 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3276 32.883 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7771 23.123 Td [(6)]TJ /F8 1 Tf 1.1867 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.086 32.883 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.4036 23.123 Td [(prouv)30(e)]TJ /F8 1 Tf 3.3614 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.948 32.883 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.06 23.123 Td [(\241)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.336 32.883 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.566 23.123 Td [(2)-415(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 26 32.883 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.247 23.123 Td [(et)-415(donc)]TJ /F8 1 Tf 3.7229 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.026 32.883 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.295 23.123 Td [(\241)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.414 32.883 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.386 23.123 Td [(\))]TJ /F11 1 Tf 0.8012 0 Td [(>)]TJ /F2 1 Tf 1.1928 0 Td [(0,)-433(c'est-)0(\265)500(a-dire)]TJ /F8 1 Tf 6.4096 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.397 32.883 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.861 23.123 Td [(\))]TJ /F11 1 Tf 0.8012 0 Td [(6)]TJ /F8 1 Tf 1.1867 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.19 32.883 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.904 23.123 Td [(\).)-692(En)]TJ -36.572 -1.3976 Td [(passan)24(t)-337(\265)499(a)-331(la)-337(limite)-331(dans)-337(cette)-337(relation)-331(on)-337(obtien)30(t)-337(bien)]TJ /F8 1 Tf 23.807 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0602 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.103 30.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.048 21.726 Td [(\).)]TJ /F1 1 Tf -29.904 -1.9639 Td [(3.)-668(Un)-385(lemme)-385(pr)18(\266)548(eliminaire.)]TJ /F12 1 Tf -0.81325 -1.8193 Td [(Soit)]TJ /F8 1 Tf 1.994 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-283(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 13.845 25.461 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 10.337 17.942 Td [(.)-427(Notons)]TJ /F8 1 Tf 4.0723 0 Td [(A)]TJ /F12 1 Tf 1.0241 0 Td [(l'ensem)30(ble)-277(des)]TJ /F8 1 Tf 6.4217 0 Td [(x)]TJ /F12 1 Tf 0.8494 0 Td [(tels)-271(que)]TJ /F8 1 Tf 3.5843 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.66868 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(0)]TJ /F12 1 Tf 0.77711 0 Td [(et)]TJ /F8 1 Tf 1.1084 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.086 26.271 Td [(c)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.886 17.942 Td [(son)-277(compl)24(\266)469(emen)24(taire.)-427(Alors)]TJ -27.554 -1.1988 Td [(p)-30(our)-331(toute)]TJ /F8 1 Tf 4.9759 0 Td [(g)]TJ /F6 1 Tf 0.78916 0 Td [(2)-283(F)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.267 23.745 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.428 16.744 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(on)-337(a)]TJ /F8 1 Tf 2.2229 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.845 23.745 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.602 16.744 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.026 23.745 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 55.747 33.584 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.301 16.744 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)]TJ /F2 1 Tf -23.355 -2.2289 Td [(Remarquons)-343(tout)-349(d'ab)-36(ord)-343(que)-355(l'on)-349(a)]TJ /F8 1 Tf 16.355 0 Td [(g)]TJ /F2 1 Tf 0.81325 0 Td [(=)]TJ /F8 1 Tf 1.0783 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.276 20.556 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.53 14.515 Td [(+)]TJ /F8 1 Tf 1.012 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.517 20.556 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 56.434 29.138 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.645 14.515 Td [(,)-349(la)-343(sous-additivit)24(\266)469(e)-349(de)]TJ /F8 1 Tf 9.8133 0 Td [(\301)]TJ /F2 1 Tf 0.94578 0 Td [(assure)-343(alors)]TJ /F8 1 Tf -34.072 -1.1988 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F11 1 Tf 0.66868 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.181 18.84 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.265 13.316 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.371 18.84 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 31.072 26.728 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.958 13.316 Td [(\))-337(et)-337(il)-337(su\261t)-337(donc)-337(de)-331(prouv)24(er)-337(l'in)24(\266)469(egalit)24(\266)475(e)-343(in)30(v)24(erse.)]TJ -10.627 -1.8193 Td [(Supp)-24(osons)-409(dans)-409(un)-409(premier)-409(temps)]TJ /F8 1 Tf 15.705 0 Td [(g)]TJ /F6 1 Tf 0.91566 0 Td [(2)-403(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 33.69 16.237 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.614 11.497 Td [(et)-409(in)24(tro)-30(duisons)-415(les)-409(fonctions)]TJ /F8 1 Tf 12.849 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.302 16.237 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.886 11.497 Td [(=)-403(inf)-78(\()]TJ /F8 1 Tf 2.7892 0 Td [(g)-36(;)-168(nf)]TJ /F2 1 Tf 2.1506 0 Td [(\).)]TJ -38.494 -1.1988 Td [(Elles)-415(formen)24(t)-421(une)-415(suite)-415(croissan)24(te)-415(d')24(\266)469(el)24(\266)469(emen)30(ts)-421(de)]TJ /F6 1 Tf 21.855 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.655 14.521 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.789 10.298 Td [(qui)-421(tend)-415(simplemen)18(t)-415(v)24(ers)]TJ /F8 1 Tf 11.687 0 Td [(g)]TJ /F2 1 Tf 0.92771 0 Td [(lorsque)]TJ /F8 1 Tf -36.072 -1.1988 Td [(f)-397(>)]TJ /F2 1 Tf 1.9518 0 Td [(0)-343(et)-343(reste)-343(n)24(ulle)-343(lorsque)]TJ /F8 1 Tf 10.283 0 Td [(f)]TJ /F2 1 Tf 0.88554 0 Td [(=)-295(0.)-463(On)-343(a)-343(donc)]TJ /F8 1 Tf 7.2289 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.284 12.806 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.693 9.0991 Td [(=)-295(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.147 12.728 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.699 9.0991 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.612 12.806 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.717 9.0991 Td [(.)-469(P)30(ar)-343(ailleurs)]TJ /F8 1 Tf 6.0964 0 Td [(g)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.293 12.806 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.922 9.0991 Td [(est)-343(dans)]TJ /F6 1 Tf -35.59 -1.2048 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3793 11.081 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5241 7.8942 Td [(,)-331(a)24(v)24(ec)]TJ /F8 1 Tf 2.8072 0 Td [(g)]TJ /F6 1 Tf 0.73494 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.517 11.081 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 12.367 7.8942 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.741 11.081 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 30.193 15.897 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.524 7.8942 Td [(.)-445(La)-337(co)108(\304)385(\263ncidence)-331(de)]TJ /F8 1 Tf 8.7952 0 Td [(\301)]TJ /F2 1 Tf 0.92771 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(\256)]TJ /F2 1 Tf 0.9759 0 Td [(sur)]TJ /F6 1 Tf 1.6747 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 42.353 11.081 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.59 7.8942 Td [(assure)-337(par)-337(cons)24(\266)475(equen)18(t)]TJ /F6 1 Tf -23.289 -1.6084 Td [(\262)]TJ /F8 1 Tf 1.5 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.043 8.7797 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.343 6.2858 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.776 8.7797 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.747 6.2858 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.181 8.7797 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.524 6.2858 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.845 8.7797 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.88 6.2858 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.25 8.7797 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.657 6.2858 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.284 8.7797 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 79.867 12.68 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.355 6.2858 Td [(\).)]TJ -16.361 -2.3976 Td [(103)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 345 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 344 0 R /ProcSet 2 0 R >> /Contents 343 0 R >> endobj 344 0 obj << /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R >> endobj 346 0 obj << /Length 27806 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4518 Td [(On)-355(p)-30(eut)-355(alors)-337(\266)475(ecrire)]TJ /F8 1 Tf 9.1867 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.457 88.754 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.849 62.171 Td [(=)]TJ /F9 1 Tf 1.1506 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.241 87.245 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.669 62.171 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.534 88.754 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.09 62.171 Td [(en)-355(p)-30(osan)24(t)]TJ /F8 1 Tf 4.6084 0 Td [(u)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.164 88.754 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.03 62.171 Td [(=)]TJ /F8 1 Tf 1.0904 0 Td [(g)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.5 88.754 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.404 62.171 Td [(et,)-361(p)-30(our)]TJ /F8 1 Tf 3.8554 0 Td [(n)]TJ /F11 1 Tf 0.91566 0 Td [(>)]TJ /F2 1 Tf 1.0904 0 Td [(1,)]TJ /F8 1 Tf 1.1386 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.483 88.754 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.837 62.171 Td [(=)]TJ /F8 1 Tf 1.0904 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.957 88.754 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.187 62.171 Td [(\241)]TJ /F8 1 Tf 1.0121 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.207 88.754 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.247 62.171 Td [(.)-506(Les)]TJ -36.916 -2.6867 Td [(fonctions)]TJ /F8 1 Tf 4.2952 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.595 84.909 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.12 59.485 Td [(son)24(t)-379(encore)-379(dans)]TJ /F6 1 Tf 7.7229 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.724 84.909 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.41 59.485 Td [(et)-379(l'on)-379(a)]TJ /F8 1 Tf 4.0723 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.569 84.909 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.554 59.485 Td [(=)-349(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.595 84.84 Td [(s)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.614 59.485 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.06 84.909 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.982 59.485 Td [(=)-343(lim)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.931 84.84 Td [(s)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 34.036 60.593 Td [(\263)]TJ 0.8253 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.802 83.4 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.53 59.485 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.095 84.909 Td [(k)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 37.596 60.593 Td [(\264)]TJ /F2 1 Tf 0.94578 -1.1084 Td [(=)]TJ /F9 1 Tf 1.1446 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.759 83.4 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.319 59.485 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.948 84.909 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.386 59.485 Td [(,)-385(par)]TJ -37.054 -2.6867 Td [(suite)]TJ /F8 1 Tf 2.3916 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14 81.064 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.44 56.798 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F9 1 Tf 1.0783 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.405 79.556 Td [(k)]TJ /F10 1 Tf 0.62931 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.813 56.798 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22 81.064 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.861 56.798 Td [(\))-283(=)]TJ /F9 1 Tf 1.747 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.164 79.556 Td [(k)]TJ /F10 1 Tf 0.63793 0 Td [(2)]TJ /F16 1 Tf 0.76724 0 Td [(N)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.241 56.798 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.836 81.064 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.337 56.798 Td [(\))-283(=)-277(lim)]TJ /F9 1 Tf 3.2831 1.1084 Td [(\263)]TJ 0.81927 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.319 79.556 Td [(k)]TJ /F17 1 Tf 0.63793 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.108 56.798 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.095 81.064 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.211 56.798 Td [(\))]TJ /F9 1 Tf 0.38554 1.1084 Td [(\264)]TJ /F2 1 Tf 0.87952 -1.1084 Td [(=)-277(lim)]TJ /F8 1 Tf 2.6084 0 Td [(\256)]TJ /F9 1 Tf 0.64458 1.1084 Td [(\263)]TJ 0.8253 -0.15663 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.362 79.556 Td [(k)]TJ /F17 1 Tf 0.62931 0 Td [(6)]TJ /F7 1 Tf 0.89655 0 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.223 56.798 Td [(u)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.655 81.064 Td [(k)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 37.289 57.906 Td [(\264)]TJ /F2 1 Tf 0.87349 -1.1084 Td [(=)-277(lim)]TJ /F8 1 Tf 2.6145 0 Td [(\256)]TJ /F2 1 Tf 0.64457 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.509 81.064 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.831 56.798 Td [(\).)]TJ -37.5 -2.747 Td [(P)30(ar)-361(passage)-355(\265)500(a)-361(la)-355(limite)-361(la)-355(relation)]TJ /F6 1 Tf 15.235 0 Td [(\262)]TJ /F2 1 Tf 0.85542 0 Td [(donne)-361(donc)]TJ /F8 1 Tf 5.3855 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F11 1 Tf 0.70482 0 Td [(>)]TJ /F8 1 Tf 1.0964 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.034 77.133 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.819 54.051 Td [(\))-240(+)]TJ /F8 1 Tf 1.6446 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.267 77.133 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 74.253 108.2 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.548 54.051 Td [(\),)-367(ce)-361(qui)-361(prouv)30(e)]TJ -32.217 -1.1988 Td [(la)-331(relation)-337(lorsque)]TJ /F8 1 Tf 8.2108 0 Td [(g)]TJ /F6 1 Tf 0.78916 0 Td [(2)-283(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 22.612 75.418 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.47 52.852 Td [(.)]TJ -11.139 -1.988 Td [(Dans)-271(le)-271(cas)-265(g)30(\266)469(en)24(\266)475(eral)-265(\(i.e.)]TJ /F8 1 Tf 10.542 0 Td [(g)]TJ /F2 1 Tf 0.77711 0 Td [(quelconque\))-271(la)-265(d)24(\266)475(e\257nition)-271(de)]TJ /F8 1 Tf 12.181 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\))-271(comme)-265(b)-30(orne)-265(inf)24(\266)469(erieure)-271(mon)24(tre)]TJ -25 -1.4036 Td [(qu'il)-337(su\261t)-337(de)-337(prouv)24(er)]TJ /F9 1 Tf 9.7711 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.224 70.564 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.657 49.461 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F8 1 Tf 1.0602 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.121 70.564 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.096 49.461 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.31 70.564 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 54.747 99.029 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.801 49.461 Td [(\))-337(lorsque)-337(\()]TJ /F8 1 Tf 4.5422 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.103 70.564 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.253 49.461 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.138 70.564 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.259 49.461 Td [(est)-337(une)-337(famille)-337(ma)-60(jo-)]TJ -29.928 -1.8012 Td [(ran)30(te)-397(p)-30(our)]TJ /F8 1 Tf 5.0663 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(.)-620(On)-397(a)-391(alors)]TJ /F8 1 Tf 5.9759 0 Td [(g)]TJ /F11 1 Tf 0.88554 0 Td [(6)]TJ /F9 1 Tf 1.1566 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.216 67.987 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.446 47.659 Td [(,)-409(d'o)-30(\265)530(u)]TJ /F8 1 Tf 2.9699 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.474 67.987 Td [(A)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 26.512 47.659 Td [(6)]TJ /F9 1 Tf 1.1566 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.724 67.987 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.187 47.659 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.009 67.987 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.795 47.659 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.198 67.987 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.729 47.659 Td [(\))]TJ /F11 1 Tf 0.76506 0 Td [(6)]TJ /F8 1 Tf 1.1566 0 Td [(\301)]TJ /F9 1 Tf 0.59638 1.1145 Td [(\263)]TJ 0.76506 -0.16265 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.957 67.987 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.53 47.659 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.233 67.987 Td [(A)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 42.747 48.774 Td [(\264)]TJ /F11 1 Tf 0.96988 -1.1145 Td [(6)]TJ /F9 1 Tf -38.386 -0.8494 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.172 65.409 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8373 45.858 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.448 65.409 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.048 45.858 Td [(\).)-451(De)-337(m)24(^)475(eme)-337(on)-331(a)]TJ /F8 1 Tf 7.7711 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.448 65.409 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 40.952 91.825 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.904 45.858 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.922 65.409 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.133 45.858 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.198 65.409 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 54.59 91.825 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.723 45.858 Td [(\).)]TJ -22.392 -2.1566 Td [(P)30(ar)-427(suite)]TJ /F8 1 Tf 4.4639 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.966 62.323 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.506 43.701 Td [(\))-289(+)]TJ /F8 1 Tf 1.741 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.336 62.323 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 33.819 87.499 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.331 43.701 Td [(\))]TJ /F11 1 Tf 0.8253 0 Td [(6)]TJ /F9 1 Tf 1.2169 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.267 62.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.88 43.701 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.543 62.323 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.096 43.701 Td [(\))-289(+)]TJ /F9 1 Tf 1.7349 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.509 62.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.337 43.701 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.793 62.323 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 61 87.499 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.928 43.701 Td [(\))-439(=)]TJ /F9 1 Tf 2.0422 0.95181 Td [(X)]TJ 1.6084 -0.13855 Td [(\241)]TJ /F8 1 Tf 0.46386 -0.81325 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.379 62.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.934 43.701 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.655 62.323 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.145 43.701 Td [(\))-295(+)]TJ /F8 1 Tf 1.741 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.31 62.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.783 43.701 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.595 62.323 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 85.892 87.499 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.367 43.701 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(.)]TJ -38.886 -1.3976 Td [(Cep)-24(endan)24(t,)-325(puisque)]TJ /F8 1 Tf 8.9518 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.259 60.323 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.47 42.304 Td [(2)-283(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 24.241 60.323 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.608 42.304 Td [(,)-319(le)-319(cas)-319(pr)24(\266)475(ec)24(\266)469(eden)24(t)-319(assure)]TJ /F8 1 Tf 10.801 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.897 60.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.307 42.304 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.172 60.323 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.518 42.304 Td [(\))-192(+)]TJ /F8 1 Tf 1.5422 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.552 60.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.958 42.304 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.828 60.323 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 72.241 84.704 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.542 42.304 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57 60.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.163 42.304 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(h)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.241 60.323 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.825 42.304 Td [(\),)]TJ -38.494 -1.2048 Td [(la)-331(relation)-337(souhait)24(\266)469(ee)-337(en)-331(d)24(\266)475(ecoule.)]TJ /F1 1 Tf 0.81325 -1.6566 Td [(4.)-668(Les)-361(\266)548(el)18(\266)548(emen)30(ts)-385(de)]TJ /F6 1 Tf 9.9217 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 23.741 56.228 Td [(+)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 17.639 39.442 Td [(son)24(t)-385(mesurables.)]TJ /F2 1 Tf -12.307 -1.6566 Td [(Soit)]TJ /F8 1 Tf 2 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 13.853 53.857 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.343 37.786 Td [(,)-289(il)-283(s'agit)-277(de)-283(prouv)30(er,)-295(p)-30(our)-283(tout)]TJ /F8 1 Tf 13.633 0 Td [(a)]TJ /F6 1 Tf 0.8012 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-289(que)-283(l'ensem)24(ble)]TJ /F8 1 Tf 7.1325 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.129 53.857 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.09 37.786 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(>)-277(a)]TJ /F6 1 Tf 1.5843 0 Td [(g)]TJ /F2 1 Tf -38.663 -1.2048 Td [(est)]TJ /F6 1 Tf 1.5602 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(-mesurable.)]TJ -2.3614 -1.6566 Td [(C'est)-307(\266)469(eviden)24(t)-337(lorsque)]TJ /F8 1 Tf 9.488 0 Td [(a)-283(<)]TJ /F2 1 Tf 1.8614 0 Td [(0)-337(puisqu'alors)]TJ /F8 1 Tf 6.3795 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.078 49.762 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.572 34.924 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-337(p)-30(ositiv)30(e\).)]TJ -22.861 -1.6566 Td [(Consid)30(\266)469(erons)-445(le)-451(cas)]TJ /F8 1 Tf 8.6867 0 Td [(a)]TJ /F2 1 Tf 0.99397 0 Td [(=)-463(0.)-783(Nous)-445(dev)24(ons)-451(donc)-445(prouv)24(er)-445(p)-30(our)]TJ /F8 1 Tf 17.602 0 Td [(E)]TJ /F2 1 Tf 1.241 0 Td [(partie)-451(quelconque)-451(de)]TJ /F8 1 Tf 9.7349 0 Td [(X)]TJ /F2 1 Tf -38.259 -1.2048 Td [(l')30(\266)469(egalit)24(\266)469(e)]TJ /F8 1 Tf 3.6205 0 Td [(\271)]TJ /F2 1 Tf 0.60843 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(E)]TJ /F6 1 Tf 0.81928 0 Td [(\\)]TJ /F8 1 Tf 0.69277 0 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 22.491 45.668 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.169 32.063 Td [(\))-30(+)]TJ /F8 1 Tf 1.2169 0 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)]TJ /F6 1 Tf 0.81927 0 Td [(\\)]TJ /F8 1 Tf 0.69277 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.534 46.478 Td [(c)]TJ /F4 1 Tf 0 -0.94828 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.084 32.063 Td [(\))-240(\(o)-30(\265)530(u)]TJ /F8 1 Tf 2.3072 0 Td [(\271)]TJ /F2 1 Tf 0.83735 0 Td [(est)-234(la)-234(pseudo-mesure)-240(d)30(\266)469(e\257nie)-234(par)]TJ /F8 1 Tf 13.861 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\).)-415(Autremen)18(t)]TJ -33.355 -1.1988 Td [(dit,)-524(il)-481(faut)-487(d)24(\266)469(emon)24(trer)]TJ /F8 1 Tf 10.265 0 Td [(\301)]TJ /F2 1 Tf 0.59036 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.526 43.952 Td [(E)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.813 30.864 Td [(\))-536(=)]TJ /F8 1 Tf 2.2289 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.897 43.952 Td [(E)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.265 30.864 Td [(:)]TJ /F2 1 Tf 0.27711 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.06 43.952 Td [(A)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 47.41 61.222 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.145 30.864 Td [(\))-331(+)]TJ /F8 1 Tf 1.8133 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.362 43.952 Td [(E)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.181 30.864 Td [(:)]TJ /F2 1 Tf 0.27711 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.534 43.952 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 59.253 61.909 Td [(c)]TJ /F19 1 Tf 0 -0.96385 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.066 30.864 Td [(\).)-897(Mais)-493(cela)-487(d)24(\266)475(ecoule)-493(du)-487(lemme)]TJ -24.735 -1.1988 Td [(pr)30(\266)469(ec)24(\266)469(eden)30(t)-337(a)24(v)24(ec)]TJ /F8 1 Tf 6.7289 0 Td [(g)]TJ /F2 1 Tf 0.78916 0 Td [(=)-277(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.698 42.237 Td [(E)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.145 29.665 Td [(.)]TJ -9.8133 -1.6627 Td [(Dans)-331(le)-331(cas)]TJ /F8 1 Tf 5.259 0 Td [(a)-283(>)]TJ /F2 1 Tf 1.8614 0 Td [(0)-325(quelconque)-331(la)-331(propri)24(\266)469(et)24(\266)475(e)-331(de)-331(Stone)-331(assure)-331(que)]TJ /F6 1 Tf 20.343 0 Td [(E)]TJ /F2 1 Tf 0.94578 0 Td [(p)-24(oss)24(\265)475(ede)-337(inf)-78(\()]TJ /F8 1 Tf 5.247 0 Td [(a;)-174(f)]TJ /F2 1 Tf 1.5723 0 Td [(\))-331(et)-331(donc)]TJ /F8 1 Tf -35.229 -1.1988 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4828 38.952 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6024 26.804 Td [(=)]TJ /F8 1 Tf 1.1747 0 Td [(f)]TJ /F6 1 Tf 0.86747 0 Td [(\241)]TJ /F2 1 Tf 1.0482 0 Td [(inf)-78(\()]TJ /F8 1 Tf 1.6084 0 Td [(a;)-174(f)]TJ /F2 1 Tf 1.5723 0 Td [(\),)-427(mais)-403(alors)]TJ /F8 1 Tf 5.9699 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.819 38.952 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.127 26.804 Td [(est)-409(\265)500(a)-403(v)48(aleurs)-409(p)-30(ositiv)30(es)-409(et)]TJ /F8 1 Tf 11.386 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.78313 0 Td [(>)-397(a)]TJ /F6 1 Tf 2.1024 0 Td [(\()168(\))]TJ /F8 1 Tf 2.2289 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.5 38.952 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.066 26.804 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.78313 0 Td [(>)]TJ /F2 1 Tf 1.1747 0 Td [(0.)-662(Le)]TJ -36.657 -1.1988 Td [(r)30(\266)469(esultat)-337(d)24(\266)475(ecoule)-343(donc)-331(du)-337(cas)-337(pr)24(\266)475(ec)24(\266)469(eden)30(t.)]TJ /F1 1 Tf 0.81325 -1.6627 Td [(5.)-668(Conclusion.)]TJ /F2 1 Tf -0.81325 -1.6566 Td [(P)30(our)]TJ /F8 1 Tf 2.3554 0 Td [(f)]TJ /F6 1 Tf 0.87952 0 Td [(2)-277(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.362 31.676 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.699 22.286 Td [(,)-277(nous)-259(v)24(enons)-259(de)-259(v)24(oir)-259(que)]TJ /F8 1 Tf 10.994 0 Td [(f)]TJ /F2 1 Tf 0.85542 0 Td [(est)-265(mesurable)-259(et)]TJ /F8 1 Tf 7.2349 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\),)-271(la)-259(prop)-36(osition)-259(pr)24(\266)475(ec)24(\266)469(e-)]TJ -29.392 -1.7108 Td [(den)30(te)-403(assure)-397(donc)]TJ /F9 1 Tf 8.2952 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.5482 0 Td [(=)]TJ /F8 1 Tf 1.1626 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-385(=)]TJ /F8 1 Tf 1.9337 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\).)-644(Cep)-24(endan)24(t,)-421(comme)]TJ /F8 1 Tf 10.175 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-403(est,)-409(par)-403(h)30(yp)-36(oth)24(\265)469(ese,)]TJ -29.964 -2.3253 Td [(\257ni,)-343(on)-337(conclut)-337(que)]TJ /F8 1 Tf 8.7651 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(est)-343(in)24(t)24(\266)475(egrable)-337(a)24(v)24(ec)]TJ /F9 1 Tf 8.4096 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf 2.4458 0 Td [(=)]TJ /F8 1 Tf 1.0663 0 Td [(\256)]TJ /F2 1 Tf 0.63856 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\).)-457(En)-313(\266)475(ecriv)48(an)30(t)-343(toute)-337(fonction)-343(de)]TJ /F6 1 Tf 14.301 0 Td [(E)]TJ /F2 1 Tf -38.548 -1.6566 Td [(comme)-295(di\256)24(\266)475(erence)-295(de)-295(sa)-295(partie)-301(p)-30(ositiv)30(e)-301(et)-295(de)-289(sa)-301(partie)-295(n)24(\266)469(egativ)30(e)-301(on)-295(v)24(oit)-295(que)-295(ce)-301(r)30(\266)469(esultat)-295(reste)]TJ 0 -1.1988 Td [(v)54(alable)-337(p)-30(our)-337(tous)-331(les)-313(\266)475(el)24(\266)469(emen)24(ts)-331(de)]TJ /F6 1 Tf 14.777 0 Td [(E)]TJ /F2 1 Tf 0.61446 0 Td [(.)]TJ ET 512.64 182.77 m 519.84 182.77 l 519.84 191.19 l 512.64 191.19 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 12.75 Td [(Remarque.)]TJ /F2 1 Tf 6.5602 0 Td [(Dans)-415(ce)-415(th)18(\266)475(eor)24(\265)469(eme)-409(on)-415(p)-30(eut)-415(se)-409(demander)-421(s'il)-415(y)-415(a)-409(unicit)24(\266)475(e)-415(du)-415(couple)-415(\()]TJ /F6 1 Tf 30.09 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3253 0 Td [(\))]TJ -38.777 -1.2048 Td [(a)30(y)24(an)24(t)-337(les)-337(propri)24(\266)475(et)24(\266)469(es)-337(demand)30(\266)469(ees.)]TJ 0 -1.6566 Td [(Remarquons)-530(tout)-536(d'ab)-30(ord)-536(que)-530(les)-506(\266)469(el)24(\266)469(emen)24(ts)-530(de)]TJ /F6 1 Tf 21.759 0 Td [(E)]TJ /F2 1 Tf 1.1506 0 Td [(dev)54(an)24(t)-506(^)469(etre)-530(in)24(t)24(\266)475(egrables)-536(doiv)24(en)30(t)-506(^)469(etre)]TJ /F6 1 Tf -22.91 -1.1988 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurables)-6(.)-644(Cela)-403(imp)-36(ose)-403(en)-397(particulier)-403(que,)-421(p)-30(our)-403(tout)]TJ /F8 1 Tf 25.036 0 Td [(f)]TJ /F6 1 Tf 0.98796 0 Td [(2)-397(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.276 12.219 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.398 8.6894 Td [(,)]TJ /F6 1 Tf 0.6988 0 Td [(T)]TJ /F2 1 Tf 1.1988 0 Td [(p)-30(oss)30(\265)469(ede)-409(l'ensem)30(ble)]TJ /F8 1 Tf -30.964 -1.1988 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 10.504 Td [(f)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0783 7.4906 Td [(=)]TJ /F6 1 Tf 1.259 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 1.0542 0 Td [(2)]TJ /F8 1 Tf 1.1446 0 Td [(X)]TJ /F6 1 Tf 1.3916 0 Td [(j)]TJ /F8 1 Tf 0.75301 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.86747 0 Td [(>)]TJ /F2 1 Tf 1.259 0 Td [(1)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(.)-813(Ainsi)-457(la)-457(tribu)]TJ /F6 1 Tf 7.6626 0 Td [(T)]TJ /F2 1 Tf 1.253 0 Td [(doit)-451(con)24(tenir)-457(la)-457(tribu)]TJ /F6 1 Tf 10.024 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 53.448 10.504 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.253 7.4906 Td [(engendr)24(\266)469(ee)-457(par)]TJ -32.922 -1.2048 Td [(l'ensem)30(ble)]TJ /F6 1 Tf 4.8133 0 Td [(S)]TJ /F2 1 Tf 1.012 0 Td [(de)-337(tous)-331(les)]TJ /F8 1 Tf 4.9578 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.138 8.7797 Td [(f)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.386 6.2858 Td [(.)]TJ /F5 1 Tf -12.054 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(104)]TJ ET endstream endobj 348 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 347 0 R /ProcSet 2 0 R >> /Contents 346 0 R >> endobj 347 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F13 39 0 R /F16 81 0 R /F17 97 0 R /F19 276 0 R >> endobj 349 0 obj << /Length 20354 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4036 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-361(si)-361(une)-349(tribu)]TJ /F6 1 Tf 13.193 0 Td [(T)]TJ /F2 1 Tf 1.1506 0 Td [(con)24(tien)24(t)]TJ /F6 1 Tf 3.8554 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 34.457 88.823 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.524 62.22 Td [(,)-361(mon)24(trons)-355(que)-355(tout)]TJ /F8 1 Tf 9.1506 0 Td [(f)]TJ /F6 1 Tf 0.90964 0 Td [(2)-313(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 51.638 88.823 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.102 62.22 Td [(est)]TJ /F6 1 Tf 1.5843 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable.)]TJ -34.151 -1.1988 Td [(P)30(our)]TJ /F8 1 Tf 2.4578 0 Td [(a)-325(>)]TJ /F2 1 Tf 1.9458 0 Td [(0,)]TJ /F6 1 Tf 1.1446 0 Td [(T)]TJ /F2 1 Tf 1.1566 0 Td [(p)-30(oss)24(\265)475(ede)]TJ /F8 1 Tf 3.6807 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.56 87.064 Td [(f)-120(=a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.145 61.021 Td [(=)]TJ /F6 1 Tf 1.1024 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.89157 0 Td [(2)]TJ /F8 1 Tf 0.98795 0 Td [(X)]TJ /F6 1 Tf 1.2229 0 Td [(j)]TJ /F8 1 Tf 0.60241 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.71084 0 Td [(>)-319(a)]TJ /F6 1 Tf 1.6265 0 Td [(g)]TJ /F2 1 Tf 0.86144 0 Td [(;)-367(il)-361(p)-30(oss)24(\265)475(ede)-367(aussi)]TJ /F6 1 Tf 7.7229 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.89157 0 Td [(2)]TJ /F8 1 Tf 0.98193 0 Td [(X)]TJ /F6 1 Tf 1.2289 0 Td [(j)]TJ /F8 1 Tf 0.59639 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.71085 0 Td [(>)]TJ /F2 1 Tf 1.0964 0 Td [(0)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf -38.663 -1.1988 Td [(qui)-307(est)-313(la)-313(r)30(\266)469(eunion)-313(des)]TJ /F6 1 Tf 9.5843 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(j)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(=n)]TJ /F6 1 Tf 1.1024 0 Td [(g)]TJ /F2 1 Tf 0.80723 0 Td [(et)]TJ /F6 1 Tf 1.1446 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F6 1 Tf 1.1807 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.66868 0 Td [(>)-277(a)]TJ /F6 1 Tf 1.5843 0 Td [(g)]TJ /F2 1 Tf 0.80723 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3434 0 Td [(a)-283(<)]TJ /F2 1 Tf 1.8614 0 Td [(0)-307(qui)-313(est)-313(tout)]TJ /F8 1 Tf -33.313 -1.2048 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(.)-451(Autremen)24(t)-343(dit)-337(tout)-313(\266)475(el)18(\266)475(emen)24(t)-337(de)]TJ /F6 1 Tf 14.554 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 30.509 83.668 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.313 58.617 Td [(est)]TJ /F6 1 Tf 1.5663 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(-mesurable.)-451(Plus)-343(g)30(\266)469(en)30(\266)469(eralemen)24(t,)-337(un)-313(\266)475(el)24(\266)469(emen)24(t)]TJ /F8 1 Tf 19.217 0 Td [(f)]TJ /F2 1 Tf -38.566 -1.1988 Td [(de)]TJ /F6 1 Tf 1.3313 0 Td [(E)]TJ /F2 1 Tf 0.95181 0 Td [(s')24(\266)469(ecriv)54(an)24(t)]TJ /F8 1 Tf 4.4578 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.974 81.952 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.446 57.418 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.371 81.952 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.946 57.418 Td [(est)-313(\266)475(egalemen)24(t)]TJ /F6 1 Tf 6.259 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable.)]TJ -17.675 -1.7952 Td [(Ainsi)]TJ /F6 1 Tf 2.6988 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 12.276 79.383 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.4699 55.623 Td [(est)-451(la)-445(plus)-445(p)-30(etite)-451(des)-445(tribus)-451(qui)-445(renden)30(t)-451(les)-421(\266)475(el)18(\266)475(emen)24(ts)-445(de)]TJ /F6 1 Tf 25.723 0 Td [(E)]TJ /F2 1 Tf 1.0602 0 Td [(mesurables.)-783(Si)-451(un)]TJ -30.922 -1.1988 Td [(couple)-289(\()]TJ /F6 1 Tf 3.4518 0 Td [(T)]TJ /F8 1 Tf 0.8012 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3193 0 Td [(\))-295(a)-283(les)-289(propri)24(\266)469(et)24(\266)475(es)-289(demand)24(\266)475(ees)-289(dans)-283(le)-295(th)24(\266)469(eor)24(\265)469(eme)-283(de)-289(Daniell-Stone,)-301(on)-283(p)-30(eut)-289(a)24(v)24(oir)]TJ /F6 1 Tf -5.5723 -1.2048 Td [(T)]TJ /F2 1 Tf 1.0964 0 Td [(strictemen)24(t)-301(plus)-295(grand)-301(que)]TJ /F6 1 Tf 11.928 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.043 75.943 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.349 53.22 Td [(,)-301(mais,)-307(par)-295(restriction,)-307(on)-295(p)-30(eut)-301(toujours)-295(se)-301(ramener)-295(au)-301(cas)]TJ /F6 1 Tf -14.018 -1.1988 Td [(T)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 11.457 74.228 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4578 52.021 Td [(.)-445(La)-337(question)-337(est)-337(alors)-337(de)-331(sa)24(v)24(oir)-337(s'il)-337(y)-337(a)-337(unicit)30(\266)469(e)-337(de)]TJ /F8 1 Tf 22.199 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(sur)]TJ /F6 1 Tf 1.6747 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.776 74.228 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.536 52.021 Td [(.)]TJ -29.205 -1.7952 Td [(P)30(our)]TJ /F8 1 Tf 2.5542 0 Td [(f)]TJ /F6 1 Tf 1.0783 0 Td [(2)-487(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.233 71.659 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.307 50.226 Td [(,)-487(on)-457(a)-457(vu)-457(dans)-457(la)-457(d)24(\266)475(emonstration)-463(que)-457(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.595 71.659 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 59.337 99.945 Td [(f)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.627 50.226 Td [(est)-463(la)-451(limite)-457(simple)-457(de)-457(la)-457(suite)]TJ -25.295 -1.1988 Td [(croissan)30(tes)-349(des)-343(fonctions)]TJ /F8 1 Tf 11.006 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.06 69.943 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.651 49.027 Td [(=)-289(inf)-78(\(1)]TJ /F8 1 Tf 3.1747 0 Td [(;)-168(ng)]TJ /F2 1 Tf 1.5542 0 Td [(\))-349(o)-24(\265)524(u)]TJ /F8 1 Tf 2.1325 0 Td [(g)]TJ /F2 1 Tf 0.80121 0 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(f)]TJ /F6 1 Tf 0.8253 0 Td [(\241)]TJ /F2 1 Tf 1.006 0 Td [(inf)-78(\(1)]TJ /F8 1 Tf 2.1084 0 Td [(;)-162(f)]TJ /F2 1 Tf 1.0422 0 Td [(\).)-469(T)78(outes)-343(ces)-343(fonctions)-343(appar-)]TJ -26.036 -1.7108 Td [(tiennen)30(t)-379(\265)500(a)]TJ /F6 1 Tf 4.8313 0 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.293 67.495 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.355 47.316 Td [(.)-560(On)-379(en)-373(d)30(\266)469(eduit)-373(que)]TJ /F8 1 Tf 8.9759 0 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.983 67.495 Td [(f)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.867 47.316 Td [(\))-343(=)]TJ /F9 1 Tf 1.8554 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.612 67.495 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 53.771 94.114 Td [(f)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.554 47.316 Td [(dm)]TJ /F2 1 Tf 1.7711 0 Td [(est)-379(la)-373(limite)-373(des)]TJ /F9 1 Tf 7.3916 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.664 67.495 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.91 47.316 Td [(dm)]TJ /F2 1 Tf 1.741 0 Td [(=)]TJ /F8 1 Tf 1.1205 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.94 67.495 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.825 47.316 Td [(\).)]TJ -38.494 -1.6566 Td [(Cela)-385(prouv)30(e)-385(que)-385(la)-379(v)54(aleur)-379(de)]TJ /F8 1 Tf 13.114 0 Td [(m)]TJ /F2 1 Tf 1.259 0 Td [(est)-385(imp)-30(os)24(\266)469(ee)-379(sur)-385(les)]TJ /F8 1 Tf 8.6807 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.698 65.125 Td [(f)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.657 45.659 Td [(,)-391(c'est-)0(\265)500(a-dire)-385(sur)-385(l'ensem)30(ble)]TJ /F6 1 Tf 12.416 0 Td [(S)]TJ /F2 1 Tf 1.0602 0 Td [(qui)]TJ -37.801 -1.2048 Td [(engendre)]TJ /F6 1 Tf 4.2229 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.457 63.4 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.548 44.454 Td [(.)]TJ -5.2169 -1.7952 Td [(Remarquons)-222(alors)-228(que)-228(l'ensem)30(ble)]TJ /F6 1 Tf 14.404 0 Td [(S)]TJ /F2 1 Tf 0.90361 0 Td [(qui)-222(engendre)]TJ /F6 1 Tf 5.7048 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.474 60.831 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.554 42.659 Td [(est)-228(stable)-228(par)-222(in)24(tersection)-222(car)]TJ /F8 1 Tf 12.729 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.724 60.831 Td [(f)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.554 42.659 Td [(\\)]TJ /F8 1 Tf 0.66867 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.5 60.831 Td [(g)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.717 42.659 Td [(=)]TJ /F8 1 Tf -38.386 -1.747 Td [(A)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 58.288 Td [(sup)-8(\()]TJ /F7 1 Tf 2.1724 0 Td [(f)-51(;g)]TJ /F4 1 Tf 1.5431 0 Td [(\))]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0361 40.912 Td [(.)-668(P)30(ar)-409(ailleurs,)-427(la)-409(ma)-60(joration)-415(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.526 58.331 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 46.663 81.319 Td [(f)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 24.235 40.912 Td [(6)]TJ /F8 1 Tf 1.1807 0 Td [(f)]TJ /F2 1 Tf 1.006 0 Td [(assure)]TJ /F8 1 Tf 3.0904 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.121 58.331 Td [(f)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.048 40.912 Td [(\))]TJ /F11 1 Tf 0.78915 0 Td [(6)]TJ /F9 1 Tf 1.1807 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)-271(dm)]TJ /F2 1 Tf 2.5663 0 Td [(=)]TJ /F8 1 Tf 1.1807 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-415(et)-409(donc)]TJ /F8 1 Tf -35.066 -1.7108 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.517 55.883 Td [(f)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8675 39.201 Td [(\))-325(est)-325(\257ni.)-439(Alors,)-325(si)]TJ /F8 1 Tf 8.2711 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(est)-325(r)24(\266)475(eunion)-325(d'une)-325(famille)-319(d)24(\266)469(enom)24(brable)-319(d')24(\266)469(el)24(\266)469(emen)30(ts)]TJ /F8 1 Tf 21.922 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.293 55.883 Td [(f)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 80.855 77.909 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.114 39.201 Td [(de)]TJ /F6 1 Tf 1.3193 0 Td [(S)]TJ /F2 1 Tf 0.68073 0 Td [(,)-325(la)]TJ -37.783 -1.1988 Td [(prop)-30(osition)-337(d'unicit)24(\266)469(e)]TJ /F1 1 Tf 9.3735 0 Td [(I)-36(I)-30(I.4.5.)]TJ /F2 1 Tf 3.8133 0 Td [(assurera)-337(l'unicit)24(\266)475(e)-343(de)]TJ /F8 1 Tf 9.0783 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(sur)]TJ /F6 1 Tf 1.6747 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.405 54.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.476 38.003 Td [(.)]TJ -26.145 -1.9398 Td [(La)-331(condition)]TJ /F8 1 Tf 5.8373 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.095 51.392 Td [(f)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 31.663 71.62 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.524 36.063 Td [(signi\257e)-325(qu'il)-331(n'y)-331(a)-325(aucun)-331(p)-30(oin)24(t)-331(o)-24(\265)524(u)-325(toutes)-337(les)]TJ /F8 1 Tf 19.47 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.216 51.392 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.145 36.063 Td [(soien)18(t)-331(inf)30(\266)469(erieures)]TJ -31.813 -1.3494 Td [(ou)-469(\266)469(egales)-500(\265)499(a)-500(1.)-939(En)-500(particulier)-500(cela)-506(imp)-30(ose)-500(qu'il)-499(n'y)-500(ait)-506(aucun)-499(p)-30(oin)24(ts)-500(o)-24(\265)524(u)-500(toutes)-499(les)]TJ /F8 1 Tf 38.343 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.198 49.469 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 33.515 Td [(s'ann)30(ulen)24(t.)-439(R)30(\266)469(ecipro)-30(quemen)30(t,)-313(si)-301(l'on)-301(a)-301(une)-301(famille)-301(d)24(\266)475(enom)24(brable)-301(de)]TJ /F8 1 Tf 28.789 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.526 47.745 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 35.223 33.515 Td [(2)-283(E)]TJ /F2 1 Tf 1.8614 0 Td [(qui)-295(ne)-301(s'ann)24(ulen)24(t)]TJ -31.753 -1.1988 Td [(sim)30(ultan)24(\266)475(emen)24(t)-409(en)-403(aucun)-403(p)-30(oin)24(t,)-427(les)]TJ /F8 1 Tf 15.717 0 Td [(n)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.078 46.03 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.747 32.316 Td [(j)]TJ /F2 1 Tf 0.68072 0 Td [(formen)24(t)-409(une)-403(famille)-403(d)24(\266)469(enom)24(brable)-403(d')24(\266)475(el)18(\266)475(emen)24(ts)-403(de)]TJ /F6 1 Tf -18.096 -1.1988 Td [(E)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3793 44.314 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8554 31.117 Td [(qui)-331(en)-337(aucun)-337(p)-30(oin)24(t)-337(ne)-331(son)24(t)-337(tous)-337(inf)30(\266)469(erieurs)-337(ou)-307(\266)469(egaux)-337(\265)500(a)-331(1.)-445(On)-337(p)-30(eut)-337(donc)-307(\266)469(enoncer)-331(:)]TJ /F12 1 Tf -1.5241 -2.1928 Td [(S'il)-409(y)-403(a)-409(une)-403(famille)-409(d)30(\266)469(enom)24(brable)-409(d')30(\266)469(el)24(\266)469(emen)24(ts)-403(de)]TJ /F6 1 Tf 21.307 0 Td [(E)]TJ /F12 1 Tf 1.0241 0 Td [(sans)-403(z)18(\266)475(ero)-409(comm)24(un,)-427(alors)-409(la)-403(mesure)]TJ /F8 1 Tf 15.952 0 Td [(m)]TJ /F12 1 Tf -38.283 -1.2048 Td [(du)-415(th)24(\266)475(eor)18(\265)475(eme)-415(pr)30(\266)469(ec)24(\266)469(eden)30(t)-415(est)-415(uniquemen)24(t)-415(d)24(\266)469(etermin)30(\266)469(ee)-415(sur)-415(la)-409(tribu)]TJ /F6 1 Tf 28.434 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.095 39.452 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 35.169 27.72 Td [(qui)-409(est)-415(la)-415(plus)-409(p)-30(etite)]TJ -29.837 -1.1988 Td [(rendan)24(t)-337(mesurables)-337(les)-307(\266)469(el)24(\266)469(emen)30(ts)-337(de)]TJ /F6 1 Tf 15.699 0 Td [(E)]TJ /F12 1 Tf 0.62048 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 18.168 Td [(VI)-35(I.4.)-236(Les)-376(mesures)-381(de)-376(Radon.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 19.985 Td [(Consid)30(\266)469(erons)-265(un)-265(espace)-259(top)-36(ologique)]TJ /F8 1 Tf 15.289 0 Td [(X)]TJ /F2 1 Tf 1.1687 0 Td [(lo)-30(calemen)30(t)-265(compact.)-421(Notons)]TJ /F6 1 Tf 12.645 0 Td [(B)]TJ /F2 1 Tf 0.94578 0 Td [(sa)-265(tribu)-265(des)-265(b)-24(or)24(\266)469(eliens)]TJ -30.048 -1.1988 Td [(et)]TJ /F6 1 Tf 1.1506 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.034 26.668 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.4157 18.786 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F3 1 Tf 1.2952 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))-325(l'espace)-325(des)-319(fonctions)-325(con)30(tin)18(ues)-319(sur)]TJ /F8 1 Tf 16.373 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-325(\265)499(a)-319(v)54(aleurs)-325(r)24(\266)475(eelles)-325(et)-319(\265)500(a)-319(supp)-30(ort)-325(compact.)]TJ -21.777 -1.7952 Td [(Av)30(ec)-391(ces)-385(notations)-391(on)-391(app)-30(elle)]TJ /F1 1 Tf 13.56 0 Td [(mesure)-445(b)-30(or)18(\266)548(elienne)]TJ /F2 1 Tf 9.6325 0 Td [(toute)-385(mesure)-391(sur)-391(l'espace)-391(mesurable)]TJ -23.193 -1.1988 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(B)]TJ /F2 1 Tf 0.68675 0 Td [(\).)-674(On)-415(dit)-415(qu'une)-409(mesure)-415(b)-30(or)30(\266)469(elienne)]TJ /F8 1 Tf 16.404 0 Td [(m)]TJ /F2 1 Tf 1.2892 0 Td [(est)]TJ /F1 1 Tf 1.6386 0 Td [(lo)-36(calemen)30(t)-469(\257nie)]TJ /F2 1 Tf 8.3916 0 Td [(si,)-433(p)-30(our)-409(toute)-409(partie)]TJ -30.096 -1.1988 Td [(compacte)]TJ /F8 1 Tf 4.4458 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(,)-331(on)-337(a)]TJ /F8 1 Tf 2.8313 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(\))]TJ /F8 1 Tf 0.66265 0 Td [(<)]TJ /F2 1 Tf 1.0602 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)]TJ -13.892 -2.5904 Td [(Soit)]TJ /F8 1 Tf 2.0422 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.905 16.961 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.127 12.003 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F3 1 Tf 1.2952 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))-319(de)-319(supp)-30(ort)]TJ /F8 1 Tf 5.7169 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(.)-439(On)-319(sait)-325(que,)-325(sur)-325(le)-325(compact)]TJ /F8 1 Tf 13.072 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(,)]TJ /F8 1 Tf 0.59639 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(est)-325(b)-30(orn)24(\266)469(ee,)-325(d'o)-30(\265)530(u)-319(l'exis-)]TJ -29.349 -1.2048 Td [(tence)-331(de)]TJ /F8 1 Tf 3.9398 0 Td [(M)]TJ /F2 1 Tf 1.4096 0 Td [(ma)-60(joran)30(t)]TJ /F6 1 Tf 4.3373 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.60843 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(K)]TJ /F2 1 Tf 1.253 0 Td [(et)]TJ /F14 1 Tf 1.1626 0 Td [(a)-361(fortiori)]TJ /F2 1 Tf 4.3193 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(.)-445(Alors)]TJ /F8 1 Tf 3.3675 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(est)-337(b)-30(or)24(\266)475(elienne)-331(et)]TJ /F6 1 Tf 7.5361 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.55421 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.172 15.245 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 10.798 Td [(.)]TJ -38.886 -1.1988 Td [(Il)-337(s'ensuit)-343(que,)-349(si)]TJ /F8 1 Tf 7.7711 0 Td [(m)]TJ /F2 1 Tf 1.2229 0 Td [(est)-343(une)-343(mesure)-349(b)-30(or)30(\266)469(elienne)-343(lo)-24(calemen)24(t)-343(\257nie)-343(sur)]TJ /F8 1 Tf 20.56 0 Td [(X)]TJ /F2 1 Tf 0.90362 0 Td [(,)]TJ /F8 1 Tf 0.6265 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(est)-349(mesurable)-343(et)]TJ /F9 1 Tf -32.018 -0.29518 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(M)-54(:m)]TJ /F2 1 Tf 2.1807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(\))]TJ /F8 1 Tf 0.66868 0 Td [(<)]TJ /F2 1 Tf 1.0542 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)-433(Autremen)24(t)-289(dit)-295(les)-265(\266)469(el)24(\266)469(emen)30(ts)-289(de)]TJ /F6 1 Tf 13.976 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.621 11.15 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.283 7.9424 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F3 1 Tf 1.2952 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))-289(son)18(t)-289(in)24(t)24(\266)475(egrables)-295(p)-30(our)]TJ -29.361 -1.6566 Td [(toute)-331(mesure)-343(b)-24(or)24(\266)469(elienne)-331(lo)-30(calemen)24(t)-337(\257nie.)]TJ 18.663 -2.3976 Td [(105)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 351 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 350 0 R /ProcSet 2 0 R >> /Contents 349 0 R >> endobj 350 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F14 43 0 R >> endobj 352 0 obj << /Length 19705 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.7651 Td [(Une)-355(mesure)-355(b)-30(or)24(\266)475(elienne)-355(lo)-30(calemen)30(t)-361(\257nie)]TJ /F8 1 Tf 17.56 0 Td [(m)]TJ /F2 1 Tf 1.2289 0 Td [(sur)]TJ /F8 1 Tf 1.6928 0 Td [(X)]TJ /F2 1 Tf 1.2349 0 Td [(\266)475(etan)24(t)-361(donn)24(\266)475(ee,)-361(l'application)]TJ /F8 1 Tf 12.066 0 Td [(f)]TJ /F6 1 Tf 0.90964 0 Td [(7!)]TJ /F9 1 Tf 1.3072 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(dm)]TJ /F2 1 Tf -37 -1.7108 Td [(est)-283(clairemen)24(t)-283(une)-283(forme)-277(lin)24(\266)469(eaire)-283(p)-30(ositiv)30(e)-283(sur)]TJ /F6 1 Tf 19.627 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.474 85.857 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.898 60.147 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F3 1 Tf 1.2952 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\).)-433(Le)-283(th)24(\266)469(eor)24(\265)469(eme)-277(suiv)54(an)24(t)-283(v)54(a)-283(fournir)-283(une)]TJ -22.97 -1.1988 Td [(r)30(\266)469(ecipro)-30(que)-337(de)-331(ce)-337(fait.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 46.721 Td [(VI)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8193 56.009 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084339 Td [(\266)]TJ 0 -0.084339 Td [(eor)]TJ 1.7952 0.084339 Td [(\265)]TJ 0 -0.084339 Td [(eme.)]TJ /F12 1 Tf 2.6386 0 Td [(\()]TJ /F1 1 Tf 0.39157 0 Td [(de)-265(Radon-Riesz)]TJ /F12 1 Tf 7.7651 0 Td [(\))-1000(Soit)]TJ /F8 1 Tf 3.3373 0 Td [(X)]TJ /F12 1 Tf 1.1386 0 Td [(un)-234(espace)-228(m)24(\266)475(etrique)-234(lo)-24(calemen)24(t)-234(com-)]TJ -23.705 -1.1988 Td [(pact)-265(et)-265(d)24(\266)475(enom)24(brable)-265(\265)499(a)-265(l'in\257ni)-259(et)]TJ /F8 1 Tf 13.994 0 Td [(\256)]TJ /F12 1 Tf 0.90964 0 Td [(est)-265(une)-265(forme)-265(lin)30(\266)469(eaire)-265(sur)-271(l'espace)]TJ /F6 1 Tf 14.687 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.724 78.219 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.855 54.81 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F3 1 Tf 1.3012 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ /F12 1 Tf 0.6506 0 Td [(des)-265(fonctions)]TJ -33.584 -1.1988 Td [(con)30(tin)18(ues)-385(\265)499(a)-385(supp)-30(ort)-391(compact.)-596(Alors,)-403(si)]TJ /F8 1 Tf 17.681 0 Td [(\256)]TJ /F12 1 Tf 1.0301 0 Td [(est)-391(p)-30(ositiv)30(e)-391(\(i.e.)]TJ /F8 1 Tf 7.6325 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ /F12 1 Tf 0.77108 0 Td [(est)-391(p)-30(ositif)-385(lorsque)]TJ /F8 1 Tf 8.2048 0 Td [(f)]TJ /F12 1 Tf 0.98193 0 Td [(est)]TJ -37.934 -1.1988 Td [(p)-30(ositiv)30(e\),)-337(il)-337(existe)-331(une)-337(mesure)-337(b)-30(or)24(\266)475(elienne)-337(lo)-30(calemen)30(t)-337(\257nie)]TJ /F8 1 Tf 25.337 0 Td [(m)]TJ /F12 1 Tf 1.2108 0 Td [(et)-337(une)-337(seule)-337(p)-30(our)-331(laquelle)-343(:)]TJ /F6 1 Tf -14.084 -1.7169 Td [(8)]TJ /F8 1 Tf 0.55422 0 Td [(f)]TJ /F6 1 Tf 0.87349 0 Td [(2)-283(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.621 72.331 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.102 50.695 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F3 1 Tf 1.2952 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ /F8 1 Tf 1.3916 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3615 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3615 Td [(f)-271(dm)]TJ /F12 1 Tf 2.494 0 Td [(.)]TJ -26.422 -2.0663 Td [(P)30(our)-343(cette)-349(raison)-343(une)-343(forme)-343(lin)24(\266)475(eaire)-349(p)-30(ositiv)30(e)-349(sur)]TJ /F6 1 Tf 21.488 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.129 69.375 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.753 48.629 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F3 1 Tf 1.2952 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ /F12 1 Tf 0.73494 0 Td [(s'app)-30(elle)-343(aussi)-349(une)]TJ /F1 1 Tf 8.4639 0 Td [(mesure)-397(de)]TJ -34.024 -1.1988 Td [(Radon)]TJ /F12 1 Tf 3.6084 0 Td [(\(p)-30(ositiv)24(e\).)]TJ /F2 1 Tf -3.6084 -2.0663 Td [(Commen)-6(\273)439(cons)-337(par)-331(un)-337(lemme)-337(pr)24(\266)475(eliminaire.)]TJ /F15 1 Tf 0 -1.8133 Td [(Lemme.)]TJ /F12 1 Tf 4.3795 0 Td [(Soit)]TJ /F8 1 Tf 2.1446 0 Td [(K)]TJ /F12 1 Tf 1.3373 0 Td [(une)-421(partie)-427(compacte)-421(dans)-421(un)-427(espace)-421(m)30(\266)469(etrique)-421(lo)-30(calemen)30(t)-427(compact)]TJ /F8 1 Tf 30.114 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(.)]TJ -38.886 -1.1988 Td [(Alors,)-355(il)-355(existe)-355(une)-355(fonction)]TJ /F8 1 Tf 12.476 0 Td [(')]TJ /F12 1 Tf 1.006 0 Td [(con)24(tin)24(ue)-355(sur)]TJ /F8 1 Tf 5.7169 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-355(qui)-355(est)-355(\265)500(a)-349(supp)-30(ort)-361(compact,)-355(\265)500(a)-349(v)48(aleurs)-355(dans)]TJ /F2 1 Tf -20.108 -1.2048 Td [([0;)-168(1])]TJ /F12 1 Tf 2.3313 0 Td [(et)-337(qui)-331(prend)-337(la)-337(v)54(aleur)-337(1)-331(exactemen)24(t)-337(sur)-337(les)-337(p)-30(oin)24(ts)-331(de)]TJ /F8 1 Tf 23.373 0 Td [(K)]TJ /F12 1 Tf 0.92169 0 Td [(.)]TJ /F1 1 Tf -26.627 -1.8072 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Si)]TJ /F8 1 Tf 1.1627 0 Td [(K)]TJ /F2 1 Tf 1.2048 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(;)]TJ /F2 1 Tf 0.5 0 Td [(,)-331(il)-337(su\261t)-337(de)-337(prendre)-331(la)-337(fonction)-337(n)24(ulle.)]TJ -12.741 -1.5602 Td [(Si)]TJ /F8 1 Tf 1.2048 0 Td [(K)]TJ /F6 1 Tf 1.2711 0 Td [(6)]TJ /F2 1 Tf [(=)]TJ /F6 1 Tf 1.1205 0 Td [(;)]TJ /F2 1 Tf 0.87952 0 Td [(on)-373(p)-30(eut)-379(in)24(tro)-30(duire)-379(la)-373(fonction)]TJ /F8 1 Tf 13.548 0 Td [(d)]TJ /F2 1 Tf 0.89759 0 Td [(\(distance)-373(\265)500(a)]TJ /F8 1 Tf 5.1988 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(\))-379(d)24(\266)475(e\257nie)-379(par)]TJ /F8 1 Tf 5.7952 0 Td [(d)]TJ /F2 1 Tf 0.5241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-349(=)-590(inf)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.543 53.142 Td [(a)]TJ /F10 1 Tf 0.62069 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(K)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.373 37.78 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-174(x)]TJ /F2 1 Tf 1.5422 0 Td [(\).)]TJ -38.494 -1.6446 Td [(Comme)-289(c)24(haque)-289(fonction)]TJ /F8 1 Tf 10.723 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(d)]TJ /F2 1 Tf 0.5241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a;)-174(x)]TJ /F2 1 Tf 1.5482 0 Td [(\))-289(est)-295(lipsc)24(hitzienne)-283(de)-289(rapp)-30(ort)-295(1,)-295(il)-289(en)-283(v)48(a)-283(de)-289(m)24(^)475(eme)-289(p)-30(our)]TJ -15.307 -1.2048 Td [(la)-367(b)-30(orne)-373(inf)30(\266)469(erieure)]TJ /F8 1 Tf 8.4518 0 Td [(d)]TJ /F2 1 Tf 0.89157 0 Td [(qui)-373(est)-373(donc)-373(con)24(tin)24(ue.)-554(P)24(ar)-367(ailleurs)-373(le)-373(fait)-373(que)]TJ /F8 1 Tf 20.5 0 Td [(K)]TJ /F2 1 Tf 1.2952 0 Td [(soit)-367(compact)-379(donc)]TJ -31.139 -1.1988 Td [(ferm)30(\266)469(e)-337(assure)-331(que)]TJ /F8 1 Tf 7.6325 0 Td [(d)]TJ /F2 1 Tf 0.8494 0 Td [(s'ann)24(ule)-337(exactemen)30(t)-337(sur)-337(les)-337(p)-30(oin)24(ts)-331(de)]TJ /F8 1 Tf 16.62 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(.)]TJ -26.024 -1.5542 Td [(Comme)]TJ /F8 1 Tf 3.741 0 Td [(X)]TJ /F2 1 Tf 1.3133 0 Td [(est)-409(lo)-30(calemen)30(t)-409(compact,)-427(p)-30(our)-403(c)18(haque)]TJ /F8 1 Tf 16.922 0 Td [(a)]TJ /F6 1 Tf 0.92771 0 Td [(2)]TJ /F8 1 Tf 1.0663 0 Td [(X)]TJ /F2 1 Tf 1.3133 0 Td [(on)-403(p)-30(eut)-409(trouv)24(er)]TJ /F8 1 Tf 7.4157 0 Td [(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.069 45.831 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.367 32.177 Td [(>)]TJ /F2 1 Tf 1.1747 0 Td [(0)-409(tel)-403(que)]TJ -35.211 -1.2048 Td [(la)-355(b)-30(oule)-355(ferm)24(\266)475(ee)]TJ ET 0.504 w 148.25 380.08 m 157.97 380.08 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.404 30.973 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-168(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.5 44.107 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.512 30.973 Td [(\))-355(soit)-355(compacte.)-518(Alors)-361(les)-355(b)-30(oules)-361(ouv)24(ertes)]TJ /F8 1 Tf 18.795 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-168(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.845 44.107 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.416 30.973 Td [(=)]TJ /F2 1 Tf 0.5 0 Td [(2\),)-361(p)-30(our)]TJ /F8 1 Tf 3.9157 0 Td [(a)]TJ /F2 1 Tf 0.87952 0 Td [(par-)]TJ -37.38 -1.1988 Td [(couran)24(t)]TJ /F8 1 Tf 3.6084 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(,)-307(formen)30(t)-301(un)-307(recouvremen)24(t)-301(par)-301(ouv)24(erts)-307(du)-301(compact)]TJ /F8 1 Tf 22.482 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(,)-301(on)-301(p)-30(eut)-301(donc)-301(en)-301(extraire)]TJ -27.934 -1.1988 Td [(un)-355(recouvremen)18(t)-355(\257ni)-361(corresp)-30(ondan)24(t)-361(aux)-361(p)-30(oin)30(ts)]TJ /F8 1 Tf 20.97 0 Td [(a)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.388 40.676 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.277 28.575 Td [(;)-168(:)-162(:)-168(:)-168(;)-168(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.974 40.676 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.572 28.575 Td [(.)-512(Notons)]TJ /F8 1 Tf 4.2349 0 Td [(r)]TJ /F2 1 Tf 0.83133 0 Td [(le)-361(minim)24(um)-361(des)]TJ /F8 1 Tf 7.3253 0 Td [(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.129 40.676 Td [(a)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 87.699 56.644 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 28.575 Td [(.)]TJ -38.886 -1.2048 Td [(Si)-319(un)-319(p)-24(oin)24(t)]TJ /F8 1 Tf 5.1687 0 Td [(x)]TJ /F2 1 Tf 0.89157 0 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(X)]TJ /F2 1 Tf 1.2229 0 Td [(v)48(\266)475(eri\257e)]TJ /F8 1 Tf 2.9337 0 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.66265 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(r)-30(=)]TJ /F2 1 Tf 0.9759 0 Td [(3)-319(alors)-319(il)-319(existe)]TJ /F8 1 Tf 6.8675 0 Td [(a)]TJ /F2 1 Tf 0.84337 0 Td [(dans)]TJ /F8 1 Tf 2.3193 0 Td [(K)]TJ /F2 1 Tf 1.241 0 Td [(tel)-319(que)]TJ /F8 1 Tf 3.2711 0 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-174(x)]TJ /F2 1 Tf 1.5422 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(<)-283(r)-30(=)]TJ /F2 1 Tf 2.0361 0 Td [(2,)-319(puis)-319(il)]TJ -35.41 -1.5181 Td [(existe)]TJ /F8 1 Tf 2.7952 0 Td [(p)]TJ /F2 1 Tf 0.8253 0 Td [(\(1)]TJ /F11 1 Tf 1.1687 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(p)]TJ /F11 1 Tf 0.78313 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(\))-319(tel)-325(que)]TJ /F8 1 Tf 3.988 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.164 36.78 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.542 25.852 Td [(;)-168(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.112 36.78 Td [(a)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 45.747 51.21 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.343 25.852 Td [(=)]TJ /F2 1 Tf 0.5 0 Td [(2\);)-331(on)-319(a)-319(donc)]TJ /F8 1 Tf 6.0663 0 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.862 36.78 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.807 25.852 Td [(;)-168(x)]TJ /F2 1 Tf 1.0181 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(<)]TJ 1.1747 0.74096 Td [(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.259 37.84 Td [(a)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 71.108 52.68 Td [(p)]TJ ET 414.36 312.04 m 430.63 312.04 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.096 25.165 Td [(2)]TJ 1.241 0.68675 Td [(+)]TJ /F8 1 Tf 1.1084 0.68072 Td [(r)]TJ ET 447.41 312.04 m 453.46 312.04 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.434 25.165 Td [(2)]TJ 0.62048 0.68675 Td [(+)]TJ /F11 1 Tf 1.0542 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.121 36.78 Td [(a)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 82.096 51.21 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.518 25.852 Td [(.)-445(Ainsi)]TJ -36.187 -1.6687 Td [(l'ensem)30(ble)-349(des)]TJ /F8 1 Tf 6.5723 0 Td [(x)]TJ /F2 1 Tf 0.91566 0 Td [(tels)-349(que)]TJ /F8 1 Tf 3.7349 0 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.68675 0 Td [(6)]TJ /F8 1 Tf 1.0843 0 Td [(r)-30(=)]TJ /F2 1 Tf 0.9759 0 Td [(3)-349(est)-355(con)30(ten)24(u)-349(dans)-349(la)-349(r)30(\266)469(eunion)]TJ /F8 1 Tf 13.277 0 Td [(L)]TJ /F2 1 Tf 1.0241 0 Td [(des)]TJ ET 440.21 298.94 m 449.93 298.94 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.831 24.183 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.181 34.392 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.018 24.183 Td [(;)-168(r)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.121 34.392 Td [(a)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 80.699 47.873 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.819 24.183 Td [(\))-349(qui)-349(est)]TJ -35.488 -1.1988 Td [(compact)-325(\(parce)-325(que)-325(r)24(\266)475(eunion)-325(\257nie)-325(de)-325(compacts\).)-439(Alors)-325(la)-319(fonction)-331(\()]TJ /F8 1 Tf 29.114 0 Td [(r)-36(=)]TJ /F2 1 Tf 0.98193 0 Td [(3)]TJ /F6 1 Tf 0.6988 0 Td [(\241)]TJ /F8 1 Tf 0.9759 0 Td [(d)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.397 32.676 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39 22.985 Td [(est)-325(con)24(tin)24(ue,)]TJ -33.669 -1.1988 Td [(n)30(ulle)-343(en)-343(dehors)-349(de)]TJ /F8 1 Tf 8.2952 0 Td [(L)]TJ /F2 1 Tf 1.0181 0 Td [(donc)-343(\265)500(a)-343(supp)-30(ort)-349(compact)-343(\(con)24(ten)24(u)-343(dans)]TJ /F8 1 Tf 17.434 0 Td [(L)]TJ /F2 1 Tf 0.68073 0 Td [(\),)-349(\265)500(a)-343(v)54(aleurs)-349(dans)-337([0)]TJ /F8 1 Tf 8.3614 0 Td [(;)-162(r)-36(=)]TJ /F2 1 Tf 1.4217 0 Td [(3])-343(et)]TJ -37.211 -1.2048 Td [(prend)-343(la)-343(v)54(aleur)]TJ /F8 1 Tf 6.9458 0 Td [(r)-36(=)]TJ /F2 1 Tf 0.98193 0 Td [(3)-343(exactemen)30(t)-349(lorsque)]TJ /F8 1 Tf 9.5663 0 Td [(d)]TJ /F2 1 Tf 0.86144 0 Td [(s'ann)30(ule,)-349(c'est-)0(\265)500(a-dire)-343(sur)-349(les)-343(p)-30(oin)24(ts)-337(de)]TJ /F8 1 Tf 16.783 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(.)-469(En)-343(la)]TJ -36.06 -1.1988 Td [(m)24(ultiplian)24(t)-337(par)-337(3)]TJ /F8 1 Tf 7.4518 0 Td [(=r)]TJ /F2 1 Tf 1.3133 0 Td [(on)-331(obtien)24(t)-337(une)-331(fonction)-343(a)30(y)24(an)24(t)-337(les)-337(propri)24(\266)475(et)24(\266)469(es)-331(v)24(oulues.)]TJ ET 512.64 230.43 m 519.84 230.43 l 519.84 238.86 l 512.64 238.86 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 17.316 Td [(D)24(\266)548(emonstration)-385(du)-385(th)18(\266)554(eor)18(\265)548(eme)-385(de)-385(Radon-Riesz.)]TJ /F2 1 Tf 0 -1.5602 Td [(L'espace)]TJ /F6 1 Tf 4.0422 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.164 22.34 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.307 15.756 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F3 1 Tf 1.2952 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))-361(est)-355(clairemen)24(t)-355(un)-361(espace)-349(v)18(ectoriel)-355(r)30(\266)469(eticul)24(\266)475(e.)-512(Prouv)30(ons)-361(alors)-355(la)-355(propri)24(\266)469(et)24(\266)475(e)]TJ -7.3795 -1.1988 Td [(de)-331(Daniell)-337(p)-30(our)]TJ /F8 1 Tf 7.1325 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(.)]TJ -7.7711 -1.5542 Td [(Soit)-433(donc)-433(\()]TJ /F8 1 Tf 5.0361 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.534 18.392 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.404 13.003 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.871 18.392 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(2)]TJ /F16 1 Tf 0.77586 0 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.813 13.003 Td [(une)-439(suite)-433(d)24(\266)475(ecroissan)24(te)-433(de)-433(fonctions)-439(con)30(tin)18(ues)-433(\265)499(a)-433(supp)-30(ort)-439(compact)-439(qui)]TJ -8.4819 -1.2048 Td [(con)30(v)24(erge)-337(simplemen)24(t)-337(v)24(ers)-337(0,)-337(il)-331(s'agit)-337(de)-337(prouv)30(er)]TJ /F8 1 Tf 21.151 0 Td [(\256)]TJ /F2 1 Tf 0.63856 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.069 16.676 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.548 11.798 Td [(\))]TJ /F6 1 Tf 0.66265 0 Td [(!)]TJ /F2 1 Tf 1.2831 0 Td [(0.)]TJ -25.163 -1.5542 Td [(Soit)]TJ /F8 1 Tf 2.0723 0 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 11.81 14.443 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0602 10.244 Td [(le)-355(supp)-30(ort)-355(\(compact\))-355(de)]TJ /F8 1 Tf 10.952 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 29.336 14.443 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.952 10.244 Td [(.)-500(Utilisan)24(t)-355(le)-361(lemme)-355(on)-355(p)-30(eut)-355(trouv)24(er)]TJ /F8 1 Tf 16.446 0 Td [(')]TJ /F2 1 Tf 1.006 0 Td [(dans)]TJ /F6 1 Tf 2.3614 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.086 14.443 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.699 10.244 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F3 1 Tf 1.3012 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ -38.777 -1.1988 Td [(p)-30(ositiv)30(e)-337(sur)]TJ /F8 1 Tf 5.3735 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(et)-337(qui)-337(v)54(aut)-337(1)-337(sur)]TJ /F8 1 Tf 7.6265 0 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 29.224 12.728 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.205 9.0448 Td [(donc)]TJ /F8 1 Tf 2.3916 0 Td [(')]TJ /F11 1 Tf 0.92771 0 Td [(>)]TJ /F2 1 Tf 1.0602 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.414 12.728 Td [(K)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 53.627 17.584 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.259 9.0448 Td [(.)]TJ -21.928 -1.5542 Td [(Fixons)-481(alors)]TJ /F8 1 Tf 5.9398 0 Td [(")-530(>)]TJ /F2 1 Tf 2.3012 0 Td [(0)-481(et)-487(notons)]TJ /F8 1 Tf 5.6807 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.526 10.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.964 7.4906 Td [(l'ensem)30(ble)-487(des)]TJ /F8 1 Tf 6.8373 0 Td [(x)]TJ /F2 1 Tf 1.0602 0 Td [(p)-30(our)-481(lesquels)]TJ /F8 1 Tf 6.3133 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.034 10.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.211 7.4906 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.92169 0 Td [(<)-524(")]TJ /F2 1 Tf 1.7711 0 Td [(.)-891(Il)-487(s'agit)]TJ -34.53 -1.2048 Td [(clairemen)24(t)-403(d'une)-409(suite)-403(croissan)24(te)-403(d'ouv)24(erts)-409(de)]TJ /F8 1 Tf 20.681 0 Td [(X)]TJ /F2 1 Tf 1.3072 0 Td [(don)24(t)-409(la)-403(r)30(\266)469(eunion)-409(est)-409(tout)]TJ /F8 1 Tf 11.114 0 Td [(X)]TJ /F2 1 Tf 1.3133 0 Td [(puisque)]TJ /F8 1 Tf 3.7108 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.897 8.7797 Td [(n)]TJ /F5 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 3.8882 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(106)]TJ ET endstream endobj 354 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 353 0 R /ProcSet 2 0 R >> /Contents 352 0 R >> endobj 353 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F16 81 0 R /F19 276 0 R >> endobj 355 0 obj << /Length 15734 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-10512(VI)-36(I.)-385(Construction)-385(de)-391(mesures)]TJ /F2 1 Tf 0 -2.4036 Td [(tend)-355(simplemen)24(t)-355(v)24(ers)-349(0.)-493(En)-355(particulier)-349(ces)-349(ouv)18(erts)-355(formen)30(t)-355(un)-355(recouvremen)24(t)-355(du)-349(compact)]TJ /F8 1 Tf 0 -1.1988 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8448 87.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0301 61.021 Td [(supp)-30(ort)-409(de)]TJ /F8 1 Tf 5.1807 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.181 87.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.151 61.021 Td [(;)-439(on)-409(p)-30(eut)-409(donc)-403(trouv)24(er)-409(un)-409(recouvremen)24(t)-409(de)]TJ /F8 1 Tf 19.819 0 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.397 87.107 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.675 61.021 Td [(par)-403(un)-409(nom)24(bre)-403(\257ni)-409(de)]TJ -29.343 -1.1988 Td [(ces)]TJ /F8 1 Tf 1.5904 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.879 85.392 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1506 59.822 Td [(,)-313(mais,)-313(comme)-313(la)-307(suite)-307(de)-307(ces)-313(ouv)24(erts)-313(est)-313(croissan)24(te,)-313(celui)-313(de)-307(plus)-307(grand)-313(indice)-307(su\261t)]TJ -2.8193 -1.2048 Td [(p)-30(our)-331(recouvrir)]TJ /F8 1 Tf 6.6205 0 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 18.319 83.668 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.247 58.617 Td [(.)-445(On)-331(a)-331(donc)]TJ /F8 1 Tf 5.6024 0 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.19 83.668 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.428 58.617 Td [(\275)]TJ /F8 1 Tf 1.0542 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.724 83.668 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.253 58.617 Td [(p)-30(our)-331(un)-337(certain)-331(en)24(tier)]TJ /F8 1 Tf 9.9458 0 Td [(N)]TJ /F2 1 Tf 0.91566 0 Td [(.)-445(Alors,)-331(si)]TJ /F8 1 Tf 4.6446 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 59.25 83.668 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.855 58.617 Td [(,)-331(p)-30(our)]TJ /F8 1 Tf -36.524 -1.1988 Td [(n)]TJ /F11 1 Tf 0.94578 0 Td [(>)]TJ /F8 1 Tf 1.1265 0 Td [(N)]TJ /F2 1 Tf 1.2892 0 Td [(on)-373(a)]TJ /F8 1 Tf 2.3072 0 Td [(x)]TJ /F6 1 Tf 0.91566 0 Td [(2)]TJ /F8 1 Tf 1.0181 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.483 81.952 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.747 57.418 Td [(d'o)-30(\265)524(u)]TJ /F8 1 Tf 2.2651 0 Td [(")-355(>)-343(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.828 81.952 Td [(N)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.205 57.418 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.74096 0 Td [(>)]TJ /F8 1 Tf 1.1205 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.655 81.952 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.06 57.418 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)-572(P)30(ar)-379(ailleurs,)-385(comme)-379(les)-373(fonctions)-379(tenden)24(t)]TJ -19.693 -1.1988 Td [(en)-397(d)24(\266)475(ecroissan)24(t)-397(v)24(ers)-403(0,)-415(elles)-397(son)18(t)-397(p)-30(ositiv)24(es)-397(et,)-415(si)]TJ /F8 1 Tf 21.145 0 Td [(x)-499(=)]TJ /F6 1 Tf 0.95783 0 Td [(2)]TJ /F8 1 Tf 1.0542 0 Td [(K)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.983 80.237 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.181 56.22 Td [(on)-403(a)]TJ /F8 1 Tf 2.3554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.259 80.237 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.572 56.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F11 1 Tf 0.77711 0 Td [(6)]TJ /F8 1 Tf 1.1627 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.888 80.237 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.41 56.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-391(=)-385(0)-397(et)-403(donc)]TJ /F8 1 Tf -33.036 -1.2048 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3276 78.512 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.3614 55.015 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)-277(0.)-445(En)-337(conclusion,)-331(p)-30(our)]TJ /F8 1 Tf 12 0 Td [(n)]TJ /F11 1 Tf 0.87952 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(N)]TJ /F2 1 Tf 0.91566 0 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.302 78.512 Td [(n)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 24.096 55.015 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(")]TJ /F2 1 Tf 0.46988 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.466 78.512 Td [(K)]TJ /F19 1 Tf 5.976 0 0 5.976 0 0 Tm 53.711 109.54 Td [(0)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 27.572 55.015 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [("')]TJ /F2 1 Tf 1.1205 0 Td [(.)]TJ -24.416 -1.5964 Td [(Comme)-253(la)-253(p)-30(ositivit)24(\266)469(e)-253(de)]TJ /F8 1 Tf 10.181 0 Td [(\256)]TJ /F2 1 Tf 0.89759 0 Td [(en)30(tra)108(^)385(\263ne)-253(sa)-253(croissance)-259(on)-253(conclut)-259(0)]TJ /F11 1 Tf 15.036 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.69 76.228 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.566 53.418 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [("\256)]TJ /F2 1 Tf 1.1145 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(')]TJ /F2 1 Tf 0.65662 0 Td [(\))-253(p)-30(our)]TJ /F8 1 Tf 2.9217 0 Td [(n)]TJ /F11 1 Tf 0.87952 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(N)]TJ /F2 1 Tf 0.91566 0 Td [(.)]TJ -38.886 -1.2048 Td [(Cela)-307(\266)469(etan)24(t)-337(v)48(alable)-331(p)-30(our)]TJ /F8 1 Tf 10.59 0 Td [(")-277(>)]TJ /F2 1 Tf 1.8012 0 Td [(0)-331(quelconque)-337(on)-337(a)-331(bien)-337(prouv)54(\266)469(e)]TJ /F8 1 Tf 13.675 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.112 74.504 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.464 52.214 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(!)]TJ /F2 1 Tf 1.2771 0 Td [(0.)]TJ -30.078 -1.8976 Td [(Nous)-499(p)-30(ouv)18(ons)-506(donc)-500(appliquer)-506(le)-506(th)18(\266)475(eor)24(\265)469(eme)-499(de)-506(Daniell-Stone)-506(p)-24(our)-506(repr)24(\266)475(esen)24(ter)-506(la)-500(forme)]TJ 0 -1.6566 Td [(lin)24(\266)475(eaire)]TJ /F8 1 Tf 3.6265 0 Td [(\256)]TJ /F2 1 Tf 1.1024 0 Td [(par)]TJ /F8 1 Tf 1.9096 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\))-493(=)]TJ /F9 1 Tf 2.1446 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)-271(dm)]TJ /F2 1 Tf 2.6205 0 Td [(p)-30(our)-457(une)-463(mesure)]TJ /F8 1 Tf 8.0301 0 Td [(m)]TJ /F2 1 Tf 1.3373 0 Td [(con)30(v)24(enable.)-825(Prouv)30(ons)-463(donc)-463(que)]TJ /F8 1 Tf 14.886 0 Td [(m)]TJ /F2 1 Tf -38.283 -1.6566 Td [(est)-337(d)24(\266)475(e\257nie)-337(sur)-337(la)-337(tribu)-331(des)-337(b)-30(or)24(\266)475(eliens)-337(de)]TJ /F8 1 Tf 17.289 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(et)-331(que)-343(la)-331(mesure)-337(d'un)-343(compact)-337(est)-337(\257nie.)]TJ -18.53 -1.6024 Td [(T)78(out)-439(d'ab)-30(ord,)-463(p)-30(our)]TJ /F8 1 Tf 9.0843 0 Td [(K)]TJ /F2 1 Tf 1.3554 0 Td [(compact)-445(dans)]TJ /F8 1 Tf 6.5482 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-463(le)-439(lemme)-439(p)-30(ermet)-433(de)-439(trouv)24(er)]TJ /F8 1 Tf 13.735 0 Td [(')]TJ /F2 1 Tf 1.0843 0 Td [(dans)]TJ /F6 1 Tf 2.4458 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.69 64.754 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.422 45.4 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F3 1 Tf 1.3012 0 Td [(R)]TJ /F2 1 Tf 0.71687 0 Td [(\),)]TJ -38.494 -1.1988 Td [(\265)499(a)-415(v)54(aleurs)-421(dans)-409([0;)-168(1],)-439(telle)-415(que)]TJ /F8 1 Tf 13.687 0 Td [(K)]TJ /F2 1 Tf 1.3373 0 Td [(=)]TJ /F8 1 Tf 1.1867 0 Td [(')]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.767 63.84 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.271 44.201 Td [(\(1\).)-692(Les)-409(fonctions)]TJ /F8 1 Tf 8.4639 0 Td [(')]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.345 63.84 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.343 44.201 Td [(appartiennen)18(t)-415(\265)500(a)]TJ /F6 1 Tf 7.4217 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.086 63.038 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.699 44.201 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F3 1 Tf 1.3012 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ -38.777 -1.1988 Td [(et)-397(formen)24(t)-397(une)-403(suite)-397(d)24(\266)475(ecroissan)24(te)-397(de)-397(fonctions)-403(in)24(t)24(\266)475(egrables)-403(p)-30(ositiv)30(es)-397(qui)-403(tend)-397(simplemen)18(t)]TJ 0 -1.2048 Td [(v)24(ers)-379(la)-373(fonction)-385(caract)30(\266)469(eristique)-379(de)]TJ /F8 1 Tf 15.295 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(.)-572(Il)-373(s'ensuit)-379(que)-379(cette)-379(limite)-379(est)-379(mesurable)-379(et)-379(m)24(^)475(eme)]TJ -16.217 -1.6566 Td [(in)24(t)24(\266)475(egrable)-337(a)24(v)24(ec)]TJ /F9 1 Tf 6.8434 1.3614 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.664 57.228 Td [(K)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.675 40.141 Td [(dm)]TJ /F2 1 Tf 1.6867 0 Td [(=)-283(lim)]TJ /F9 1 Tf 2.6205 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(')]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.526 58.038 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.343 40.141 Td [(dm)]TJ /F2 1 Tf 1.6867 0 Td [(=)-283(lim)]TJ /F8 1 Tf 2.6145 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(')]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.121 58.038 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.88 40.141 Td [(\).)-457(On)-343(en)-337(d)24(\266)475(eduit)-337(que)]TJ /F8 1 Tf 9.1145 0 Td [(K)]TJ /F2 1 Tf 1.259 0 Td [(est)-343(mesurable,)]TJ -32.922 -1.7108 Td [(de)-331(mesure)-337(\257nie)-337(et)-337(m)24(^)475(eme)-337(que)-337(la)-331(v)48(aleur)-337(de)-331(sa)-337(mesure)]TJ /F8 1 Tf 22.867 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(\))-337(est)-343(imp)-30(os)24(\266)469(ee)-337(par)]TJ /F8 1 Tf 7.8795 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(.)]TJ -33.578 -1.6024 Td [(Ainsi,)-427(la)-403(mesure)]TJ /F8 1 Tf 7.6084 0 Td [(m)]TJ /F2 1 Tf 1.2831 0 Td [(est)-409(d)24(\266)475(e\257nie)-409(en)-403(particulier)-403(sur)-409(la)-409(tribu)]TJ /F6 1 Tf 16.693 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.026 52.487 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.313 36.828 Td [(engendr)24(\266)469(ee)-409(par)-403(les)-409(compacts)]TJ -26.982 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)-457(En)-337(outre,)-349(l'ensem)30(ble)-337(des)-343(compacts)-343(est)-343(stable)-337(par)-343(in)24(tersection)-343(et)]TJ /F8 1 Tf 28.964 0 Td [(X)]TJ /F2 1 Tf 0.90963 0 Td [(,)-337(d)24(\266)475(enom)24(brable)-343(\265)500(a)]TJ -32.12 -1.1988 Td [(l'in\257ni,)-325(est)-331(r)30(\266)469(eunion)-331(d)30(\266)469(enom)24(brable)-325(de)-325(compacts.)-445(Comme)-325(la)-325(v)48(aleur)-325(de)]TJ /F8 1 Tf 29.614 0 Td [(m)]TJ /F2 1 Tf 1.1988 0 Td [(est)-331(imp)-36(os)30(\266)469(ee)-331(et)-325(\257nie)]TJ -30.813 -1.2048 Td [(sur)-373(les)-373(compacts,)-379(la)-373(prop)-30(osition)-373(d'unicit)24(\266)475(e)]TJ /F1 1 Tf 18.506 0 Td [(I)-36(I)-36(I.4.5.)]TJ /F2 1 Tf 3.8554 0 Td [(assure)-367(que)-373(la)-373(v)54(aleur)-373(de)]TJ /F8 1 Tf 10.476 0 Td [(m)]TJ /F2 1 Tf 1.247 0 Td [(est)-379(imp)-30(os)24(\266)469(ee)]TJ -34.084 -1.1988 Td [(sur)-337(toute)-337(la)-331(tribu)]TJ /F6 1 Tf 7.9036 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 19.716 45.616 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.229 32.027 Td [(.)-445(P)30(our)-337(conclure)-331(il)-337(su\261t)-337(de)-337(prouv)30(er)-337(que)]TJ /F6 1 Tf 17.373 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 46 45.616 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.928 32.027 Td [(est)-337(la)-337(tribu)-331(des)-337(b)-30(or)24(\266)475(eliens.)]TJ -27.596 -1.6024 Td [(Comme)]TJ /F8 1 Tf 3.753 0 Td [(X)]TJ /F2 1 Tf 1.3313 0 Td [(est)-421(d)24(\266)469(enom)24(brable)-421(\265)500(a)-421(l'in\257ni,)-445(on)-421(p)-30(eut)-421(l')24(\266)469(ecrire)-421(comme)-427(r)30(\266)469(eunion)-421(de)-421(compacts)]TJ /F8 1 Tf 32.41 0 Td [(K)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.5 43.331 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 30.424 Td [(.)]TJ -38.886 -1.1988 Td [(Alors,)-409(p)-30(our)-397(tout)-403(ferm)30(\266)469(e)]TJ /F8 1 Tf 10.476 0 Td [(F)]TJ /F2 1 Tf 1.1807 0 Td [(de)]TJ /F8 1 Tf 1.3916 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)]TJ /F8 1 Tf 0.69277 0 Td [(F)]TJ /F6 1 Tf 1.0422 0 Td [(\\)]TJ /F8 1 Tf 0.93374 0 Td [(K)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.638 41.607 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.747 29.226 Td [(est)-403(compact)-397(donc)-397(appartien)24(t)-397(\265)499(a)]TJ /F6 1 Tf 13.964 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.75 41.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.102 29.226 Td [(et)]TJ /F8 1 Tf 1.2289 0 Td [(F)]TJ /F2 1 Tf 1.1807 0 Td [(qui)-391(est)]TJ -36.181 -1.1988 Td [(la)-433(r)30(\266)469(eunion)-433(de)-433(ces)-433(parties)-433(\(en)-439(quan)30(tit)24(\266)469(e)-433(d)24(\266)469(enom)24(brable\))-427(est)-439(aussi)-433(dans)]TJ /F6 1 Tf 30.392 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 51.905 39.892 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.717 28.027 Td [(.)-746(P)30(ar)-433(passage)-433(au)]TJ -31.386 -1.1988 Td [(compl)24(\266)475(emen)24(taire)-325(on)-325(conclut)-325(que)]TJ /F6 1 Tf 13.904 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.302 38.176 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.548 26.828 Td [(p)-30(oss)24(\265)475(ede)-301(\266)475(egalemen)24(t)-325(les)-325(ouv)24(erts)-331(de)]TJ /F8 1 Tf 14.584 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(et)-325(con)30(tien)24(t)-325(donc)-325(la)]TJ -31.03 -1.2048 Td [(tribu)-385(qu'ils)-385(engendren)24(t)-391(c'est-)0(\265)500(a-dire)-391(celle)-385(des)-385(b)-30(or)24(\266)475(eliens.)-602(L'inclusion)-391(in)30(v)24(erse)-385(r)24(\266)475(esulte)-385(du)-391(fait)]TJ 0 -1.1988 Td [(que)-337(les)-337(compacts)-331(son)18(t)-337(des)-337(b)-24(or)24(\266)469(eliens.)]TJ ET 512.64 290.7 m 519.84 290.7 l 519.84 299.12 l 512.64 299.12 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 22.226 Td [(Exemple.)]TJ /F2 1 Tf 5.0964 0 Td [(Dans)-331(le)-337(cas)]TJ /F8 1 Tf 5.259 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)]TJ /F3 1 Tf 1.0542 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-331(en)-331(prenan)24(t)-331(p)-30(our)]TJ /F8 1 Tf 7.994 0 Td [(\256)]TJ /F2 1 Tf 0.96988 0 Td [(la)-331(forme)-325(lin)24(\266)469(eaire)-331(donn)24(\266)469(ee)-331(par)-331(l'in)24(t)24(\266)475(egrale)]TJ -22.283 -1.2048 Td [(de)-385(Riemann)-385(on)-385(obtien)24(t)-391(la)-385(mesure)-391(de)-385(Borel,)-403(c'est-)0(\265)499(a-dire)-391(la)-385(restriction)-391(aux)-391(b)-30(or)24(\266)475(eliens)-391(de)-385(la)]TJ 0 -1.1988 Td [(mesure)-337(de)-331(Leb)-36(esgue.)]TJ 18.663 -15.934 Td [(107)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 357 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 356 0 R /ProcSet 2 0 R >> /Contents 355 0 R >> endobj 356 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F19 276 0 R >> endobj 358 0 obj << /Length 226 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)]TJ /F5 1 Tf 0 -60.735 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(108)]TJ ET endstream endobj 360 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 359 0 R /ProcSet 2 0 R >> /Contents 358 0 R >> endobj 359 0 obj << /F1 12 0 R /F2 13 0 R /F5 16 0 R >> endobj 361 0 obj << /Length 19446 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F0 1 Tf 20.592 0 0 20.592 0 0 Tm 5.049 32.152 Td [(Chapitre)-377(VI)-31(I)-31(I.)-1132(Les)-377(mesures)-381(pro)-31(duits)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 40.183 Td [(VI)-35(I)-35(I.1.)-236(Sections)-376(et)-376(applications)-381(partielles.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 46.605 Td [(Consid)30(\266)469(erons)-385(deux)-379(ensem)24(bles)]TJ /F8 1 Tf 12.837 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.293 67.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.735 46.605 Td [(et)]TJ /F8 1 Tf 1.2108 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.276 67.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.747 46.605 Td [(et)-379(leur)-385(pro)-30(duit)]TJ /F8 1 Tf 6.8976 0 Td [(X)]TJ /F2 1 Tf 1.259 0 Td [(=)]TJ /F8 1 Tf 1.1325 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.147 67.288 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.482 46.605 Td [(\243)]TJ /F8 1 Tf 1.0301 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.69 67.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.928 46.605 Td [(.)-584(On)-385(note)]TJ /F8 1 Tf 4.8494 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.216 67.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.042 46.605 Td [(\(resp.)]TJ /F8 1 Tf -36.711 -1.1988 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4914 65.564 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4458 45.406 Td [(\))-337(la)-337(pro)-54(jection)-337(de)]TJ /F8 1 Tf 7.9277 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.034 65.564 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.807 45.406 Td [(\(resp.)]TJ /F8 1 Tf 2.8976 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.362 65.564 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.127 45.406 Td [(\).)-451(P)30(our)]TJ /F8 1 Tf 3.5482 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-337(on)-331(a)-337(donc)]TJ /F8 1 Tf 5.2229 0 Td [(x)]TJ /F2 1 Tf 0.85542 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.767 65.564 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.849 45.406 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(;)-168(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.603 65.564 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.765 45.406 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39156 0.80723 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.80723 Td [(.)]TJ -37.241 -1.5663 Td [(P)30(our)]TJ /F8 1 Tf 2.5964 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.164 63.331 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3253 43.84 Td [(2)]TJ /F8 1 Tf 1.2169 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.388 63.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.229 43.84 Td [(on)-499(d)30(\266)469(e\257nit)-493(une)-500(application)]TJ /F8 1 Tf 12.211 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.888 62.521 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 51.072 87.776 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.349 43.84 Td [(de)]TJ /F8 1 Tf 1.5 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.147 63.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.759 43.84 Td [(dans)]TJ /F8 1 Tf 2.506 0 Td [(X)]TJ /F2 1 Tf 1.4036 0 Td [(par)]TJ /F8 1 Tf 1.9398 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.776 63.331 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.241 43.84 Td [(7!)]TJ /F2 1 Tf 1.5542 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.888 63.331 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.036 43.84 Td [(;)-162(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.741 63.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.56 43.84 Td [(\).)-939(On)]TJ -36.229 -1.1988 Td [(remarquera)-409(que)-415(la)-409(comp)-30(os)24(\266)469(ee)]TJ /F8 1 Tf 13.072 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.207 61.607 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.566 42.641 Td [(\261)]TJ /F8 1 Tf 0.77109 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.017 60.806 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 42.867 85.379 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.163 42.641 Td [(est)-415(l'application)-415(constan)24(te)-409(de)-409(v)48(aleur)]TJ /F8 1 Tf 16.458 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.086 61.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.886 42.641 Td [(tandis)-409(que)]TJ /F8 1 Tf -34.554 -1.2048 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4914 59.892 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.494 41.436 Td [(\261)]TJ /F8 1 Tf 0.54819 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.991 59.081 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 16.265 82.981 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.7048 41.436 Td [(est)-246(l'application)-253(iden)24(tique)-246(de)]TJ /F8 1 Tf 12.602 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.793 59.892 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.729 41.436 Td [(.)-415(Cette)-246(application)-253(d)30(\266)469(e\257nit)-246(clairemen)24(t)-246(une)-246(bijection)]TJ -17.398 -1.1988 Td [(de)]TJ /F8 1 Tf 1.2771 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.759 58.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3072 40.238 Td [(sur)-283(son)-283(image,)-295(la)-283(bijection)-283(in)24(v)24(erse)-253(\266)475(etan)24(t)-283(la)-283(restriction)-283(de)]TJ /F8 1 Tf 24.434 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.716 58.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.855 40.238 Td [(.)-427(Cette)-283(image)-283(est)-283(form)24(\266)469(ee)]TJ -28.524 -1.1988 Td [(par)-379(les)-355(\266)475(el)18(\266)475(emen)24(ts)-379(de)]TJ /F8 1 Tf 8.8373 0 Td [(X)]TJ /F2 1 Tf 1.2892 0 Td [(qui)-379(on)30(t)]TJ /F8 1 Tf 3.5361 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28 56.452 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.223 39.039 Td [(p)-30(our)-385(premi)24(\265)475(ere)-385(comp)-30(osan)24(te,)-397(on)-379(dit)-385(que)-385(c'est)-379(la)]TJ /F1 1 Tf 20.795 0 Td [(section)]TJ /F2 1 Tf -35.687 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3554 0 Td [(X)]TJ /F2 1 Tf 1.259 0 Td [(d)30(\266)469(e\257nie)-355(par)]TJ /F8 1 Tf 4.988 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.328 54.737 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.139 37.834 Td [(\(les)-325(\266)469(el)24(\266)469(emen)24(ts)-355(de)-349(cette)-361(section)-349(s'obtiennen)24(t)-355(donc)-355(en)-355(\257xan)24(t)-355(la)-355(premi)24(\265)469(ere)]TJ -8.8072 -1.1988 Td [(comp)-30(osan)24(te)-337(et)-337(en)-331(laissan)24(t)-337(v)54(arier)-337(l'autre\).)]TJ 0 -1.5663 Td [(Plus)-415(g)30(\266)469(en)24(\266)475(eralemen)24(t,)-433(si)]TJ /F8 1 Tf 9.8675 0 Td [(A)]TJ /F2 1 Tf 1.1627 0 Td [(est)-415(une)-415(partie)-415(de)]TJ /F8 1 Tf 7.9879 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-433(l'in)24(tersection)-415(de)]TJ /F8 1 Tf 8.1265 0 Td [(A)]TJ /F2 1 Tf 1.1627 0 Td [(a)30(v)24(ec)-415(cette)-415(section)-415(sera)]TJ -29.217 -1.1988 Td [(app)-30(el)24(\266)469(ee)-325(la)]TJ /F1 1 Tf 4.6687 0 Td [(section)]TJ /F2 1 Tf 3.7952 0 Td [(de)]TJ /F8 1 Tf 1.3193 0 Td [(A)]TJ /F2 1 Tf 1.0723 0 Td [(d)24(\266)475(e\257nie)-325(par)]TJ /F8 1 Tf 4.9217 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.026 49.056 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.964 33.87 Td [(.)-439(C'est)-325(donc)-325(l'ensem)30(ble)-325(des)-325(\()]TJ /F8 1 Tf 12.548 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.207 49.056 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.367 33.87 Td [(;)-162(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.06 49.056 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.892 33.87 Td [(\))-325(qui)-319(son)18(t)-319(dans)]TJ /F8 1 Tf 6.8554 0 Td [(A)]TJ /F2 1 Tf -38.416 -1.1988 Td [(\(p)-30(our)]TJ /F8 1 Tf 2.7952 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.448 47.34 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.5904 32.671 Td [(v)54(arian)24(t)-385(dans)]TJ /F8 1 Tf 5.8253 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.353 47.34 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.831 32.671 Td [(\).)-584(En)-379(utilisan)24(t)-379(alors)-385(la)-379(pro)-54(jection)]TJ /F8 1 Tf 15.229 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.75 47.34 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.181 32.671 Td [(,)-391(on)-379(iden)24(ti\257era)-379(souv)24(en)24(t)-379(la)]TJ -27.849 -1.2048 Td [(section)-301(de)]TJ /F8 1 Tf 4.6145 0 Td [(X)]TJ /F2 1 Tf 1.2108 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1687 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.362 45.625 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.048 31.467 Td [(et)-307(la)-301(section)-307(de)]TJ /F8 1 Tf 6.8373 0 Td [(A)]TJ /F2 1 Tf 1.0542 0 Td [(a)30(v)24(ec)-307(son)-313(image)-307(par)]TJ /F8 1 Tf 8.5301 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.905 45.625 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.59 31.467 Td [(.)-433(La)-313(section)-301(de)]TJ /F8 1 Tf 6.759 0 Td [(A)]TJ /F2 1 Tf 1.0542 0 Td [(sera)-307(donc)]TJ -35.072 -1.1988 Td [(iden)24(ti\257)24(\266)469(ee)-337(\265)500(a)]TJ /F6 1 Tf 5.0843 0 Td [(f)]TJ /F8 1 Tf 0.5 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.44 43.909 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.277 30.268 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.216 43.909 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.916 30.268 Td [(j)]TJ /F2 1 Tf 0.56024 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.517 43.909 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.717 30.268 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.379 43.909 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.241 30.268 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(A)]TJ /F6 1 Tf 0.75301 0 Td [(g)]TJ /F2 1 Tf 0.77711 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.034 43.926 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F7 1 Tf -0.90517 -1.0345 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 47.072 60.355 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.157 30.268 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-337(que)-337(l'on)-337(notera)]TJ /F8 1 Tf 7.6506 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.216 43.099 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 68.289 60.632 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.47 30.268 Td [(.)]TJ -29.139 -1.5663 Td [(Si)]TJ /F8 1 Tf 1.1867 0 Td [(f)]TJ /F2 1 Tf 0.94578 0 Td [(est)-361(une)-349(application)-361(d)30(\266)469(e\257nie)-355(sur)]TJ /F8 1 Tf 13.56 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-361(on)-355(d)30(\266)469(e\257nit)-349(l')]TJ /F1 1 Tf 5.7349 0 Td [(application)-409(partielle)]TJ /F8 1 Tf 10.416 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.724 41.668 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.916 28.701 Td [(;)-168(:)]TJ /F2 1 Tf 0.72289 0 Td [(\))-355(sur)]TJ /F8 1 Tf 2.4398 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.94 41.668 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 27.503 Td [(par)]TJ /F8 1 Tf 1.8976 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.164 39.943 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.7831 27.503 Td [(7!)]TJ /F8 1 Tf 1.4699 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.905 39.943 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.096 27.503 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.767 39.943 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.62 27.503 Td [(\))-457(\(autremen)24(t)-451(dit)-457(on)-451(\257xe)-451(le)-457(premi)24(\265)469(ere)-451(comp)-30(osan)18(te)-451(et)-451(on)-451(prend)-451(l'autre)]TJ -8.2892 -1.2048 Td [(comme)-337(v)54(ariable\).)-445(Cette)-337(application)-337(partielle)-343(est)-337(en)-331(fait)-337(la)-337(comp)-30(os)24(\266)475(ee)]TJ /F8 1 Tf 30 0 Td [(f)]TJ /F6 1 Tf 0.81927 0 Td [(\261)]TJ /F8 1 Tf 0.72289 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.69 37.426 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 75.94 52.704 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.289 26.298 Td [(.)]TJ -32.958 -1.8253 Td [(Il)-397(est)-409(clair)-397(que)-403(l'on)-403(p)-30(eut)-403(donner)-409(des)-403(d)30(\266)469(e\257nitions)-397(analogues)-403(en)-373(\266)475(ec)24(hangean)18(t)-403(les)-397(r)-6(^)500(oles)-397(de)]TJ /F8 1 Tf 37.976 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 35.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 23.268 Td [(et)-337(de)]TJ /F8 1 Tf 2.5 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.509 33.892 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.253 23.268 Td [(.)-445(A\257n)-331(d')24(\266)469(eviter)-331(des)-337(confusions,)-337(les)-337(sections)-337(que)-337(l'on)-337(vien)18(t)-337(de)-331(d)24(\266)475(e\257nir)-337(seron)24(t)-337(parfois)]TJ -3.9217 -1.1988 Td [(app)-30(el)24(\266)469(ees)]TJ /F1 1 Tf 3.9518 0 Td [(sections)-349(v)30(erticales)]TJ /F2 1 Tf 8.988 0 Td [(,)-313(tandis)-307(que)-307(celles)-307(d)24(\266)475(e\257nies)-313(en)-301(\257xan)24(t)-307(la)-307(deuxi)30(\265)469(eme)-301(comp)-36(osan)24(te)]TJ /F8 1 Tf -12.94 -1.1988 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4483 30.452 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.747 20.87 Td [(seron)24(t)-337(app)-30(el)24(\266)469(ees)]TJ /F1 1 Tf 6.9639 0 Td [(sections)-379(horizon)24(tales)]TJ /F2 1 Tf 10.319 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 13.756 Td [(VI)-35(I)-35(I.2.)-236(Espace)-376(mesurable)-376(pro)-35(duit.)]TJ 0 -2.4623 Td [(VI)-35(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 13.539 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(D)]TJ 0.83132 0.078314 Td [(\266)]TJ 0 -0.078314 Td [(efinition.)]TJ /F12 1 Tf 5.494 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.1024 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.241 19.961 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.114 13.539 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.422 19.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.639 13.539 Td [(\))]TJ /F12 1 Tf 0.79518 0 Td [(et)]TJ /F2 1 Tf 1.241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.595 19.961 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.482 13.539 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.103 19.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.235 13.539 Td [(\))]TJ /F12 1 Tf 0.79518 0 Td [(des)-409(espaces)-409(mesurables.)-668(Dans)-409(le)]TJ -24.699 -1.2048 Td [(pro)-30(duit)]TJ /F8 1 Tf 3.5361 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.198 18.245 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.428 12.334 Td [(\243)]TJ /F8 1 Tf 0.90361 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.379 18.245 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.753 12.334 Td [(,)-295(on)-289(app)-30(elle)]TJ /F1 1 Tf 5.2892 0 Td [(pa)30(v)48(\266)548(e)-331(mesurable)]TJ /F12 1 Tf 7.9819 0 Td [(tout)-289(pro)-30(duit)]TJ /F8 1 Tf 5.6626 0 Td [(A)]TJ /F2 1 Tf 1.0241 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.259 18.245 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 36.928 12.334 Td [(\243)]TJ /F8 1 Tf 0.90361 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.216 18.245 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 39.38 12.334 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.3434 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.353 18.245 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 42.042 12.334 Td [(\(resp.)]TJ /F8 1 Tf -36.711 -1.1988 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 16.53 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5904 11.135 Td [(\))-337(est)]TJ /F6 1 Tf 2.2831 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.845 16.53 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 9.9578 11.135 Td [(-mesurable)-337(\(resp.)]TJ /F6 1 Tf 7.9639 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.793 16.53 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 19.229 11.135 Td [(-mesurable\).)]TJ -13.898 -1.5663 Td [(La)-313(tribu)]TJ /F6 1 Tf 3.9157 0 Td [(T)]TJ /F12 1 Tf 1.1084 0 Td [(engendr)24(\266)469(ee)-313(par)-313(les)-313(pa)24(v)54(\266)469(es)-313(mesurables)-313(s'app)-30(elle)-313(la)-313(tribu)]TJ /F1 1 Tf 23.259 0 Td [(pro)-30(duit)]TJ /F12 1 Tf 4.0723 0 Td [(des)-319(tribus)]TJ /F6 1 Tf 4.5843 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.638 14.288 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 43.663 9.5689 Td [(et)]TJ /F6 1 Tf -38.331 -1.1988 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7672 12.564 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6386 8.3701 Td [(.)-445(On)-337(la)-337(note)]TJ /F6 1 Tf 5.7289 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.836 12.564 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.669 8.3701 Td [(\243)-222(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.138 12.564 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 15.976 8.3701 Td [(.)]TJ -10.645 -1.5663 Td [(L'espace)-331(mesurable)]TJ /F2 1 Tf 8.753 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-343(alors)-337(le)]TJ /F1 1 Tf 5.0181 0 Td [(pro)-36(duit)]TJ /F12 1 Tf 4.1024 0 Td [(des)-337(espaces)]TJ /F2 1 Tf 5.241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.147 10.323 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.223 6.8039 Td [(;)]TJ /F6 1 Tf 0.44579 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.328 10.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.753 6.8039 Td [(\))]TJ /F12 1 Tf 0.71687 0 Td [(et)]TJ /F2 1 Tf 1.1687 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.284 10.323 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.446 6.8039 Td [(;)]TJ /F6 1 Tf 0.44579 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.802 10.323 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.205 6.8039 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)]TJ /F2 1 Tf -16.596 -2.9157 Td [(109)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 363 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 362 0 R /ProcSet 2 0 R >> /Contents 361 0 R >> endobj 362 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R /F18 107 0 R >> endobj 364 0 obj << /Length 21224 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ 0 -2.4036 Td [(Remarque.)]TJ /F2 1 Tf 6.6627 0 Td [(On)-439(v)54(\266)469(eri\257era)-433(que)-439(l'ensem)24(ble)-433(des)-439(pa)24(v)54(\266)469(es)-439(mesurables)-439(est)-439(un)-439(semi-anneau)-439(de)]TJ -6.6627 -1.1988 Td [(parties)-337(de)]TJ /F8 1 Tf 4.6205 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 48.279 Td [(VI)-35(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 57.876 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Av)24(ec)-337(les)-337(notations)-337(pr)30(\266)469(ec)24(\266)469(eden)30(tes)-337(:)]TJ /F1 1 Tf -11.831 -1.6747 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(Les)-331(applications)]TJ /F8 1 Tf 7.3614 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.431 81.012 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 16.289 56.201 Td [(et)]TJ /F8 1 Tf 1.1626 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.845 81.012 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 18.904 56.201 Td [(son)18(t)-331(mesurables;)]TJ /F1 1 Tf -12.759 -1.6807 Td [(2.)]TJ /F12 1 Tf 1.5843 0 Td [(Si)]TJ /F2 1 Tf 1.1747 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.491 77.806 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.572 54.521 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.543 77.806 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.012 54.521 Td [(\))]TJ /F12 1 Tf 0.73494 0 Td [(est)-349(un)-343(espace)-349(mesurable,)-349(une)-349(application)]TJ /F8 1 Tf 18.265 0 Td [(f)]TJ /F12 1 Tf 0.93976 0 Td [(de)]TJ /F8 1 Tf 1.3434 0 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.836 77.806 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 34.922 54.521 Td [(dans)]TJ /F8 1 Tf 2.3494 0 Td [(X)]TJ /F12 1 Tf 1.253 0 Td [(est)-349(mesurable)]TJ -33.193 -1.1988 Td [(si)-337(et)-337(seulemen)30(t)-337(si)-337(c)24(hacune)-337(de)-331(ses)-343(comp)-30(osan)24(tes)]TJ /F8 1 Tf 20.38 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.664 76.892 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.819 53.322 Td [(\261)]TJ /F8 1 Tf 0.72289 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.284 76.892 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.982 53.322 Td [(\261)]TJ /F8 1 Tf 0.72289 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-343(mesurable.)]TJ /F1 1 Tf -27.301 -2.0181 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(P)30(our)]TJ /F8 1 Tf 2.3735 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.716 74.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.825 51.304 Td [(mesurable)-277(dans)]TJ /F8 1 Tf 6.9578 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.759 74.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.241 51.304 Td [(l'image)-277(r)24(\266)469(ecipro)-24(que)]TJ /F8 1 Tf 8.2229 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.19 74.012 Td [(0\241)]TJ /F4 1 Tf 1.2241 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.38 51.304 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.69 74.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.795 51.304 Td [(\))-277(est)-283(clairemen)24(t)]TJ -32.464 -1.1988 Td [(le)-379(pro)-30(duit)]TJ /F8 1 Tf 4.7289 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.466 72.288 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.337 50.105 Td [(\243)]TJ /F8 1 Tf 1.0301 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19 72.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.789 50.105 Td [(.)-572(C'est)-379(un)-379(pa)30(v)48(\266)469(e)-379(mesurable)-373(donc)-379(une)-379(partie)]TJ /F6 1 Tf 19.361 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable,)-391(par)-373(suite)]TJ /F8 1 Tf 9.6566 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 72.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 48.9 Td [(est)-337(mesurable.)-445(On)-337(raisonne)-337(de)-331(m)24(^)475(eme)-337(p)-30(our)-331(prouv)24(er)-337(que)]TJ /F8 1 Tf 24.428 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.457 70.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.211 48.9 Td [(est)-343(mesurable.)]TJ -25.88 -1.6747 Td [(Soit)]TJ /F8 1 Tf 2.1205 0 Td [(f)]TJ /F2 1 Tf 0.99398 0 Td [(de)]TJ /F8 1 Tf 1.3976 0 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 15.267 67.366 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.512 47.226 Td [(dans)]TJ /F8 1 Tf 2.4036 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)-632(Si)]TJ /F8 1 Tf 2.1446 0 Td [(f)]TJ /F2 1 Tf 0.99398 0 Td [(est)-397(mesurable)-397(lorsque)]TJ /F8 1 Tf 9.9096 0 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.069 67.366 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.548 47.226 Td [(est)-397(m)24(uni)-403(de)-397(la)-397(tribu)]TJ /F6 1 Tf 9.3554 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.448 67.366 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.289 47.226 Td [(alors,)-421(par)]TJ -34.958 -1.2048 Td [(comp)-30(osition,)-337(on)-337(conclut)-337(que)]TJ /F8 1 Tf 12.62 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.56 66.452 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.06 46.021 Td [(\261)]TJ /F8 1 Tf 0.72289 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.181 66.452 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.223 46.021 Td [(\261)]TJ /F8 1 Tf 0.72289 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(son)18(t)-337(encore)-337(mesurables.)]TJ -19.542 -1.6747 Td [(R)30(\266)469(ecipro)-30(quemen)30(t,)-289(si)]TJ /F8 1 Tf 8.6024 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.802 64.047 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.91 44.346 Td [(\261)]TJ /F8 1 Tf 0.59036 0 Td [(f)]TJ /F2 1 Tf 0.86747 0 Td [(et)]TJ /F8 1 Tf 1.1024 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.871 64.047 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.681 44.346 Td [(\261)]TJ /F8 1 Tf 0.59036 0 Td [(f)]TJ /F2 1 Tf 0.86747 0 Td [(son)24(t)-271(mesurables,)-283(p)-30(our)-271(tout)-271(pa)24(v)54(\266)469(e)-271(mesurable)]TJ /F8 1 Tf 18.741 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.698 64.047 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.084 44.346 Td [(\243)]TJ /F8 1 Tf 0.87349 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.543 64.047 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 44.346 Td [(,)]TJ -38.886 -1.1988 Td [(l'image)-295(r)30(\266)469(ecipro)-30(que)]TJ /F8 1 Tf 8.259 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.302 62.331 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.259 43.147 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-295(est)-301(clairemen)24(t)-295(l'in)24(tersection)-295(de)-289(\()]TJ /F8 1 Tf 14.596 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.216 62.331 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.018 43.147 Td [(\261)]TJ /F8 1 Tf 0.63856 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.147 62.331 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.717 43.147 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.31 62.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.139 43.147 Td [(\))-295(et)-295(\()]TJ /F8 1 Tf 2.1928 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.724 62.331 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.675 43.147 Td [(\261)]TJ /F8 1 Tf 0.71687 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\))]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.216 62.331 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.458 43.147 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.388 62.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 43.147 Td [(\))]TJ -38.777 -1.2048 Td [(qui)-409(son)24(t)]TJ /F6 1 Tf 3.994 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.121 59.806 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.319 41.942 Td [(-mesurables.)-680(Comme)-409(les)-415(pa)24(v)54(\266)469(es)-409(mesurables)-415(engendren)24(t)-415(la)-409(tribu)]TJ /F6 1 Tf 28.331 0 Td [(T)]TJ /F2 1 Tf 1.2108 0 Td [(on)-409(conclut)]TJ -34.53 -1.1988 Td [(g)0(^)500(ace)-337(\265)500(a)-331(la)-337(prop)-30(osition)]TJ /F1 1 Tf 9.4277 0 Td [(I)-36(I)-36(I.5.3.)]TJ /F2 1 Tf 3.8133 0 Td [(que)]TJ /F8 1 Tf 1.8675 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.103 58.892 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.651 40.744 Td [(est)-337(mesurable.)]TJ ET 512.64 485.74 m 519.84 485.74 l 519.84 494.17 l 512.64 494.17 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 31.369 Td [(VI)-35(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 37.605 Td [(3.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8614 0 Td [(Av)24(ec)-313(les)-313(notations)-319(pr)30(\266)469(ec)24(\266)469(eden)30(tes,)-319(m)24(unissons)]TJ /F8 1 Tf 18.639 0 Td [(X)]TJ /F12 1 Tf 1.2169 0 Td [(de)-313(la)-313(tribu)-313(pro-)]TJ -32.494 -1.2048 Td [(duit)]TJ /F6 1 Tf 2.1084 0 Td [(T)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.845 52.685 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.88 36.4 Td [(\243)-222(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.138 52.685 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 13.187 36.4 Td [(.)-445(Alors,)-337(p)-30(our)-337(tout)]TJ /F8 1 Tf 8.1807 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.397 52.685 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 22.554 36.4 Td [(dans)]TJ /F8 1 Tf 2.3434 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.922 52.685 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.416 36.4 Td [(:)]TJ -21.084 -1.6747 Td [({)-668(Si)]TJ /F8 1 Tf 2.3313 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(est)]TJ /F6 1 Tf 1.5602 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurable,)-337(la)-337(section)]TJ /F8 1 Tf 9.7952 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.991 49.478 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 44.217 69.548 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 22.765 34.726 Td [(est)]TJ /F6 1 Tf 1.5602 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.957 50.28 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.639 34.726 Td [(-mesurable.)]TJ -20.307 -1.6807 Td [({)-746(Si)]TJ /F8 1 Tf 2.4578 0 Td [(f)]TJ /F12 1 Tf 0.96988 0 Td [(est)-379(mesurable)-373(de)]TJ /F2 1 Tf 7.753 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F12 1 Tf 0.76506 0 Td [(dans)-373(un)-379(espace)-373(mesurable,)-385(l'application)-379(partielle)]TJ /F8 1 Tf 21.783 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.681 47.883 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.386 33.045 Td [(;)-162(:)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ /F12 1 Tf -38.777 -1.1988 Td [(est)-337(mesurable.)]TJ 0 -1.6807 Td [(On)-337(a)-331(un)-307(\266)469(enonc)30(\266)469(e)-337(analogue)-337(en)-307(\266)469(ec)24(hangean)24(t)]TJ /F8 1 Tf 17.898 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.534 43.762 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.747 30.165 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.388 43.762 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 27.337 30.165 Td [(.)]TJ /F1 1 Tf -22.006 -2.0181 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6807 Td [(Nous)-259(a)30(v)18(ons)-259(remarqu)24(\266)469(e)-259(que)-265(la)-259(comp)-30(os)24(\266)469(ee)]TJ /F8 1 Tf 16.717 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.422 38.469 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.006 26.467 Td [(\261)]TJ /F8 1 Tf 0.57831 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.664 37.668 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 49.349 53.042 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.253 26.467 Td [(\(resp.)]TJ /F8 1 Tf 2.8735 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.121 38.469 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.319 26.467 Td [(\261)]TJ /F8 1 Tf 0.57229 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.69 37.668 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 61.964 53.042 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.307 26.467 Td [(\))-259(est)-265(une)-259(application)-265(constan)24(te)]TJ -25.976 -1.1988 Td [(\(resp.)-548(l'iden)24(tit)30(\266)469(e)-373(de)]TJ /F8 1 Tf 8.5964 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.224 36.754 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.343 25.268 Td [(\).)-548(Il)-367(s'agit)-373(toujours)-367(d'une)-373(application)-367(mesurable,)-379(par)-367(suite)]TJ /F8 1 Tf 26.145 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.284 35.943 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 85.157 50.644 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 25.268 Td [(est)]TJ -37.934 -1.1988 Td [(mesurable)-331(d'apr)18(\265)475(es)-337(la)-331(prop)-36(osition)]TJ /F1 1 Tf 14.56 0 Td [(VI)-36(I)-36(I.2.2.)]TJ /F2 1 Tf 4.3494 0 Td [(.)]TJ -18.91 -1.6807 Td [(Nous)-313(a)24(v)24(ons)-289(\266)469(egalemen)30(t)-319(remarqu)24(\266)475(e)]TJ /F8 1 Tf 14.331 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.216 31.823 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 41.735 44.885 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.464 22.388 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(\256)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.147 32.65 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F7 1 Tf -0.89655 -1.0259 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 47.229 44.596 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.241 22.388 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-319(et)]TJ /F8 1 Tf 1.8494 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.198 32.633 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.066 22.388 Td [(;)-168(:)]TJ /F2 1 Tf 0.72289 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(f)]TJ /F6 1 Tf 0.78313 0 Td [(\261)]TJ /F8 1 Tf 0.68072 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.103 31.823 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 68.145 44.885 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.392 22.388 Td [(,)-319(le)-319(r)30(\266)469(esultat)-319(en)-313(d)24(\266)475(ecoule.)]TJ ET 512.64 252.03 m 519.84 252.03 l 519.84 260.46 l 512.64 260.46 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 15.058 Td [(VI)-35(I)-35(I.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 18.051 Td [(4.)-283(Exemples.)]TJ /F2 1 Tf -4.0301 -1.6807 Td [(Consid)30(\266)469(erons)-463(deux)-457(espaces)-463(top)-36(ologiques)]TJ /F8 1 Tf 17.849 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.466 24.021 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.825 16.37 Td [(et)]TJ /F8 1 Tf 1.2952 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.672 24.021 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28 16.37 Td [(et)-457(l'espace)-463(top)-36(ologique)-457(pro)-30(duit)]TJ /F8 1 Tf 14.319 0 Td [(X)]TJ /F2 1 Tf 1.3976 0 Td [(=)]TJ /F8 1 Tf -38.386 -1.1988 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.9224 22.297 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6747 15.171 Td [(\243)]TJ /F8 1 Tf 0.92771 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.181 22.297 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3253 15.171 Td [(don)24(t)-301(les)-301(ouv)24(erts)-307(son)24(t)-307(les)-301(r)30(\266)469(eunions)-301(\(quelconques\))-301(de)-301(pro)-30(duits)]TJ /F8 1 Tf 26.223 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.009 22.297 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 36.777 15.171 Td [(\243)]TJ /F8 1 Tf 0.93373 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.095 22.297 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.313 15.171 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.3554 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.336 22.297 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.042 15.171 Td [(\(resp.)]TJ /F8 1 Tf -36.711 -1.2048 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7586 20.581 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6325 13.967 Td [(\))-349(est)-349(un)-343(ouv)18(ert)-343(de)]TJ /F8 1 Tf 8.2289 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.569 20.581 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.398 13.967 Td [(\(resp.)]TJ /F8 1 Tf 2.9277 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.957 20.581 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.747 13.967 Td [(\).)-475(On)-349(p)-30(eut)-349(alors)-343(m)24(unir)]TJ /F8 1 Tf 10.476 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)]TJ /F8 1 Tf 0.62048 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.172 20.581 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.289 13.967 Td [(et)]TJ /F8 1 Tf 1.1747 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.052 20.581 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.229 13.967 Td [(de)-343(leur)-343(tribu)-349(des)]TJ -31.898 -1.1988 Td [(b)-30(or)30(\266)469(eliens)]TJ /F6 1 Tf 4.1867 0 Td [(B)]TJ /F2 1 Tf 0.69277 0 Td [(,)]TJ /F6 1 Tf 0.59639 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.448 18.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.096 12.768 Td [(et)]TJ /F6 1 Tf 1.1566 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.948 18.866 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.452 12.768 Td [(,)-319(mais)-325(on)-319(p)-30(eut)-325(aussi)-319(m)24(unir)]TJ /F8 1 Tf 11.946 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(de)-319(la)-319(tribu)-319(pro)-30(duit)]TJ /F6 1 Tf 8.4819 0 Td [(T)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.707 18.866 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 39.404 12.768 Td [(\243)-198(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.767 18.866 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.578 12.768 Td [(.)-439(Nous)]TJ -36.247 -1.1988 Td [(allons)-307(\266)475(etudier)-343(les)-331(liens)-343(qu'il)-331(y)-337(a)-337(en)30(tre)]TJ /F6 1 Tf 16.446 0 Td [(T)]TJ /F2 1 Tf 1.1325 0 Td [(et)]TJ /F6 1 Tf 1.1687 0 Td [(B)]TJ /F2 1 Tf 0.68675 0 Td [(.)]TJ -19.434 -1.6807 Td [(T)78(out)-271(d'ab)-36(ord,)-283(les)-271(pro)-54(jections)]TJ /F8 1 Tf 12.747 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.741 14.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.235 9.8882 Td [(et)]TJ /F8 1 Tf 1.1024 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.966 14.745 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.723 9.8882 Td [(de)]TJ /F8 1 Tf 1.2651 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(sur)]TJ /F8 1 Tf 1.6084 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.19 14.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.235 9.8882 Td [(et)]TJ /F8 1 Tf 1.1024 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.845 14.745 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.024 9.8882 Td [(son)24(t)-271(con)24(tin)24(ues)-271(donc)-271(mesurables)-277(si)]TJ -24.693 -1.1988 Td [(l'on)-283(m)24(unit)]TJ /F8 1 Tf 4.753 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(de)-283(la)-283(tribu)-283(des)-283(b)-30(or)30(\266)469(eliens)]TJ /F6 1 Tf 10.62 0 Td [(B)]TJ /F2 1 Tf 0.68674 0 Td [(.)-427(D'apr)24(\265)475(es)-283(la)-283(prop)-30(osition)]TJ /F1 1 Tf 10.578 0 Td [(VI)-36(I)-36(I.2.2.)]TJ /F2 1 Tf 4.6325 0 Td [(cela)-283(prouv)24(e)-283(que)]TJ -32.458 -1.1988 Td [(l'application)-409(iden)24(tique)-403(de)-409(l'espace)-403(\()]TJ /F8 1 Tf 15.741 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(B)]TJ /F2 1 Tf 0.68675 0 Td [(\))-409(dans)-403(l'espace)-409(\()]TJ /F8 1 Tf 7.3434 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(\))-403(est)-409(mesurable,)-427(autremen)24(t)]TJ -27.163 -1.2048 Td [(dit)-415(tout)-391(\266)469(el)24(\266)469(emen)30(t)-415(de)]TJ /F6 1 Tf 9.0723 0 Td [(T)]TJ /F2 1 Tf 1.2169 0 Td [(se)-415(trouv)24(e)-415(aussi)-415(dans)]TJ /F6 1 Tf 9.4036 0 Td [(B)]TJ /F2 1 Tf 0.68674 0 Td [(.)-686(Cep)-30(endan)24(t)-415(il)-415(est)-415(p)-30(ossible)-415(de)-415(donner)-421(des)]TJ /F5 1 Tf -20.38 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(110)]TJ ET endstream endobj 366 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 365 0 R /ProcSet 2 0 R >> /Contents 364 0 R >> endobj 365 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R /F18 107 0 R >> endobj 367 0 obj << /Length 18721 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(exemples)-301(p)-30(our)-301(lesquels)-307(la)-301(r)30(\266)469(ecipro)-30(que)-301(n'est)-301(pas)-307(vraie,)-313(toutefois)-301(nous)-301(allons)-301(pr)24(\266)475(esen)24(ter)-301(un)-307(cas)]TJ 0 -1.1988 Td [(imp)-30(ortan)18(t)-331(o)-30(\265)524(u)-331(l'on)-337(a)]TJ /F6 1 Tf 8.8916 0 Td [(T)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(B)]TJ /F2 1 Tf 0.68675 0 Td [(.)]TJ -11.717 -1.759 Td [(P)30(ar)-277(d)24(\266)469(e\257nition)-277(l'ensem)24(ble)]TJ /F6 1 Tf 10.97 0 Td [(T)]TJ /F2 1 Tf 1.0783 0 Td [(p)-30(oss)30(\265)469(ede)-283(en)-277(particulier)-283(tous)-277(les)-283(pro)-30(duits)]TJ /F8 1 Tf 17.102 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.483 85.392 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 35.669 59.262 Td [(\243)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.448 85.392 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.139 59.262 Td [(d'un)-283(ouv)24(ert)]TJ /F8 1 Tf 5.2831 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 85.392 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 58.063 Td [(de)]TJ /F8 1 Tf 1.2651 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.741 83.676 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.0482 58.063 Td [(et)-271(d'un)-271(ouv)24(ert)]TJ /F8 1 Tf 6.3614 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.75 83.676 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.982 58.063 Td [(de)]TJ /F8 1 Tf 1.2651 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.974 83.676 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.663 58.063 Td [(.)-421(Il)-271(p)-30(oss)30(\265)469(ede)-271(donc)-265(les)-271(r)30(\266)469(eunions)-265(d)24(\266)469(enom)24(brables)-265(de)-265(ces)-265(pro)-30(duits,)]TJ -13.331 -1.2048 Td [(mais)-373(p)-30(our)-379(obtenir)-373(n'imp)-30(orte)-379(quel)-379(ouv)24(ert)-379(de)]TJ /F8 1 Tf 19.627 0 Td [(X)]TJ /F2 1 Tf 1.2831 0 Td [(on)-373(doit)-379(faire)-379(des)-379(r)30(\266)469(eunions)-379(quelconques)-373(de)]TJ -20.91 -1.1988 Td [(ces)-289(pro)-30(duits.)-439(On)-289(comprend)-295(donc)-295(qu'une)-295(h)24(yp)-30(oth)24(\265)469(ese)-289(de)-295(d)30(\266)469(enom)24(brabilit)24(\266)469(e)-295(sur)-295(les)-295(top)-30(ologies)-295(de)]TJ /F8 1 Tf 0 -1.1988 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.9224 78.521 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8494 54.461 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.776 78.521 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7711 54.461 Td [(soit)-337(utile)-337(p)-30(our)-331(obtenir)-337(l')24(\266)475(egalit)24(\266)469(e)-337(de)]TJ /F6 1 Tf 15.048 0 Td [(B)]TJ /F2 1 Tf 1.0181 0 Td [(et)-337(de)]TJ /F6 1 Tf 2.5 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)]TJ -23.807 -1.759 Td [(Plus)-289(pr)30(\266)469(ecis)24(\266)475(emen)24(t,)-295(supp)-30(osons)-289(que)-289(les)-283(espaces)-289(top)-30(ologiques)]TJ /F8 1 Tf 25.042 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.767 76.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.843 52.701 Td [(et)]TJ /F8 1 Tf 1.1205 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.466 76.004 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.663 52.701 Td [(soien)24(t)-289(\265)499(a)]TJ /F1 1 Tf 3.6024 0 Td [(base)-331(d)24(\266)548(enom-)]TJ -32.934 -1.2048 Td [(brable)]TJ /F2 1 Tf 3.1566 0 Td [(,)-560(c'est-)0(\265)500(a-dire)-512(que)-518(l'on)-518(puisse)-512(trouv)24(er)-518(un)-518(ensem)24(ble)-512(d)24(\266)475(enom)24(brable)]TJ /F6 1 Tf 29.283 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.19 74.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.361 51.497 Td [(\(resp.)]TJ /F6 1 Tf 3.4337 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.388 74.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 51.497 Td [(\))]TJ -38.777 -1.1988 Td [(d'ouv)30(erts)-421(de)]TJ /F8 1 Tf 5.8373 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.284 72.573 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.771 50.298 Td [(\(resp.)]TJ /F8 1 Tf 3.1446 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.069 72.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.331 50.298 Td [(\))-421(de)-415(fa)0(\273)439(con)-415(que)-421(tout)-415(ouv)18(ert)-415(de)]TJ /F8 1 Tf 13.747 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.767 72.573 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.681 50.298 Td [(\(resp.)]TJ /F8 1 Tf 3.1386 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.56 72.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.241 50.298 Td [(\))-415(soit)-415(la)-415(r)24(\266)475(eunion)]TJ -31.91 -1.1988 Td [(de)-379(certains)-379(des)-355(\266)475(el)24(\266)469(emen)24(ts)-379(de)]TJ /F6 1 Tf 12.446 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.578 70.849 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.235 49.099 Td [(\(resp.)]TJ /F6 1 Tf 3.0301 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.009 70.849 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.578 49.099 Td [(\).)-584(Il)-379(est)-385(alors)-379(clair)-379(que)-385(tout)-379(ouv)18(ert)-379(de)]TJ /F8 1 Tf 17.247 0 Td [(X)]TJ /F2 1 Tf 1.2831 0 Td [(est)-385(la)]TJ -36.777 -1.1988 Td [(r)30(\266)469(eunion)-409(de)-403(certains)-403(des)]TJ /F8 1 Tf 10.693 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.06 69.133 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.367 47.9 Td [(\243)]TJ /F8 1 Tf 1.0482 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.483 69.133 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.12 47.9 Td [(p)-30(our)]TJ /F8 1 Tf 2.4337 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.414 69.133 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 24.024 47.9 Td [(2)-403(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.043 69.133 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.572 47.9 Td [(et)]TJ /F8 1 Tf 1.241 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.931 69.133 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.512 47.9 Td [(2)-397(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.897 69.133 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.886 47.9 Td [(.)-656(Cep)-24(endan)24(t)-409(il)-403(n'y)-409(a)-403(qu'une)]TJ -26.554 -1.2048 Td [(quan)30(tit)24(\266)469(e)-361(d)30(\266)469(enom)24(brable)-361(de)-355(tels)-355(pro)-30(duits)-361(et)-361(ceux-ci)-361(appartiennen)24(t,)-367(par)-355(d)24(\266)469(e\257nition,)-361(\265)500(a)-361(la)-355(tribu)]TJ 0 -1.1988 Td [(pro)-30(duit)]TJ /F6 1 Tf 3.5723 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(.)-439(Il)-319(s'ensuit)-325(que)]TJ /F6 1 Tf 7.1386 0 Td [(T)]TJ /F2 1 Tf 1.1145 0 Td [(p)-30(oss)24(\265)475(ede)-325(tous)-325(les)-325(ouv)24(erts)-325(de)]TJ /F8 1 Tf 12.054 0 Td [(X)]TJ /F2 1 Tf 1.2289 0 Td [(et)-325(donc)-319(con)24(tien)24(t)-325(la)-319(tribu)-319(qu'ils)]TJ -25.91 -1.1988 Td [(engendren)24(t.)-445(Autremen)24(t)-337(dit)]TJ /F6 1 Tf 12.151 0 Td [(B)-307(\275)-277(T)]TJ /F2 1 Tf 2.8193 0 Td [(.)-445(L'autre)-337(inclusion)-337(a)30(y)24(an)24(t)-307(\266)469(et)24(\266)475(e)-313(\266)469(etablie)-337(on)-331(en)-337(d)30(\266)469(eduit)-337(:)]TJ /F15 1 Tf -14.97 -2.1386 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8313 0 Td [(Si)]TJ /F8 1 Tf 1.0723 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.241 60.918 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.657 42.159 Td [(et)]TJ /F8 1 Tf 1.0723 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.802 60.918 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.38 42.159 Td [(son)18(t)-234(des)-240(espaces)-240(top)-30(ologiques)-240(\265)500(a)-234(base)-240(d)24(\266)475(enom)24(brable)-240(alors)-240(l'espace)]TJ -12.048 -1.1988 Td [(top)-30(ologique)-301(pro)-30(duit)]TJ /F8 1 Tf 8.9157 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.888 59.202 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.831 40.961 Td [(\243)]TJ /F8 1 Tf 0.93976 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.155 59.202 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.488 40.961 Td [(est)-307(aussi)-307(\265)500(a)-301(base)-301(d)24(\266)469(enom)24(brable)-301(et)-307(la)-301(tribu)-301(de)-301(b)-30(or)24(\266)475(eliens)-307(de)]TJ /F8 1 Tf -15.157 -1.2048 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(est)-337(le)-337(pro)-30(duit)-337(des)-337(tribus)-337(de)-331(b)-30(or)24(\266)475(eliens)-337(de)]TJ /F8 1 Tf 17.705 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.043 57.487 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.795 39.756 Td [(et)-337(de)]TJ /F8 1 Tf 2.506 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.793 57.487 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 29.717 39.756 Td [(.)]TJ /F2 1 Tf -24.386 -2.5181 Td [(En)-331(particulier)-331(nous)-331(a)30(v)24(ons)-337(observ)54(\266)475(e)-337(que)-331(tout)-337(ouv)24(ert)-331(de)]TJ /F3 1 Tf 23.518 0 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(\(resp.)]TJ ET 0.504 w 392.04 455.04 m 400.68 455.04 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 32.801 37.238 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))-331(est)-337(une)-331(r)30(\266)469(eunion)-331(d'in)24(ter-)]TJ -28.193 -1.1988 Td [(v)54(alles)-331(ouv)18(erts)-331(\265)499(a)-331(extr)30(\266)469(emit)24(\266)469(es)-331(rationnelles)-331(\(resp.)-445(rationnelles)-337(ou)-331(in\257nies\).)-439(Cela)-337(prouv)30(e)-337(que)]TJ /F3 1 Tf 38.44 0 Td [(R)]TJ /F2 1 Tf -38.44 -1.1988 Td [(et)]TJ ET 77.688 426.31 m 86.328 426.31 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5 34.84 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(son)24(t)-337(des)-337(espaces)-337(top)-36(ologiques)-337(\265)500(a)-331(base)-337(d)24(\266)475(enom)24(brable.)]TJ -2.2229 -1.759 Td [(De)-415(m)24(^)469(eme,)-433(tout)-415(ouv)18(ert)-415(d'un)-421(espace)-415(v)24(ectoriel)-415(de)-415(dimension)-421(\257nie)-415(est)-421(la)-415(r)30(\266)469(eunion)-421(de)-415(b)-24(oules)]TJ 0 -1.1988 Td [(ouv)24(ertes)-289(de)-289(ra)24(y)24(on)-289(rationnel)-289(et)-289(de)-283(cen)18(tre)-289(\265)500(a)-283(co)-36(ordonn)24(\266)469(ees)-289(rationnelles.)-433(Un)-289(tel)-289(espace)-289(est)-295(donc)]TJ 0 -1.2048 Td [(\265)499(a)-331(base)-337(d)24(\266)475(enom)24(brable.)]TJ /F15 1 Tf 0 -2.1386 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8855 0 Td [(Le)-337(pro)-30(duit)]TJ /F2 1 Tf 5 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a;)-174(b)]TJ /F2 1 Tf 1.4036 0 Td [(\))]TJ /F6 1 Tf 0.6747 0 Td [(7!)]TJ /F8 1 Tf 1.2892 0 Td [(a)]TJ /F6 1 Tf 0.5241 0 Td [(\242)]TJ /F8 1 Tf 0.27711 0 Td [(b)]TJ /F12 1 Tf 0.77108 0 Td [(est)-343(une)-337(application)-343(b)-30(or)24(\266)475(elienne)-337(de)]TJ ET 446.18 351.07 m 454.82 351.07 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 37.331 28.539 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 54.457 41.762 Td [(2)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.837 28.539 Td [(dans)]TJ ET 492.19 351.07 m 500.83 351.07 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 41.181 28.539 Td [(R)]TJ /F12 1 Tf 1.0602 0 Td [(\(a)18(v)24(ec)]TJ -36.91 -1.1988 Td [(la)-331(con)24(v)24(en)30(tion)]TJ /F2 1 Tf 6.1145 0 Td [(0)]TJ /F6 1 Tf 0.72289 0 Td [(\242)-222(\2471)]TJ /F2 1 Tf 2.5542 0 Td [(=)-277(0)]TJ /F12 1 Tf 1.5542 0 Td [(\).)]TJ -10.946 -2 Td [(L'ensem)24(ble)]TJ /F8 1 Tf 5.1446 0 Td [(X)]TJ /F12 1 Tf 1.2229 0 Td [(des)-319(p)-24(oin)24(ts)-319(o)-24(\265)524(u)-319(la)-313(somme)-319(est)-319(d)24(\266)475(e\257nie)-319(\()]TJ /F8 1 Tf 15.572 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(=)]TJ ET 352.73 312.76 m 361.37 312.76 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 29.512 25.34 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.267 37.176 Td [(2)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.867 25.34 Td [(n)-186(f)]TJ /F2 1 Tf 1.1867 0 Td [(\()]TJ /F6 1 Tf 0.39156 0 Td [(\2411)]TJ /F8 1 Tf 1.7771 0 Td [(;)]TJ /F2 1 Tf 0.44578 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F8 1 Tf 0.39156 0 Td [(;)]TJ /F2 1 Tf 0.44579 0 Td [(\(+)]TJ /F6 1 Tf 1.1626 0 Td [(1)]TJ /F8 1 Tf 1.006 0 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(\2411)]TJ /F2 1 Tf 1.7831 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(g)]TJ /F12 1 Tf 0.5 0 Td [(\))-319(est)]TJ -37.229 -1.1988 Td [(b)-30(or)30(\266)469(elien)-331(et)-337(la)-337(somme)]TJ /F2 1 Tf 9.4277 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a;)-168(b)]TJ /F2 1 Tf 1.4036 0 Td [(\))]TJ /F6 1 Tf 0.66265 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(a)]TJ /F2 1 Tf 0.75301 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(b)]TJ /F12 1 Tf 0.76506 0 Td [(est)-337(une)-337(application)-337(b)-30(or)30(\266)469(elienne)-337(de)]TJ /F8 1 Tf 14.765 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(dans)]TJ ET 470.38 298.44 m 479.02 298.44 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 39.355 24.141 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F1 1 Tf -34.747 -2.1386 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Soit)]TJ /F8 1 Tf 2.1988 0 Td [(U)]TJ /F2 1 Tf 1.2651 0 Td [(un)-481(ouv)24(ert)-481(de)]TJ ET 286.13 272.88 m 294.77 272.88 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 23.94 22.003 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(,)-512(il)-475(s'agit)-475(de)-481(prouv)30(er)-481(que)-481(l'ensem)30(ble)]TJ /F8 1 Tf 16.886 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.595 32.073 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.102 22.003 Td [(des)]TJ -37.771 -1.3916 Td [(couples)-337(\()]TJ /F8 1 Tf 3.8976 0 Td [(a;)-168(b)]TJ /F2 1 Tf 1.3976 0 Td [(\))-337(tels)-337(que)]TJ /F8 1 Tf 4.4277 0 Td [(a)]TJ /F6 1 Tf 0.53012 0 Td [(\242)]TJ /F8 1 Tf 0.27711 0 Td [(b)]TJ /F2 1 Tf 0.75904 0 Td [(appartienne)-337(\265)500(a)]TJ /F8 1 Tf 6.3434 0 Td [(U)]TJ /F2 1 Tf 1.1265 0 Td [(est)-337(un)-337(b)-30(or)30(\266)469(elien)-331(de)]TJ ET 385.34 256.32 m 393.98 256.32 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 32.241 20.611 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.172 30.418 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.41 20.611 Td [(.)]TJ -28.078 -2 Td [(P)30(osons)]TJ /F8 1 Tf 3.2892 0 Td [(F)]TJ /F2 1 Tf 1.0602 0 Td [(=)-277(\()]TJ /F6 1 Tf 1.4458 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(0)]TJ /F6 1 Tf 0.5 0 Td [(g)-138(\243)]TJ ET 163.58 232.34 m 172.22 232.34 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 13.687 18.611 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ /F6 1 Tf 0.53012 0 Td [([)]TJ /F2 1 Tf 0.80723 0 Td [(\()]TJ ET 192.82 232.34 m 201.46 232.34 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 16.133 18.611 Td [(R)]TJ /F6 1 Tf 0.86747 0 Td [(\243)-138(f)]TJ /F2 1 Tf 1.4157 0 Td [(0)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.5 0 Td [(\).)-433(C'est)-301(un)-295(ferm)30(\266)469(e)-295(de)]TJ ET 339.98 232.34 m 348.62 232.34 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 28.446 18.611 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.741 27.556 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.62 18.611 Td [(;)-307(c'est)-295(aussi)-295(l'ensem)30(ble)-295(des)-295(couples)]TJ -24.289 -1.1988 Td [(de)-331(pro)-30(duit)-331(n)30(ul.)-451(Il)-331(s'ensuit)-331(que)]TJ /F8 1 Tf 13.446 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.009 25.504 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 20.072 17.412 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(F)]TJ /F2 1 Tf 1.1145 0 Td [(est)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F2 1 Tf 1.1145 0 Td [(ou)-331(vide)-337(suiv)54(an)24(t)-331(que)-337(0)-331(appartien)30(t)-337(ou)-331(non)-337(\265)500(a)]TJ /F8 1 Tf 18.687 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(,)]TJ -38.886 -1.2048 Td [(c'est)-337(de)-331(toute)-337(fa)-6(\273)439(con)-331(un)-337(ferm)24(\266)475(e.)]TJ 0 -1.753 Td [(P)30(ar)-319(ailleurs,)-319(sur)-319(le)-319(compl)24(\266)469(emen)30(taire)]TJ /F8 1 Tf 15.452 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.862 21.271 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.289 14.454 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(F)]TJ /F2 1 Tf 1.0964 0 Td [(le)-325(pro)-30(duit)-313(est)-319(con)24(tin)24(u.)-439(Il)-319(s'ensuit)-313(que)]TJ /F8 1 Tf 16.494 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.086 21.271 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 42.452 14.454 Td [(\\)]TJ /F8 1 Tf 0.85542 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.095 21.271 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 13.063 Td [(est)-337(un)-337(ouv)24(ert)-337(de)]TJ /F8 1 Tf 7.4578 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.414 19.288 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.307 13.063 Td [(donc)-337(de)]TJ ET 215.57 166.1 m 224.21 166.1 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 18.036 13.063 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 26.845 19.616 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.536 13.063 Td [(puisque)]TJ /F8 1 Tf 3.6506 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.302 19.288 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.711 13.063 Td [(est)-337(ouv)24(ert.)]TJ -19.38 -1.759 Td [(P)30(ar)-337(suite)]TJ /F8 1 Tf 4.2711 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.879 16.771 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.012 11.304 Td [(est)-337(r)30(\266)469(eunion)-337(d'un)-343(ouv)24(ert)-337(et)-331(d'un)-343(ferm)30(\266)469(e,)-337(c'est)-337(b)-30(or)24(\266)475(elien.)]TJ -5.6807 -2.1386 Td [(L'ensem)24(ble)]TJ /F8 1 Tf 5.1687 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(a)-343(un)-343(compl)24(\266)469(emen)24(taire)-343(\257ni)-343(donc)-343(ferm)24(\266)475(e,)-349(c'est)-349(un)-343(ouv)18(ert)-343(donc)-343(un)-343(b)-30(or)24(\266)475(elien.)-469(La)]TJ -6.4157 -1.1988 Td [(con)30(tin)18(uit)24(\266)469(e)-433(de)-433(la)-439(somme)-433(de)]TJ /F8 1 Tf 12.289 0 Td [(X)]TJ /F2 1 Tf 1.3373 0 Td [(dans)]TJ ET 255.74 105.12 m 264.38 105.12 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 21.398 7.9665 Td [(R)]TJ /F2 1 Tf 1.1566 0 Td [(en)24(tra)108(^)385(\263ne)-433(qu'il)-433(s'agit)-433(d'une)-439(application)-439(b)-30(or)30(\266)469(elienne.)]TJ ET 512.64 79.664 m 519.84 79.664 l 519.84 88.088 l 512.64 88.088 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.994 3.8882 Td [(111)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 369 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 368 0 R /ProcSet 2 0 R >> /Contents 367 0 R >> endobj 368 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 370 0 obj << /Length 18714 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ 0 -2.4036 Td [(Remarque.)]TJ /F2 1 Tf 6.253 0 Td [(Bien)-337(que)-337(non)-337(con)24(tin)24(ue)-331(l'application)-343(pro)-30(duit)-331(est)-343(b)-30(or)30(\266)469(elienne.)]TJ -6.253 -2.0964 Td [(Cette)-427(prop)-30(osition)-433(p)-30(ermet)-427(de)-427(donner)-427(une)-427(autre)-433(preuv)30(e)-433(du)-427(fait)-427(que)-433(la)-427(somme)-427(\(si)-427(elle)-427(est)]TJ 0 -1.1988 Td [(d)30(\266)469(e\257nie\))-337(et)-337(le)-337(pro)-30(duit)-337(de)-331(deux)-337(applications)-331(mesurables)]TJ /F8 1 Tf 23.807 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(g)]TJ /F2 1 Tf 0.84337 0 Td [(son)18(t)-331(mesurables.)]TJ -26.753 -1.8072 Td [(En)-463(e\256et,)-493(la)-457(mesurabilit)24(\266)469(e)-463(de)]TJ /F8 1 Tf 12.831 0 Td [(f)]TJ /F2 1 Tf 1.0602 0 Td [(et)]TJ /F8 1 Tf 1.2952 0 Td [(g)]TJ /F2 1 Tf 0.9759 0 Td [(en)30(tra)108(^)385(\263ne)-463(celle)-463(de)-457(\()]TJ /F8 1 Tf 8.1928 0 Td [(f)-54(;)-168(g)]TJ /F2 1 Tf 1.5 0 Td [(\))-463(\()-6(\265)500(a)-457(v)48(aleurs)-463(dans)]TJ ET 0.504 w 470.38 692.56 m 479.02 692.56 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 39.355 57.117 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.345 82.65 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.524 57.117 Td [(\))-463(d'apr)24(\265)469(es)]TJ -35.193 -1.1988 Td [(la)-499(prop)-30(osition)]TJ /F1 1 Tf 6.7048 0 Td [(VI)-36(I)-36(I.2.2.)]TJ /F2 1 Tf 4.3494 0 Td [(.)-945(Elle)-500(est)-506(en)-500(outre)-506(\265)500(a)-500(v)48(aleurs)-500(dans)]TJ /F8 1 Tf 16.458 0 Td [(X)]TJ /F2 1 Tf 1.4096 0 Td [(si)-506(la)-500(somme)]TJ /F8 1 Tf 5.9578 0 Td [(f)]TJ /F2 1 Tf 0.93374 0 Td [(+)]TJ /F8 1 Tf 1.1084 0 Td [(g)]TJ /F2 1 Tf 1.012 0 Td [(est)]TJ -37.934 -1.2048 Td [(d)30(\266)469(e\257nie.)-427(Le)-271(r)30(\266)469(esultat)-271(pr)24(\266)469(ec)24(\266)475(eden)24(t)-271(donne)-271(alors)-277(la)-265(mesurabilit)24(\266)469(e)-271(de)]TJ /F8 1 Tf 26.096 0 Td [(f)-108(g)]TJ /F2 1 Tf 1.3795 0 Td [(et)-271(de)]TJ /F8 1 Tf 2.3735 0 Td [(f)]TJ /F2 1 Tf 0.68674 0 Td [(+)]TJ /F8 1 Tf 0.87349 0 Td [(g)]TJ /F2 1 Tf 0.77711 0 Td [(par)-271(comp)-30(osition)]TJ -32.187 -1.1988 Td [(d'applications)-337(mesurables.)]TJ /F1 1 Tf 0 -2.0181 Td [(Remarque.)]TJ /F2 1 Tf 6.2651 0 Td [(La)-343(d)24(\266)475(emonstration)-343(pr)24(\266)475(ec)24(\266)469(eden)30(te)-343(a)-337(utilis)24(\266)475(e)-343(sur)]TJ ET 360.22 625.39 m 368.86 625.39 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 30.139 51.497 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.164 74.616 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.645 51.497 Td [(la)-337(tribu)-343(pro)-30(duit)-337(des)-337(tribus)-343(de)]TJ -26.313 -1.1988 Td [(b)-30(or)30(\266)469(eliens)-325(sur)-325(c)24(haque)-319(facteur)-325(p)-30(our)-319(prouv)24(er)-325(la)-319(mesurabilit)24(\266)469(e)-325(de)-319(\()]TJ /F8 1 Tf 27.018 0 Td [(f)-54(;)-168(g)]TJ /F2 1 Tf 1.5 0 Td [(\))-325(et)-319(la)-325(tribu)-319(de)-325(b)-24(or)24(\266)469(eliens)]TJ -28.518 -1.4458 Td [(de)]TJ ET 78.84 593.78 m 87.48 593.78 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5964 48.852 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 10.474 70.831 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.0361 48.852 Td [(p)-30(our)-265(prouv)24(er)-271(\(gr)-6(^)500(ace)-271(\265)500(a)-265(la)-271(con)30(tin)18(uit)24(\266)469(e\))-265(la)-271(mesurabilit)24(\266)475(e)-271(de)-271(la)-265(somme)-271(et)-265(du)-271(pro)-30(duit.)-427(Il)-265(est)]TJ -2.7048 -1.1988 Td [(donc)-234(essen)18(tiel)-234(de)-234(sa)24(v)24(oir)-240(que)-240(ces)-234(tribus)-240(son)24(t)-234(iden)24(tiques)-240(\(parce)-240(que)]TJ ET 393.84 579.45 m 402.48 579.45 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 32.952 47.653 Td [(R)]TJ /F2 1 Tf 0.95783 0 Td [(est)-240(\265)500(a)-234(base)-240(d)30(\266)469(enom)24(brable\).)]TJ -28.578 -1.5663 Td [(Le)-373(r)30(\266)469(esultat)-373(concernan)24(t)-373(la)-373(somme)-373(p)-30(eut)-373(d'ailleurs)-343(^)469(etre)-373(mis)-367(en)-373(d)24(\266)475(efaut)-373(si)-379(on)-367(remplace)]TJ ET 501.41 560.73 m 510.05 560.73 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 41.952 46.087 Td [(R)]TJ /F2 1 Tf 1.0964 0 Td [(par)]TJ -37.717 -1.1988 Td [(une)-331(espace)-337(v)24(ectoriel)-337(norm)24(\266)469(e)-337(don)24(t)-337(la)-331(top)-36(ologie)-331(n'est)-337(pas)-337(\265)500(a)-337(base)-337(d)30(\266)469(enom)24(brable.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 33.791 Td [(VI)-35(I)-35(I.3.)-236(Pro)-35(duit)-376(de)-381(mesures.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 38.942 Td [(Soien)24(t)-463(\()]TJ /F8 1 Tf 3.5482 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.009 56.323 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.066 38.942 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.19 56.323 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.59 38.942 Td [(;)-168(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.483 56.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.199 38.942 Td [(\))-463(et)-469(\()]TJ /F8 1 Tf 2.5361 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.819 56.323 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.157 38.942 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.328 56.323 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.91 38.942 Td [(;)-168(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.957 56.323 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.747 38.942 Td [(\))-469(des)-463(espaces)-469(mesur)24(\266)475(es.)-843(Sur)-469(l'ensem)30(ble)-469(pro)-24(duit)]TJ /F8 1 Tf 21.566 0 Td [(X)]TJ /F2 1 Tf 1.4036 0 Td [(=)]TJ /F8 1 Tf -38.386 -1.1988 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.9224 54.599 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.741 37.744 Td [(\243)]TJ /F8 1 Tf 1 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.379 54.599 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.494 37.744 Td [(on)-337(a)-331(d)24(\266)469(e\257ni)-337(une)-331(tribu)-337(pro)-30(duit)]TJ /F6 1 Tf 12.934 0 Td [(T)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.293 54.599 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.867 37.744 Td [(\243)-222(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.586 54.599 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.175 37.744 Td [(.)-451(Nous)-331(nous)-337(prop)-30(osons)-337(de)-337(d)30(\266)469(e\257nir)-337(une)]TJ -22.843 -1.1988 Td [(mesure)-337(sur)-337(l'espace)-337(mesurable)-337(\()]TJ /F8 1 Tf 13.867 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))-337(ainsi)-337(obten)24(u.)]TJ -15.964 -1.5663 Td [(Rapp)-30(elons)-373(qu'une)-367(mesure)-373(est)-373(dite)]TJ /F8 1 Tf 15.265 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nie)-373(lorsque)-367(l'ensem)24(ble)-367(sur)-373(lequel)-373(elle)-373(est)-373(d)30(\266)469(e\257nie)-367(est)]TJ -15.873 -1.2048 Td [(une)-331(r)24(\266)469(eunion)-337(d)30(\266)469(enom)24(brable)-337(de)-331(parties)-337(mesurables)-337(de)-337(mesure)-337(\257nie.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 25.711 Td [(VI)-35(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 30.822 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8434 0 Td [(Av)24(ec)-259(les)-259(notations)-259(pr)24(\266)475(ec)24(\266)469(eden)24(tes)-259(il)-259(existe)-259(sur)]TJ /F2 1 Tf 18.536 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F12 1 Tf 0.64458 0 Td [(une)-259(mesure)]TJ /F8 1 Tf -34.283 -1.1988 Td [(m)]TJ /F12 1 Tf 1.2108 0 Td [(qui)-331(est)-343(un)]TJ /F1 1 Tf 4.7048 0 Td [(pro)-30(duit)-385(tensoriel)]TJ /F12 1 Tf 8.7651 0 Td [(des)-337(mesures)]TJ /F8 1 Tf 5.5181 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.793 42.978 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 27.024 29.623 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.603 42.978 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 29.584 29.623 Td [(,)-331(c'est-)0(\265)500(a-dire)-337(telle)-337(que)-337(l'on)-337(ait)-337(:)]TJ -20.337 -1.1988 Td [(p)-30(our)-337(tout)-337(pa)30(v)48(\266)475(e)-343(mesurable)]TJ /F8 1 Tf 11.572 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.871 41.262 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.072 28.424 Td [(\243)]TJ /F8 1 Tf 1 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.095 41.262 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.337 28.424 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.578 41.262 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.861 28.424 Td [(\243)]TJ /F8 1 Tf 1 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.802 41.262 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.12 28.424 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.259 41.262 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.006 28.424 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.293 41.262 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.422 28.424 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.543 41.262 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.928 28.424 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.905 41.262 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.578 28.424 Td [(\))]TJ /F12 1 Tf 0.71687 0 Td [(.)]TJ -34.964 -1.5663 Td [(En)-367(outre)-373(si)-367(les)-367(mesures)]TJ /F8 1 Tf 10.602 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.052 39.021 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.458 26.858 Td [(et)]TJ /F8 1 Tf 1.1988 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.957 39.021 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.416 26.858 Td [(son)24(t)]TJ /F8 1 Tf 2.1747 0 Td [(\276)]TJ /F12 1 Tf 0.60843 0 Td [(-\257nies,)-379(il)-367(n'y)-367(a)-367(qu'une)-367(mesure)-373(sur)]TJ /F2 1 Tf 15.163 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(\))]TJ /F12 1 Tf 0.75301 0 Td [(qui)-367(ait)]TJ -36.265 -1.2048 Td [(cette)-259(propri)24(\266)475(et)24(\266)469(e.)-421(Elle)-253(est)]TJ /F8 1 Tf 10.428 0 Td [(\276)]TJ /F12 1 Tf 0.60843 0 Td [(-\257nie)-259(et)-259(se)-253(note)-259(alors)]TJ /F8 1 Tf 9.0663 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.655 37.306 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.657 25.653 Td [(\255)]TJ /F8 1 Tf 0.84337 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.603 37.306 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 28.886 25.653 Td [(,)-271(on)-259(dit)-259(que)-259(c'est)]TJ /F1 1 Tf 7.3253 0 Td [(le)]TJ /F12 1 Tf 1.0964 0 Td [(pro)-30(duit)-259(tensoriel)]TJ -31.976 -1.1988 Td [(des)-337(mesures)]TJ /F8 1 Tf 5.5181 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.784 35.59 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 12.343 24.454 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.595 35.59 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.904 24.454 Td [(.)]TJ /F1 1 Tf -9.5723 -1.8313 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Nous)-246(a)30(v)24(ons)-246(remarqu)24(\266)469(e)-246(que)-246(l'ensem)24(ble)-240(des)-253(pa)30(v)48(\266)475(es)-246(mesurables)-246(est)-253(un)-246(semi-)]TJ -8.8193 -1.1988 Td [(anneau)]TJ /F6 1 Tf 3.4699 0 Td [(S)]TJ /F2 1 Tf 1.0361 0 Td [(qui)-361(engendre,)-367(par)-355(d)24(\266)469(e\257nition,)-367(la)-355(tribu)]TJ /F6 1 Tf 16.482 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)-518(On)-361(p)-24(eut)-361(d)24(\266)469(e\257nir)-361(sur)]TJ /F6 1 Tf 9.6566 0 Td [(S)]TJ /F2 1 Tf 1.0361 0 Td [(une)-361(fonction)]TJ /F8 1 Tf 5.8012 0 Td [(m)]TJ /F2 1 Tf -38.283 -1.1988 Td [(par)]TJ /F8 1 Tf 1.7952 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.086 29.53 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 9.6627 20.226 Td [(\243)]TJ /F8 1 Tf 1.012 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.345 29.53 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.934 20.226 Td [(\))-313(=)]TJ /F8 1 Tf 1.7831 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.888 29.53 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.873 20.226 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.914 29.53 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.295 20.226 Td [(\))]TJ /F6 1 Tf 0.62048 0 Td [(\242)]TJ /F8 1 Tf 0.51205 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.198 29.53 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.819 20.226 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.56 29.53 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.47 20.226 Td [(\).)-500(Il)-349(s'agit)-355(de)-349(mon)18(trer)-349(que)]TJ /F8 1 Tf 11.904 0 Td [(m)]TJ /F2 1 Tf 1.2289 0 Td [(p)-30(eut)-355(se)-355(prolonger)-355(en)-349(une)]TJ -28.271 -1.2048 Td [(mesure)-337(d)24(\266)475(e\257nie)-337(sur)-337(la)-337(tribu)-331(engendr)24(\266)469(ee)-337(par)]TJ /F6 1 Tf 18.307 0 Td [(S)]TJ /F2 1 Tf 0.68674 0 Td [(.)]TJ -18.994 -1.5663 Td [(D'apr)30(\265)469(es)-451(le)-457(th)24(\266)469(eor)24(\265)475(eme)-451(de)-451(Carath)24(\266)469(eo)-30(dory)-451(\(cf.)]TJ /F1 1 Tf 19.524 0 Td [(I)-36(I)-30(I.4.6.)]TJ /F2 1 Tf 3.4819 0 Td [(\))-451(l'existence)-451(de)-451(ce)-457(prolongemen)30(t)-451(est)]TJ -23.006 -1.1988 Td [(assur)24(\266)475(ee)-487(si)-481(l'application)]TJ /F8 1 Tf 10.633 0 Td [(m)]TJ /F2 1 Tf 1.3554 0 Td [(est)-487(une)-481(pr)24(\266)475(e-mesure,)-524(c'est-)0(\265)500(a-dire)-481(si)-487(elle)-481(est)-487(d)24(\266)475(enom)24(brablemen)24(t)]TJ -11.988 -1.1988 Td [(additiv)30(e)-337(sur)]TJ /F6 1 Tf 5.512 0 Td [(S)]TJ /F2 1 Tf 0.68072 0 Td [(.)]TJ -6.1928 -1.5663 Td [(En)-421(outre,)-445(lorsque)-427(les)-421(mesures)]TJ /F8 1 Tf 13.596 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.336 19.9 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.506 13.491 Td [(et)]TJ /F8 1 Tf 1.253 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.397 19.9 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.572 13.491 Td [(son)24(t)]TJ /F8 1 Tf 2.2349 0 Td [(\276)]TJ /F2 1 Tf 0.60241 0 Td [(-\257nies,)-451(on)-421(p)-30(eut)-421(recouvrir)]TJ /F8 1 Tf 11.506 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.56 19.9 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.524 13.491 Td [(\(resp.)]TJ /F8 1 Tf 3.1626 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.388 19.9 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 13.491 Td [(\))]TJ -38.777 -1.1988 Td [(par)-475(un)-475(ensem)24(ble)-475(d)30(\266)469(enom)24(brable)]TJ /F6 1 Tf 13.964 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.759 18.176 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.849 12.292 Td [(\(resp.)]TJ /F6 1 Tf 3.3193 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.724 18.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.476 12.292 Td [(\))-475(de)-475(parties)-475(qui)-475(son)24(t)]TJ /F6 1 Tf 9.8855 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.75 18.176 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.44 12.292 Td [(-mesurables)-475(\(resp.)]TJ /F6 1 Tf -31.108 -1.2048 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7672 16.461 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6386 11.087 Td [(-mesurables\))-433(et)-433(de)-433(mesure)-433(\257nie)-433(p)-30(our)]TJ /F8 1 Tf 16.867 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.897 16.461 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.096 11.087 Td [(\(resp.)]TJ /F8 1 Tf 3.1867 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.733 16.461 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.675 11.087 Td [(\).)-734(Il)-433(est)-433(alors)-433(clair)-433(que)-433(les)-433(pa)24(v)54(\266)469(es)]TJ -24.343 -1.1988 Td [(mesurables)]TJ /F8 1 Tf 5.0422 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.922 14.745 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.464 9.8882 Td [(\243)]TJ /F8 1 Tf 0.83735 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.672 14.745 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.813 9.8882 Td [(obten)24(us)-259(en)-253(prenan)24(t)]TJ /F8 1 Tf 8.488 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.991 14.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.584 9.8882 Td [(dans)]TJ /F6 1 Tf 2.259 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.121 14.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.175 9.8882 Td [(et)]TJ /F8 1 Tf 1.0843 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.509 14.745 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.771 9.8882 Td [(dans)]TJ /F6 1 Tf 2.259 0 Td [(D)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.974 14.745 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.59 9.8882 Td [(formen)24(t)-253(un)-253(recouvremen)18(t)]TJ -28.259 -1.1988 Td [(d)30(\266)469(enom)24(brable)-361(de)]TJ /F8 1 Tf 7.3072 0 Td [(X)]TJ /F2 1 Tf 1.2711 0 Td [(par)-361(des)-337(\266)475(el)24(\266)469(emen)24(ts)-361(de)]TJ /F6 1 Tf 9.0422 0 Td [(S)]TJ /F2 1 Tf 1.0361 0 Td [(sur)-367(lesquels)]TJ /F8 1 Tf 5.3855 0 Td [(m)]TJ /F2 1 Tf 1.2349 0 Td [(prend)-367(une)-361(v)54(aleur)-361(\257nie.)-530(Dans)-367(ce)]TJ -25.277 -1.1988 Td [(cas,)-349(le)-349(th)24(\266)475(eor)18(\265)475(eme)-343(de)-349(Carath)24(\266)475(eo)-36(dory)-343(assure)-319(\266)475(egalemen)24(t)-343(l'unicit)24(\266)469(e)-349(de)-343(la)-343(mesure)]TJ /F8 1 Tf 32.524 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)-475(P)30(ar)-343(ailleurs)]TJ -33.404 -1.2048 Td [(la)-331(mesure)]TJ /F8 1 Tf 4.506 0 Td [(m)]TJ /F2 1 Tf 1.2169 0 Td [(est)-337(encore)]TJ /F8 1 Tf 4.6747 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nie.)]TJ /F5 1 Tf -11.006 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(112)]TJ ET endstream endobj 372 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 371 0 R /ProcSet 2 0 R >> /Contents 370 0 R >> endobj 371 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F8 22 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 373 0 obj << /Length 27853 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(Prouv)30(ons)-337(donc)-337(cette)-337(additivit)24(\266)469(e)-337(d)24(\266)475(enom)24(brable.)]TJ 0 -1.5602 Td [(Soit)]TJ /F8 1 Tf 2.0843 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(=)]TJ /F8 1 Tf 1.1084 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.819 87.4 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.88 60.659 Td [(\243)]TJ /F8 1 Tf 1.0241 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.103 87.4 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.53 60.659 Td [(un)-367(pa)30(v)48(\266)469(e)-367(mesurable.)-542(P)30(our)]TJ /F8 1 Tf 11.494 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.629 87.4 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.205 60.659 Td [(2)]TJ /F8 1 Tf 1 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.224 87.4 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.759 60.659 Td [(la)-361(section)]TJ /F8 1 Tf 4.512 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.69 86.59 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 68.952 121.43 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.163 60.659 Td [(est)-367(clairemen)24(t)-367(vide)-373(si)]TJ /F8 1 Tf -29.831 -1.1988 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4483 85.685 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5723 59.461 Td [(=)]TJ /F6 1 Tf -0.11446 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.672 85.685 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.7108 59.461 Td [(et)]TJ /F8 1 Tf 1.1084 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.121 85.685 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.355 59.461 Td [(si)]TJ /F8 1 Tf 0.94578 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.422 85.685 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.428 59.461 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.647 85.685 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.404 59.461 Td [(,)-289(dans)-271(tous)-277(les)-277(cas)-277(elle)-277(est)-277(mesurable.)-427(La)-283(fonction)]TJ /F8 1 Tf 21.488 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.612 85.685 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.024 59.461 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.5 85.685 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.693 59.461 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.862 84.875 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 84.566 119.02 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.602 59.461 Td [(\))-277(est)]TJ -37.271 -1.1988 Td [(donc)-331(le)-331(pro)-30(duit)-331(de)]TJ /F8 1 Tf 8.3494 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.836 83.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.066 58.262 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.198 83.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.717 58.262 Td [(\))-337(par)-331(la)-325(fonction)-337(indicatrice)-325(de)]TJ /F8 1 Tf 13.681 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.578 83.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.428 58.262 Td [(.)-445(En)-331(particulier)-331(il)-325(s'agit)-331(d'une)]TJ -26.096 -1.7169 Td [(fonction)]TJ /F6 1 Tf 3.8614 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.302 81.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.271 56.545 Td [(-mesurable)-337(sur)]TJ /F8 1 Tf 6.741 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.647 81.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.536 56.545 Td [(et)-337(l'on)-337(a)]TJ /F9 1 Tf 3.9458 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.862 81.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.873 56.545 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.224 80.711 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 52.928 113.2 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.789 56.545 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.129 81.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.024 56.545 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.905 81.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.265 56.545 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.034 81.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.38 56.545 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.397 81.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.03 56.545 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.974 81.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.301 56.545 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.009 81.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.717 56.545 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\).)]TJ -37.127 -2.012 Td [(Alors,)-319(si)-325(un)-319(pa)30(v)48(\266)475(e)-325(mesurable)]TJ /F8 1 Tf 12.337 0 Td [(A)]TJ /F2 1 Tf 1.0663 0 Td [(est)-319(la)-319(r)30(\266)469(eunion)-319(disjoin)24(te)-319(d'une)-319(famille)-319(d)24(\266)475(enom)24(brable)-319(de)-319(pa)24(v)54(\266)469(es)]TJ -13.404 -1.1988 Td [(mesurables)]TJ /F8 1 Tf 5.1928 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.138 76.107 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.741 53.334 Td [(,)-421(c)24(haque)-403(section)]TJ /F8 1 Tf 7.512 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.629 76.107 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 40.916 106.76 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.181 53.334 Td [(est)-409(la)-403(r)24(\266)475(eunion)-409(disjoin)30(te)-409(des)-403(sections)-409(\()]TJ /F8 1 Tf 16.566 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.095 76.107 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.964 53.334 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.31 76.107 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 79.614 106.76 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.53 53.334 Td [(\(qui)-403(son)18(t)]TJ -35.199 -1.4398 Td [(mesurables\).)-674(On)-409(a)-409(donc)]TJ /F8 1 Tf 11.247 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.983 74.857 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.97 51.894 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.345 74.047 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 39.12 103.88 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.886 51.894 Td [(\))-403(=)]TJ /F9 1 Tf 1.9699 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.017 72.616 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.47 51.894 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.845 74.857 Td [(00)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 24.861 52.707 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.862 74.047 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.922 51.894 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.078 74.047 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 55.53 103.88 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 28.084 52.707 Td [(\242)]TJ /F2 1 Tf 0.86747 -0.81326 Td [(et,)-427(s'agissan)24(t)-409(d'une)-415(somme)-409(d)24(\266)475(enom-)]TJ -23.62 -2.8494 Td [(brable)-343(de)-343(fonctions)-343(mesurables)-343(p)-30(ositiv)24(es,)]TJ /F9 1 Tf 18.193 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.353 70.78 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.916 49.045 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.716 69.969 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 55.012 98.198 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.831 49.045 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.621 70.78 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.066 49.045 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.397 70.78 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.307 49.045 Td [(\))-295(=)]TJ /F9 1 Tf 1.747 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.043 68.538 Td [(p)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 34.669 50.406 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.302 70.78 Td [(00)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 37.06 49.858 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.319 69.969 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.12 49.045 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.534 69.969 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 79.916 98.198 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 40.283 49.858 Td [(\242)]TJ /F8 1 Tf 0.62651 -0.81325 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.543 70.78 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.584 49.045 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.319 70.78 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.825 49.045 Td [(\),)]TJ -38.494 -2.5361 Td [(c'est-)0(\265)500(a-dire)]TJ /F8 1 Tf 5.1205 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.052 64.909 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.807 46.509 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.509 66.34 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.283 46.509 Td [(\).)]TJ ET 512.64 554.65 m 519.84 554.65 l 519.84 563.07 l 512.64 563.07 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 43.448 Td [(La)-457(d)30(\266)469(emonstration)-457(pr)24(\266)475(ec)24(\266)469(eden)24(te)-451(a)-451(utilis)18(\266)475(e)-457(le)-457(fait)-457(que)-457(la)-451(mesure)-463(des)-451(pa)24(v)54(\266)469(es)-457(mesurables)-457(p)-30(eut)]TJ 0 -1.1988 Td [(s')24(\266)475(ev)48(aluer)-337(comme)-343(l'in)24(t)24(\266)475(egrale)-343(de)-343(la)-337(mesure)-343(des)-343(sections.)-457(Nous)-343(allons)-337(v)24(oir)-343(s'il)-343(est)-343(p)-30(ossible)-337(de)]TJ 0 -1.1988 Td [(g)30(\266)469(en)30(\266)469(eraliser)-337(ce)-337(fait.)-445(Dans)-337(ce)-337(but)-337(commen)0(\273)439(cons)-337(par)-337(d)24(\266)475(egager)-343(un)-337(r)30(\266)469(esultat)-337(imp)-30(ortan)18(t.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 31.786 Td [(VI)-35(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 38.105 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8735 0 Td [(Av)24(ec)-337(les)-337(notations)-337(pr)24(\266)475(ec)24(\266)469(eden)30(tes,)-337(supp)-30(osons)-337(que)-337(la)-337(mesure)]TJ /F8 1 Tf 25.12 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.94 55.125 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 36.906 Td [(soit)]TJ /F8 1 Tf 1.8916 0 Td [(\276)]TJ /F12 1 Tf 0.60843 0 Td [(-\257nie.)-451(Alors)]TJ /F1 1 Tf -1.6867 -1.5602 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(si)]TJ /F8 1 Tf 1.006 0 Td [(B)]TJ /F12 1 Tf 1.1446 0 Td [(est)-337(une)-337(partie)]TJ /F6 1 Tf 6.3434 0 Td [(T)]TJ /F12 1 Tf 0.8012 0 Td [(-mesurable)-337(de)]TJ /F8 1 Tf 6.3976 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-331(la)-337(fonction)]TJ /F8 1 Tf 5.5843 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.595 51.176 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.024 35.346 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.483 51.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.693 35.346 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.853 50.366 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 70.578 70.788 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.614 35.346 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)]TJ /F6 1 Tf 1.5602 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.371 51.176 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.976 35.346 Td [(-mesurable)-337(;)]TJ /F1 1 Tf -32.831 -2.0723 Td [(2.)]TJ /F12 1 Tf 1.9036 0 Td [(si)]TJ /F8 1 Tf 1.1807 0 Td [(f)]TJ /F12 1 Tf 1.1084 0 Td [(est)]TJ /F6 1 Tf 1.7349 0 Td [(T)]TJ /F12 1 Tf 0.79518 0 Td [(-mesurable)-512(de)]TJ /F8 1 Tf 6.747 0 Td [(X)]TJ /F12 1 Tf 1.4157 0 Td [(dans)]TJ ET 0.504 w 281.38 407.66 m 290.02 407.66 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 23.542 33.274 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 34.724 47.4 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 24.928 33.274 Td [(,)-554(la)-506(fonction)]TJ /F8 1 Tf 6.1506 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.293 48.211 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 32.5 33.274 Td [(7!)]TJ /F9 1 Tf 1.5663 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.414 48.211 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.904 33.274 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.267 48.211 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.434 33.274 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.793 48.211 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.898 33.274 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.905 48.211 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.373 33.274 Td [(\))]TJ /F12 1 Tf 0.89157 0 Td [(est)]TJ /F6 1 Tf -37.934 -1.759 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.7672 45.685 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4096 31.515 Td [(-mesurable.)]TJ /F1 1 Tf -1.0783 -1.8193 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.5602 Td [(1.)]TJ /F2 1 Tf 1.7289 0 Td [(P)30(our)-421(toute)]TJ /F8 1 Tf 5.2169 0 Td [(B)]TJ /F2 1 Tf 1.2229 0 Td [(partie)]TJ /F6 1 Tf 2.9819 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable)-421(de)]TJ /F8 1 Tf 6.5723 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(,)-439(notons)]TJ /F8 1 Tf 4.0361 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.276 40.047 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.343 28.135 Td [(la)-421(fonction)]TJ /F8 1 Tf 5.1446 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.034 40.849 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.759 28.135 Td [(7!)]TJ /F8 1 Tf 1.4217 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.328 40.849 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.566 28.135 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.698 40.047 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 84.337 56.367 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.494 28.135 Td [(\).)-698(Il)]TJ -37.163 -1.2048 Td [(s'agit)-385(donc)-391(de)-385(prouv)24(er)-391(que)-391(l'ensem)24(ble)]TJ /F6 1 Tf 17.066 0 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(des)]TJ /F8 1 Tf 1.7831 0 Td [(B)]TJ /F2 1 Tf 1.1928 0 Td [(tels)-385(que)]TJ /F8 1 Tf 3.8072 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.276 38.331 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.012 26.93 Td [(est)]TJ /F6 1 Tf 1.6145 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.267 39.133 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.711 26.93 Td [(-mesurable)-385(con)24(tien)24(t)-391(la)]TJ -29.38 -1.1988 Td [(tribu)]TJ /F6 1 Tf 2.5 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(.)]TJ -2.3012 -1.5602 Td [(Supp)-30(osons)-331(tout)-337(d'ab)-30(ord)-337(que)-337(la)-337(mesure)]TJ /F8 1 Tf 17.169 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.888 35.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.223 24.171 Td [(soit)-337(\257nie)-337(\()]TJ /F8 1 Tf 4.4518 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.724 35.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.066 24.171 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.31 35.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.873 24.171 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(<)]TJ /F2 1 Tf 1.0542 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\).)]TJ ET 75.672 287.28 m 442.8 287.28 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 22.611 Td [(La)-265(tribu)]TJ /F6 1 Tf 3.8193 0 Td [(T)]TJ /F2 1 Tf 1.0663 0 Td [(est)-265(engendr)24(\266)469(ee)-265(par)-265(l'ensem)24(ble)]TJ /F6 1 Tf 12.548 0 Td [(S)]TJ /F2 1 Tf 0.93976 0 Td [(des)-271(pa)30(v)48(\266)475(es)-265(mesurables.)-427(Cet)-265(ensem)18(ble)]TJ /F6 1 Tf 15.789 0 Td [(S)]TJ /F2 1 Tf 0.94578 0 Td [(est)-271(stable)]TJ -35.108 -1.1988 Td [(par)-379(in)24(tersection)-385(\257nie.)-590(En\257n,)-397(si)]TJ /F8 1 Tf 14.018 0 Td [(A)]TJ /F2 1 Tf 1.1024 0 Td [(=)]TJ /F8 1 Tf 1.1386 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.966 31.237 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.873 21.412 Td [(\243)]TJ /F8 1 Tf 1.0301 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.284 31.237 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.548 21.412 Td [(est)-385(un)-385(pa)30(v)48(\266)475(e)-385(mesurable,)-397(nous)-379(a)24(v)24(ons)-385(observ)54(\266)475(e)]TJ -20.217 -1.1988 Td [(dans)-301(la)-301(preuv)30(e)-307(pr)30(\266)469(ec)24(\266)469(eden)30(te)-301(l')24(\266)469(egalit)24(\266)475(e)]TJ /F8 1 Tf 15.211 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.31 28.711 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.114 20.214 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.043 29.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.699 20.214 Td [(\))0(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.138 28.711 Td [(A)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 54.494 40.523 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.572 20.214 Td [(,)-307(par)-301(suite)]TJ /F8 1 Tf 4.6928 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.086 28.711 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.855 20.214 Td [(est)-307(mesurable)-301(et)]TJ /F8 1 Tf 7.3614 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(2)-283(U)]TJ /F2 1 Tf 1.9699 0 Td [(;)]TJ -38.886 -1.2048 Td [(autremen)30(t)-379(dit)]TJ /F6 1 Tf 6.4518 0 Td [(S)-421(\275)-349(U)]TJ /F2 1 Tf 2.8795 0 Td [(.)-572(Il)-373(su\261t)-379(donc)-379(de)-373(v)48(\266)475(eri\257er)-379(que)]TJ /F6 1 Tf 13.506 0 Td [(U)]TJ /F2 1 Tf 1.1024 0 Td [(satisfait)-373(aux)-385(h)30(yp)-36(oth)24(\265)469(eses)-379(du)-379(lemme)]TJ -23.94 -1.1988 Td [(des)-337(classes)-331(monotones)-343(\(cf.)]TJ /F1 1 Tf 11.783 0 Td [(I)-36(I)-30(I.3.4.)]TJ /F2 1 Tf 3.4819 0 Td [(\))-337(p)-30(our)-331(conclure)]TJ /F6 1 Tf 7.0361 0 Td [(U)-385(\276)-277(T)]TJ /F2 1 Tf 2.8614 0 Td [(.)]TJ -25.163 -1.5602 Td [(Comme)]TJ /F8 1 Tf 3.6687 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(est)-337(un)-337(pa)24(v)54(\266)469(e)-337(mesurable)-331(on)-337(a)]TJ /F8 1 Tf 12.265 0 Td [(X)]TJ /F6 1 Tf 1.1867 0 Td [(2)-283(U)]TJ /F2 1 Tf 1.6747 0 Td [(.)]TJ -20.036 -1.5602 Td [(P)30(our)]TJ /F8 1 Tf 2.4277 0 Td [(A)]TJ /F2 1 Tf 1.0783 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(B)]TJ /F2 1 Tf 1.1325 0 Td [(dans)]TJ /F6 1 Tf 2.3373 0 Td [(U)]TJ /F2 1 Tf 0.72289 0 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(A)]TJ /F2 1 Tf 1.0783 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(ainsi)-331(donc)-325(que)]TJ /F8 1 Tf 6.5723 0 Td [(A)]TJ /F6 1 Tf 0.96385 0 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(B)]TJ /F2 1 Tf 1.1325 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(A)]TJ /F6 1 Tf 0.95783 0 Td [(\\)]TJ /F8 1 Tf 0.87952 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.121 21.616 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.265 14.689 Td [(son)18(t)-331(mesurables,)-331(en)-331(outre)]TJ /F8 1 Tf 11.482 0 Td [(A)]TJ /F2 1 Tf -38.416 -1.1988 Td [(est)-277(la)-277(r)30(\266)469(eunion)-277(disjoin)24(te)-277(de)]TJ /F8 1 Tf 11.331 0 Td [(A)]TJ /F6 1 Tf 0.85542 0 Td [(\\)]TJ /F8 1 Tf 0.77108 0 Td [(B)]TJ /F2 1 Tf 1.0843 0 Td [(et)]TJ /F8 1 Tf 1.1084 0 Td [(A)]TJ /F6 1 Tf 0.85542 0 Td [(\\)]TJ /F8 1 Tf 0.77108 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.793 19.9 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.319 13.491 Td [(,)-289(par)-277(suite,)-289(p)-30(our)-277(tout)]TJ /F8 1 Tf 9.3313 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.543 19.9 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.777 13.491 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.991 19.9 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.91 13.491 Td [(,)-289(la)-277(section)]TJ /F8 1 Tf 4.8976 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.474 19.09 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 84.024 27.078 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.608 13.491 Td [(et)-277(la)]TJ -37.277 -1.1988 Td [(r)30(\266)469(eunion)-283(disjoin)24(te)-283(des)-283(sections)-283(\()]TJ /F8 1 Tf 13.265 0 Td [(A)]TJ /F6 1 Tf 0.86747 0 Td [(\\)]TJ /F8 1 Tf 0.78313 0 Td [(B)]TJ /F2 1 Tf 0.81325 0 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.69 17.375 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 43.807 24.68 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.5 12.292 Td [(et)-283(\()]TJ /F8 1 Tf 1.506 0 Td [(A)]TJ /F6 1 Tf 0.86747 0 Td [(\\)]TJ /F8 1 Tf 0.78313 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.879 18.176 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.873 12.292 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.017 17.375 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 55.434 24.68 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.036 12.292 Td [(.)-433(Comme)-283(ces)-283(sections)-283(son)24(t)-283(mesurables)]TJ -22.705 -1.2048 Td [(on)-445(a)]TJ /F8 1 Tf 2.4458 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.388 16.461 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1687 11.087 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.75 15.659 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 21.518 22.282 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.084 11.087 Td [(\))-463(=)]TJ /F8 1 Tf 2.0964 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.121 16.461 Td [(00)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 14.566 11.9 Td [(\241)]TJ /F2 1 Tf 0.46386 -0.81325 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F6 1 Tf 1.0482 0 Td [(\\)]TJ /F8 1 Tf 0.96386 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.655 15.659 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 38.157 22.282 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 19.404 11.9 Td [(\242)]TJ /F2 1 Tf 0.75301 -0.81325 Td [(+)]TJ /F8 1 Tf 1.0783 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.638 16.461 Td [(00)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 22.62 11.9 Td [(\241)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F6 1 Tf 1.0482 0 Td [(\\)]TJ /F8 1 Tf 0.96386 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.621 16.461 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.699 11.087 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.759 15.659 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 55.084 22.282 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 27.861 11.9 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(.)-783(Ainsi)-451(on)-445(a)]TJ /F8 1 Tf 6.2108 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.328 15.659 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.289 11.087 Td [(=)]TJ /F8 1 Tf 1.241 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.621 15.659 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [(\\)]TJ /F7 1 Tf 0.77586 0 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.295 11.087 Td [(+)]TJ /F8 1 Tf 1.0783 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.121 15.659 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [(\\)]TJ /F7 1 Tf 0.77586 0 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 87.59 22.282 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 11.087 Td [(.)]TJ -38.886 -1.1988 Td [(Cep)-24(endan)24(t)-487(les)-481(mesures)-481(des)-487(sections)-481(son)18(t)-481(ma)-60(jor)24(\266)469(ees)-475(par)]TJ /F8 1 Tf 25.078 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.776 14.745 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.795 9.8882 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.362 14.745 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.608 9.8882 Td [(\))-481(qui)-481(est)-487(supp)-30(os)30(\266)469(ee)-487(\257nie,)]TJ -28.277 -1.1988 Td [(les)-367(fonctions)-373(dans)-367(la)-367(relation)-373(pr)30(\266)469(ec)24(\266)475(eden)24(te)-373(son)24(t)-367(donc)-373(\265)500(a)-367(v)48(aleurs)-367(\257nies)-379(ce)-367(qui)-367(p)-30(ermet)-373(d')24(\266)469(ecrire)]TJ /F8 1 Tf 0 -1.1988 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.5431 10.504 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [(\\)]TJ /F7 1 Tf 0.76724 0 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 15.494 15.078 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4518 7.4906 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.526 10.504 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.861 7.4906 Td [(\241)]TJ /F8 1 Tf 0.83735 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.655 10.504 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [(\\)]TJ /F7 1 Tf 0.76724 0 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.169 7.4906 Td [(.)-415(Si)-253(alors)]TJ /F8 1 Tf 4.0964 0 Td [(A)]TJ /F6 1 Tf 0.80723 0 Td [(\\)]TJ /F8 1 Tf 0.72892 0 Td [(B)]TJ /F2 1 Tf 1.0602 0 Td [(appartien)24(t)-228(\266)475(egalemen)24(t)-253(\265)499(a)]TJ /F6 1 Tf 10.151 0 Td [(U)]TJ /F2 1 Tf 0.72892 0 Td [(,)]TJ /F8 1 Tf 0.54217 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.121 10.504 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.831 7.4906 Td [(et)]TJ /F8 1 Tf 1.0843 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.879 10.504 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [(\\)]TJ /F7 1 Tf 0.77586 0 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.639 7.4906 Td [(son)24(t)-253(mesurables)]TJ -32.307 -1.2048 Td [(ainsi)-331(donc)-337(que)]TJ /F8 1 Tf 6.5904 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.974 8.7797 Td [(A)]TJ /F10 1 Tf 0.86207 0 Td [(\\)]TJ /F7 1 Tf 0.77586 0 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 28.687 12.68 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.102 6.2858 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(A)]TJ /F6 1 Tf 0.9759 0 Td [(\\)]TJ /F8 1 Tf 0.88554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.103 9.59 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.681 6.2858 Td [(appartien)24(t)-337(encore)-337(\265)500(a)]TJ /F6 1 Tf 8.8133 0 Td [(U)]TJ /F2 1 Tf 0.72892 0 Td [(.)]TJ -5.2289 -2.3976 Td [(113)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 375 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 374 0 R /ProcSet 2 0 R >> /Contents 373 0 R >> endobj 374 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F18 107 0 R >> endobj 376 0 obj << /Length 28470 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(Consid)30(\266)469(erons)-271(une)-271(suite)-277(croissan)30(te)-271(d')24(\266)469(el)24(\266)469(emen)30(ts)]TJ /F8 1 Tf 19.114 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.069 88.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.018 62.22 Td [(de)]TJ /F6 1 Tf 1.2711 0 Td [(U)]TJ /F2 1 Tf 0.99397 0 Td [(et)-277(notons)]TJ /F8 1 Tf 4.2711 0 Td [(B)]TJ /F2 1 Tf 1.0783 0 Td [(la)-271(r)30(\266)469(eunion)-277(des)]TJ /F8 1 Tf 6.2651 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.181 88.823 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.199 62.22 Td [(.)-427(Alors,)]TJ -35.867 -1.1988 Td [(p)-30(our)-379(tout)]TJ /F8 1 Tf 4.6205 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.06 87.909 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.151 61.021 Td [(2)]TJ /F8 1 Tf 1.0181 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.716 87.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.355 61.021 Td [(,)-391(la)-379(section)]TJ /F8 1 Tf 5.2048 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.647 87.107 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 39.554 122.14 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.476 61.021 Td [(est)-379(la)-379(r)30(\266)469(eunion)-379(croissan)24(te)-379(des)-379(sections)-379(\()]TJ /F8 1 Tf 17 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.716 87.107 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.777 61.021 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.052 87.107 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 79.241 122.14 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.94 61.021 Td [(,)-391(par)-379(suite)]TJ /F8 1 Tf -34.608 -1.1988 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8879 86.193 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7169 59.822 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.259 85.392 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 16.639 119.74 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6446 59.822 Td [(\))-343(est)-349(la)-349(limite)-343(des)]TJ /F8 1 Tf 8.012 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.095 86.193 Td [(00)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 18.042 60.629 Td [(\241)]TJ /F2 1 Tf 0.46386 -0.80723 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.121 85.392 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.199 59.822 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.457 85.392 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 42.072 119.74 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 21.361 60.629 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.80723 Td [(.)-475(Autremen)24(t)-349(dit)]TJ /F8 1 Tf 7.3916 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.716 85.392 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.886 59.822 Td [(est)-349(la)-343(limite)-343(des)-349(fonctions)]TJ /F8 1 Tf 11.548 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 61.638 85.392 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 87.349 119.14 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 59.822 Td [(.)]TJ -38.886 -1.2048 Td [(Comme)-355(les)]TJ /F8 1 Tf 5.1566 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.095 83.668 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.145 58.617 Td [(appartiennen)24(t)-361(\265)500(a)]TJ /F6 1 Tf 7.3012 0 Td [(U)]TJ /F2 1 Tf 0.72289 0 Td [(,)-361(ces)-349(fonctions)]TJ /F8 1 Tf 6.5542 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.155 83.668 Td [(B)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 55.928 116.74 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.861 58.617 Td [(son)24(t)-355(mesurables)-361(ainsi)-349(donc)-355(que)-361(leur)]TJ -23.53 -1.1988 Td [(limite)]TJ /F8 1 Tf 2.8313 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.603 81.952 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.494 57.418 Td [(,)-331(par)-337(suite)]TJ /F8 1 Tf 4.7892 0 Td [(B)]TJ /F6 1 Tf 1.0843 0 Td [(2)-283(U)]TJ /F2 1 Tf 1.6687 0 Td [(.)]TJ -11.705 -1.5843 Td [(Nous)-427(a)24(v)24(ons)-433(donc)-427(mon)18(tr)24(\266)475(e)-433(que)-433(l'on)-433(p)-30(ouv)48(ait)-427(appliquer)-433(le)-433(lemme)-433(des)-433(classes)-433(monotones)-433(et)]TJ 0 -1.1988 Td [(conclure)]TJ /F6 1 Tf 3.9458 0 Td [(T)-536(\275)-277(U)]TJ /F2 1 Tf 3.1928 0 Td [(lorsque)]TJ /F8 1 Tf 3.4277 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.009 78.78 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.62 54.635 Td [(est)-337(\257nie.)]TJ -11.289 -1.8614 Td [(Rev)24(enons)-337(au)-337(cas)-337(g)30(\266)469(en)30(\266)469(eral)-331(o)-30(\265)530(u)]TJ /F8 1 Tf 12.367 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.017 76.116 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.422 52.774 Td [(est)]TJ /F8 1 Tf 1.5602 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nie.)]TJ ET 0.504 w 75.672 629.06 m 299.3 629.06 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 51.189 Td [(P)30(ar)-475(d)30(\266)469(e\257nition)]TJ /F8 1 Tf 6.6024 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.371 73.849 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.819 51.189 Td [(est)-481(la)-469(r)30(\266)469(eunion)-475(d'une)-475(suite)-475(de)-469(parties)]TJ /F8 1 Tf 17 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.397 73.849 Td [(00)]TJ /F7 1 Tf -0.11207 -0.94827 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.711 51.189 Td [(de)-469(mesure)-475(\257nie.)-861(Quitte)-475(\265)500(a)]TJ -27.38 -1.1988 Td [(remplacer)]TJ /F8 1 Tf 4.5964 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.509 72.133 Td [(00)]TJ /F7 1 Tf -0.11207 -0.94827 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.663 49.991 Td [(par)-319(la)-313(r)24(\266)469(eunion)-319(des)]TJ /F8 1 Tf 8.1687 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.672 72.133 Td [(00)]TJ /F7 1 Tf -0.11207 -0.94827 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.56 49.991 Td [(p)-30(our)]TJ /F8 1 Tf 2.3494 0 Td [(k)]TJ /F11 1 Tf 0.83133 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(n)]TJ /F2 1 Tf 0.91566 0 Td [(on)-319(p)-30(eut)-319(supp)-30(oser)-313(que)-319(la)-319(suite)-319(des)]TJ /F8 1 Tf 14.819 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.733 72.133 Td [(00)]TJ /F7 1 Tf -0.11207 -0.94827 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.265 49.991 Td [(est)]TJ -37.934 -1.1988 Td [(croissan)30(te.)]TJ 0 -1.5843 Td [(P)30(our)-319(c)24(haque)]TJ /F8 1 Tf 5.7349 0 Td [(n)]TJ /F2 1 Tf 0.91566 0 Td [(d)24(\266)469(e\257nissons)]TJ /F8 1 Tf 4.9518 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.483 68.15 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.675 47.207 Td [(de)]TJ /F6 1 Tf 1.3133 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.75 68.15 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.614 47.207 Td [(dans)]TJ ET 286.13 574.2 m 294.77 574.2 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 23.94 47.207 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.293 67.34 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.645 47.207 Td [(par)]TJ /F8 1 Tf 1.7651 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.483 68.15 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.831 47.207 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.672 68.15 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.127 47.207 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F6 1 Tf 0.98193 0 Td [(\\)]TJ /F8 1 Tf 0.85542 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.897 68.15 Td [(00)]TJ /F7 1 Tf -0.11207 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.771 47.207 Td [(\).)-445(Il)-319(est)-319(imm)18(\266)475(ediat)]TJ -31.44 -1.1988 Td [(de)-246(v)24(oir)-253(que)]TJ /F8 1 Tf 4.9458 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.966 66.435 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.952 46.009 Td [(est)-253(une)-253(mesure)-253(sur)-253(\()]TJ /F8 1 Tf 8.5783 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.672 66.435 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.946 46.009 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.19 66.435 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.705 46.009 Td [(\))-253(qui)-246(est)-253(\257nie)-253(puisque)]TJ /F8 1 Tf 9.3675 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.586 66.435 Td [(00)]TJ /F7 1 Tf 0 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.5 46.009 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.948 66.435 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.91 46.009 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.534 66.435 Td [(00)]TJ /F7 1 Tf -0.11207 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.717 46.009 Td [(\))]TJ /F8 1 Tf 0.66265 0 Td [(<)]TJ /F2 1 Tf 1.0602 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)]TJ -38.886 -1.1988 Td [(En)-361(appliquan)30(t)-367(le)-367(r)30(\266)469(esultat)-367(pr)30(\266)469(ec)24(\266)469(eden)30(t)-367(on)-361(conclut)-367(que,)-373(p)-30(our)]TJ /F8 1 Tf 25.488 0 Td [(B)]TJ /F2 1 Tf 1.1687 0 Td [(partie)]TJ /F6 1 Tf 2.9217 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(-mesurable)-361(de)]TJ /F8 1 Tf 6.4518 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-367(la)]TJ -37.741 -1.2048 Td [(fonction)]TJ /F8 1 Tf 3.8855 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.103 62.185 Td [(n;B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.596 43.605 Td [(:)]TJ /F8 1 Tf 0.59639 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.267 62.995 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.367 43.605 Td [(7!)]TJ /F8 1 Tf 1.3193 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.267 62.995 Td [(00)]TJ /F7 1 Tf 0 -0.94827 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.108 43.605 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.69 62.185 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 35.422 87.319 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.253 43.605 Td [(\))-319(=)]TJ /F8 1 Tf 1.8012 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.957 62.995 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.446 43.605 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.328 62.185 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 46.096 87.319 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.608 43.605 Td [(\\)]TJ /F8 1 Tf 0.90361 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.379 62.995 Td [(00)]TJ /F7 1 Tf -0.12069 -0.94827 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.928 43.605 Td [(\))-361(est)]TJ /F6 1 Tf 2.3373 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.586 62.995 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.343 43.605 Td [(-mesurable.)-524(Cep)-30(endan)24(t,)]TJ /F8 1 Tf 10.898 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.672 62.185 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 82.904 87.319 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.133 43.605 Td [(est)-361(la)]TJ -36.801 -1.1988 Td [(r)30(\266)469(eunion)-361(croissan)24(te)-361(des)]TJ /F8 1 Tf 10.018 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.052 60.469 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 33.12 84.909 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.12 42.406 Td [(\\)]TJ /F8 1 Tf 0.90964 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.095 61.28 Td [(00)]TJ /F7 1 Tf -0.11207 -0.94828 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.807 42.406 Td [(et)-361(la)-361(propri)24(\266)475(et)24(\266)469(e)-361(de)-361(Borel)-361(assure)-361(que)-361(la)-361(fonction)]TJ /F8 1 Tf 20.633 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.793 60.469 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.133 42.406 Td [(est)-361(la)]TJ -36.801 -1.1988 Td [(limite)-331(simple)-337(des)-337(fonctions)]TJ /F8 1 Tf 11.94 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.629 58.754 Td [(n;B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.331 41.207 Td [(,)-331(donc)-337(qu'elle)-337(est)-313(\266)469(egalemen)30(t)]TJ /F6 1 Tf 12.398 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.552 59.556 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.807 41.207 Td [(-mesurable.)]TJ /F1 1 Tf -26.663 -2.0361 Td [(2.)]TJ /F2 1 Tf 1.5783 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4398 0 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(application)-349(mesurable)-343(de)]TJ /F8 1 Tf 11.265 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(dans)]TJ ET 310.32 478.08 m 318.96 478.08 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 25.964 39.171 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.19 55.84 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.355 39.171 Td [(,)-343(notons)]TJ /F9 1 Tf 4.0482 0.26506 Td [(e)]TJ /F8 1 Tf -0.19277 -0.26506 Td [(f)]TJ /F2 1 Tf 0.93976 0 Td [(l'application)-349(de)]TJ /F8 1 Tf 7.0783 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.44 56.65 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.759 39.171 Td [(dans)]TJ ET 515.23 478.08 m 523.87 478.08 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 43.108 39.171 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 62.724 55.84 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 37.479 Td [(d)30(\266)469(e\257nie)-337(par)]TJ /F9 1 Tf 5.1386 0.25904 Td [(e)]TJ /F8 1 Tf -0.19277 -0.25904 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.94 54.219 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.12 37.479 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.466 54.219 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.681 37.479 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.328 54.219 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.211 37.479 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.853 54.219 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.675 37.479 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.966 54.219 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.145 37.479 Td [(\))-337(.)]TJ -16.813 -2.3434 Td [(Lorsque)]TJ /F8 1 Tf 3.8795 0 Td [(f)]TJ /F2 1 Tf 1.0542 0 Td [(=)-451(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.25 50.073 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.187 35.135 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.3012 0 Td [(B)]TJ /F2 1 Tf 1.247 0 Td [(partie)-445(mesurable)-439(de)]TJ /F8 1 Tf 9.2771 0 Td [(X)]TJ /F2 1 Tf 1.3494 0 Td [(on)-439(a)-439(clairemen)24(t)]TJ /F9 1 Tf 7.5301 0.24699 Td [(f)]TJ /F2 1 Tf -0.12048 -0.24699 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.56 50.073 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.476 35.135 Td [(=)]TJ /F8 1 Tf 1.2289 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.879 50.073 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.476 35.135 Td [(et)-439(donc)]TJ /F9 1 Tf 3.8916 0.24699 Td [(f)]TJ /F2 1 Tf -0.12048 -0.24699 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.69 50.073 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 33.936 Td [(mesurable)-331(d'apr)18(\265)475(es)-337(le)-337(r)24(\266)475(esultat)-337(pr)24(\266)469(ec)24(\266)475(eden)24(t.)]TJ 0 -1.7229 Td [(P)30(our)]TJ /F8 1 Tf 2.4639 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(mesurable)-373(p)-30(ositiv)30(e)-373(quelconque)-367(on)-367(p)-30(eut)-343(\266)469(ecrire)]TJ /F8 1 Tf 20.229 0 Td [(f)]TJ /F2 1 Tf 0.93373 0 Td [(=)]TJ /F9 1 Tf 1.1084 0.94579 Td [(X)]TJ /F8 1 Tf 1.6145 -0.94579 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.629 45.883 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.62 32.214 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.914 45.883 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 69.566 63.921 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.512 32.214 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.2289 0 Td [(I)]TJ /F2 1 Tf 0.88554 0 Td [(d)24(\266)469(enom)30(brable,)]TJ /F8 1 Tf -33.295 -1.4036 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6983 43.875 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7952 30.81 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(-mesurable)-385(et)]TJ /F8 1 Tf 6.3253 0 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.828 43.875 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.241 30.81 Td [(2)]TJ /F3 1 Tf 1.0241 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 24.31 43.875 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.03 30.81 Td [(\(cf.)]TJ /F1 1 Tf 2 0 Td [(I)-30(I)-36(I.5.9.)]TJ /F2 1 Tf 3.4819 0 Td [(\).)-584(Comme)-379(les)]TJ /F8 1 Tf 6.4578 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.957 43.875 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.428 30.81 Td [(son)18(t)-379(mesurables,)-397(p)-30(our)-379(c)24(haque)]TJ /F8 1 Tf -26.096 -1.1988 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4483 42.961 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.6687 29.611 Td [(2)]TJ /F8 1 Tf 1.1566 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.491 42.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.012 29.611 Td [(,)]TJ /F8 1 Tf 0.76506 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.81 42.961 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.349 29.611 Td [(7!)]TJ /F2 1 Tf 1.488 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.172 42.159 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 28 58.716 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.367 29.611 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.931 42.961 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.608 29.611 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.793 42.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.139 29.611 Td [(\))-457(est)]TJ /F6 1 Tf 2.5301 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.293 42.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.982 29.611 Td [(-mesurable)-463(et)-457(l'additivit)24(\266)475(e)-463(d)24(\266)469(enom)24(brable)-457(de)-463(l'in)24(t)24(\266)475(egrale)]TJ -15.651 -1.7108 Td [(\(sur)-355(les)-355(fonctions)-355(mesurables\))-361(assure)]TJ /F9 1 Tf 16.41 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.767 40.512 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.578 27.9 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.629 40.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.108 27.9 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.155 40.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.572 27.9 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.267 40.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.048 27.9 Td [(\))-313(=)]TJ /F9 1 Tf 1.7892 0.94579 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.716 38.211 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.452 27.9 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.784 39.711 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 34.59 29.262 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.733 39.711 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 73.506 55.295 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.12 27.9 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.491 40.512 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.361 27.9 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.353 40.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.892 27.9 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.879 40.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.355 27.9 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.991 40.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.825 27.9 Td [(\).)]TJ -38.494 -2.5 Td [(Ainsi)]TJ /F9 1 Tf 2.7771 0.26506 Td [(e)]TJ /F8 1 Tf -0.18675 -0.26506 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(\256)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.319 36.133 Td [(i)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 12.476 25.641 Td [(g)]TJ /F2 1 Tf -0.042169 -0.24096 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.603 36.133 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 27.205 50.295 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.301 25.4 Td [(donc)-337(est)-337(mesurable.)]TJ ET 512.64 302.36 m 519.84 302.36 l 519.84 310.78 l 512.64 310.78 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 23.003 Td [(On)-367(a)-337(\266)475(evidemmen)24(t)-367(un)-367(analogue)-367(du)-367(r)24(\266)475(esultat)-373(pr)30(\266)469(ec)24(\266)469(eden)30(t)-367(en)-337(\266)469(ec)24(hangean)24(t)-367(les)-367(r)0(^)500(oles)-367(de)]TJ /F8 1 Tf 35.416 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.603 33.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.295 23.003 Td [(et)-367(de)]TJ /F8 1 Tf -36.964 -1.1988 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.9224 31.788 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.747 21.804 Td [(.)-445(Nous)-337(laissons)-331(au)-337(lecteur)-337(le)-337(soin)-343(de)-331(l'expliciter.)]TJ -1.4157 -2.1446 Td [(Nous)-331(sommes)-343(alors)-337(en)-331(mesure)-337(de)-337(prouv)30(er)-343(le)-337(r)30(\266)469(esultat)-337(fondamen)18(tal)-337(de)-331(ce)-337(c)24(hapitre.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 13.907 Td [(VI)-35(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 16.671 Td [(3.)]TJ /F15 1 Tf 1.747 0 Td [(Th)]TJ 1.4036 0.084337 Td [(\266)]TJ -0.0060244 -0.084337 Td [(eor)]TJ 1.8012 0.084337 Td [(\265)]TJ 0 -0.084337 Td [(eme.)]TJ /F12 1 Tf 2.6747 0 Td [(\()]TJ /F1 1 Tf 0.39157 0 Td [(de)-385(F)96(ubini,)-385(Leb)-36(esgue,)-385(T)90(onelli)]TJ /F12 1 Tf 14.157 0 Td [(\))]TJ -26.199 -1.5783 Td [(Soien)24(t)]TJ /F2 1 Tf 2.9699 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.733 22.185 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.8795 15.093 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.914 22.185 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.404 15.093 Td [(;)-162(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.207 22.185 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.006 15.093 Td [(\))]TJ /F12 1 Tf 0.66265 0 Td [(et)]TJ /F2 1 Tf 1.1084 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23 22.185 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.584 15.093 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.517 22.185 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.337 15.093 Td [(;)-168(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.138 22.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.175 15.093 Td [(\))]TJ /F12 1 Tf 0.66265 0 Td [(des)-277(espaces)-283(mesur)30(\266)469(es.)-427(On)-277(supp)-30(ose)-277(les)-277(mesures)]TJ /F8 1 Tf 19.729 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.302 22.185 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 42 15.093 Td [(et)]TJ /F8 1 Tf 1.1024 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.94 22.185 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 13.894 Td [(\276)]TJ /F12 1 Tf 0.60843 0 Td [(-\257nies)]TJ ET 63.72 164.37 m 101.66 164.37 l S BT /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5 13.894 Td [(.)-524(On)-361(m)24(unit)]TJ /F8 1 Tf 5.4398 0 Td [(X)]TJ /F12 1 Tf 1.2711 0 Td [(de)-361(la)-361(tribu)-361(pro)-30(duit)]TJ /F6 1 Tf 8.6446 0 Td [(T)]TJ /F2 1 Tf 1.1205 0 Td [(=)]TJ /F6 1 Tf 1.1024 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.457 20.469 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.398 13.894 Td [(\243)-240(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.802 20.469 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 30.084 13.894 Td [(et)-361(de)-361(la)-361(mesure)-367(pro)-30(duit)-361(tensoriel)]TJ /F8 1 Tf -24.753 -1.2048 Td [(m)]TJ /F2 1 Tf 1.1566 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.052 18.754 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.9217 12.689 Td [(\255)]TJ /F8 1 Tf 1 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.457 18.754 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 11.313 12.689 Td [(.)-445(Alors)]TJ /F1 1 Tf -5.1687 -1.5783 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(fonction)-337(mesurable)-337(p)-24(ositiv)24(e)-337(sur)]TJ /F8 1 Tf 13.97 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)]TJ -20.62 -2.0361 Td [(les)-337(fonctions)-337(partielles)]TJ /F8 1 Tf 10 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.172 13.573 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.169 9.075 Td [(;)-168(:)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(:)-168(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.819 13.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.747 9.075 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(ainsi)-337(que)-337(les)-337(fonctions)]TJ /F8 1 Tf 9.9157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.595 13.573 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.518 9.075 Td [(7!)]TJ /F9 1 Tf 1.2831 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.897 13.573 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.639 9.075 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.75 13.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.169 9.075 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.276 13.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.633 9.075 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.388 13.573 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 9.075 Td [(\))]TJ /F12 1 Tf -38.777 -2.3253 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.112 10.245 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8554 6.7497 Td [(7!)]TJ /F9 1 Tf 1.2831 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.733 10.245 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.976 6.7497 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.595 10.245 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.506 6.7497 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.121 10.245 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.741 6.7497 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.897 10.245 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.982 6.7497 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(son)24(t)-337(mesurables)-337(p)-30(ositiv)30(es.)]TJ /F5 1 Tf -12.373 -2.8614 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(114)]TJ ET endstream endobj 378 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 377 0 R /ProcSet 2 0 R >> /Contents 376 0 R >> endobj 377 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F18 107 0 R >> endobj 379 0 obj << /Length 26821 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ 0.81325 -2.4036 Td [(2.)]TJ /F12 1 Tf 1.5602 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(in)30(t)24(\266)469(egrable)-337(sur)]TJ /F8 1 Tf 6.3193 0 Td [(X)]TJ /F12 1 Tf 0.90362 0 Td [(,)]TJ -12.964 -1.5301 Td [(la)-331(fonction)-343(partielle)]TJ /F8 1 Tf 8.8675 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.552 87.435 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.036 60.689 Td [(;)-168(:)]TJ /F2 1 Tf 0.72289 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(p)-30(our)-337(presque)-337(tout)]TJ /F8 1 Tf 14.386 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.422 87.435 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.723 60.689 Td [(,)]TJ -27.392 -1.5301 Td [(la)-331(fonction)-343(partielle)]TJ /F8 1 Tf 8.8675 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(:)-162(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.819 85.245 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.157 59.159 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(p)-30(our)-337(presque)-337(tout)]TJ /F8 1 Tf 14.386 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.991 85.245 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 33.349 59.159 Td [(,)-337(et)]TJ -28.018 -2.0482 Td [(les)-337(fonctions)]TJ /F8 1 Tf 5.7048 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.612 82.323 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 12.169 57.111 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.905 82.323 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.289 57.111 Td [(;)-162(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.759 82.323 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.813 57.111 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.293 82.323 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.283 57.111 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.397 82.323 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.753 57.111 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.655 82.323 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 25.006 57.111 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.276 82.323 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.12 57.111 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.129 82.323 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.651 57.111 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.655 82.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.886 57.111 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.431 82.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.127 57.111 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(son)24(t)-337(in)24(t)24(\266)469(egrables.)]TJ -29.518 -2.2169 Td [(En)-331(outre,)-343(dans)-331(les)-337(cas)]TJ /F1 1 Tf 9.9277 0 Td [(1.)]TJ /F12 1 Tf 1.3373 0 Td [(et)]TJ /F1 1 Tf 1.1687 0 Td [(2.)]TJ /F12 1 Tf 0.89157 0 Td [(,)-337(on)-331(a)-337(l')]TJ /F1 1 Tf 3.3675 0 Td [(\266)548(egalit)24(\266)548(e)-391(de)-385(F)96(ubini)]TJ /F12 1 Tf 8.753 0 Td [(:)]TJ /F9 1 Tf -26.892 -0.61446 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.2241 76.314 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.7289 52.918 Td [(;)-162(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.078 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.253 52.918 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.983 76.314 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.452 52.918 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.845 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.976 52.918 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ 1 0.048195 Td [(\265)]TJ 0.90361 -0.048195 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.422 76.314 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.446 52.918 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.284 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.97 52.918 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.81 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.44 52.918 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.914 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.91 52.918 Td [(\))]TJ /F9 1 Tf 0.39157 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90362 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.5 76.314 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.886 52.918 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.276 76.314 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.127 52.918 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ 1 0.048195 Td [(\265)]TJ 0.90361 -0.048195 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.534 76.314 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.59 52.918 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.388 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.12 52.918 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.914 76.314 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.355 52.918 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.69 76.314 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.596 52.918 Td [(\))]TJ /F9 1 Tf 0.39157 1.4096 Td [(\266)]TJ /F8 1 Tf 0.89759 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.948 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.801 52.918 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 64.052 76.314 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 45.271 52.918 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ /F1 1 Tf -40.663 -2.3072 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.5361 Td [(1.)]TJ /F2 1 Tf 1.6084 0 Td [(La)-361(mesurabilit)24(\266)469(e)-361(des)-361(applications)-355(partielles)-361(d)24(\266)475(ecoule)-367(de)-355(la)-361(prop)-30(osition)]TJ /F1 1 Tf 30.199 0 Td [(VI)-36(I)-36(I.2.3.)]TJ /F2 1 Tf 4.7108 0 Td [(\(elle)]TJ -37.331 -1.1988 Td [(ne)-331(n)24(\266)475(ecessite)-343(pas)-337(l'h)24(yp)-30(oth)18(\265)475(ese)-337(de)]TJ /F8 1 Tf 13.831 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nitude\),)-337(leur)-337(p)-30(ositivit)24(\266)475(e)-343(est)-307(\266)469(eviden)24(te.)]TJ -14.44 -2.0422 Td [(La)-240(mesurabilit)24(\266)475(e)-246(des)-240(fonctions)]TJ /F8 1 Tf 12.741 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.681 66.176 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 19.199 45.834 Td [(7!)]TJ /F9 1 Tf 1.2831 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.966 66.176 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.319 45.834 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.828 66.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.849 45.834 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.353 66.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.313 45.834 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.466 66.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.783 45.834 Td [(\))-246(et)]TJ /F8 1 Tf 1.7048 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.44 66.176 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.843 45.834 Td [(7!)]TJ /F9 1 Tf 1.2831 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.06 66.176 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.964 45.834 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.922 66.176 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.494 45.834 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.448 66.176 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.729 45.834 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.224 66.176 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.97 45.834 Td [(\))-240(r)30(\266)469(esulte)]TJ -35.639 -1.7651 Td [(de)-301(la)-295(prop)-36(osition)]TJ /F1 1 Tf 7.6024 0 Td [(VI)-36(I)-36(I.3.2.)]TJ /F2 1 Tf 4.6506 0 Td [(gr)0(^)500(ace)-307(\265)500(a)-301(la)]TJ /F8 1 Tf 4.4639 0 Td [(\276)]TJ /F2 1 Tf 0.60241 0 Td [(-\257nitude)-307(de)]TJ /F8 1 Tf 5.2711 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.216 63.659 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.38 44.069 Td [(\(p)-30(our)-307(la)-295(deuxi)24(\265)469(eme\))-301(et)-301(de)]TJ /F8 1 Tf 11.006 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.052 63.659 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.072 44.069 Td [(\(p)-36(our)]TJ -36.741 -1.1988 Td [(la)-331(premi)24(\265)469(ere\).)-445(Leur)-337(p)-30(ositivit)24(\266)469(e)-337(est)-313(\266)469(eviden)24(te.)]TJ 0 -1.5361 Td [(On)-337(p)-24(eut)-337(alors)-343(d)30(\266)469(e\257nir)-337(une)-331(fonctionnelle)]TJ /F8 1 Tf 17.241 0 Td [(\301)]TJ /F2 1 Tf 0.92771 0 Td [(sur)]TJ /F6 1 Tf 1.6747 0 Td [(M)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.75 58.943 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.042 41.334 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(\))-337(par)]TJ /F8 1 Tf -12.542 -1.7952 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ 1 0.048191 Td [(\265)]TJ 0.74096 -0.048191 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.483 57.168 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.681 39.539 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.345 57.168 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.211 39.539 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.871 57.168 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.675 39.539 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.974 57.168 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.145 39.539 Td [(\))]TJ /F9 1 Tf 0.39157 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90362 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.56 57.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.12 39.539 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.336 57.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.361 39.539 Td [(\).)]TJ -28.03 -2.1867 Td [(Autremen)24(t)-391(dit)-391(on)-367(\266)475(ev)48(alue)]TJ /F8 1 Tf 11.253 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-391(en)-391(prenan)24(t)-391(l'in)24(t)24(\266)469(egrale)-391(\(p)-30(our)]TJ /F8 1 Tf 13.428 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.474 54.047 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.988 37.352 Td [(\))-391(de)-391(l'application)-391(partielle)]TJ /F8 1 Tf -27.657 -1.1988 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.8534 52.323 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1687 36.153 Td [(;)-168(:)]TJ /F2 1 Tf 0.72289 0 Td [(\))-397(\(qui)-397(d)30(\266)469(ep)-30(end)-391(de)]TJ /F8 1 Tf 7.8554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.353 52.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.596 36.153 Td [(\))-397(puis)-391(on)-397(prend)-397(l'in)24(t)24(\266)469(egrale)-397(\(p)-30(our)]TJ /F8 1 Tf 14.825 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.224 52.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.578 36.153 Td [(\))-397(de)-391(la)-397(fonction)-397(de)]TJ /F8 1 Tf 8.6626 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.836 52.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.488 36.153 Td [(ainsi)]TJ -37.157 -1.2048 Td [(obten)24(ue)-409(\(don)24(t)-409(on)-409(sait)-409(qu'elle)-415(est)-415(mesurable\).)-668(On)-409(v)54(a)-409(prouv)24(er)-409(que)]TJ /F8 1 Tf 29.428 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-409(est)-415(l'in)24(t)24(\266)475(egrale)-415(de)]TJ /F8 1 Tf -31.012 -1.1988 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)-409(ce)-397(qui)-367(\266)475(etablira)-397(la)-391(premi)24(\265)469(ere)-397(partie)-397(de)-397(l')30(\266)469(egalit)24(\266)469(e)-397(de)-391(F)78(ubini)-391(p)-30(our)-397(les)-397(fonctions)-391(mesurables)]TJ -0.59639 -1.1988 Td [(p)-30(ositiv)30(es.)]TJ 0 -1.5361 Td [(Les)-481(propri)24(\266)469(et)30(\266)469(es)-481(des)-487(in)24(t)24(\266)475(egrales)-487(mon)24(tren)24(t)-481(imm)18(\266)475(ediatemen)24(t)-487(que)-481(la)-481(fonctionnelle)]TJ /F8 1 Tf 34.283 0 Td [(\301)]TJ /F2 1 Tf 1.0783 0 Td [(est)-487(p)-30(osi-)]TJ -35.361 -1.1988 Td [(tiv)24(emen)30(t)-548(homog)24(\265)469(ene,)-596(mon)24(trons)-548(qu'elle)-548(est)-548(d)24(\266)475(enom)24(brablemen)24(t)-542(additiv)24(e)-548(sur)-548(les)-548(fonctions)]TJ 0 -1.1988 Td [(mesurables.)]TJ 0 -1.5301 Td [(Soit)-385(\()]TJ /F8 1 Tf 2.5 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.905 38.547 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6506 27.087 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.94 38.547 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.711 27.087 Td [(une)-391(famille)-391(d)24(\266)475(enom)24(brable)-391(de)-391(fonctions)-391(mesurables)-391(p)-30(ositiv)30(es)-391(sur)]TJ /F8 1 Tf 27.994 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)-608(Les)-391(fonc-)]TJ -34.283 -1.2048 Td [(tions)-355(partielles)]TJ /F8 1 Tf 6.7771 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.026 36.823 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.928 25.882 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.879 37.633 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.175 25.882 Td [(;)-162(:)]TJ /F2 1 Tf 0.71687 0 Td [(\))-331(\266)469(etan)24(t)-355(mesurables)-355(p)-30(ositiv)24(es,)-355(l'additivit)24(\266)475(e)-361(d)30(\266)469(enom)24(brable)-355(de)-355(l'in)24(t)24(\266)475(egrale)]TJ -9.5602 -1.7108 Td [(sur)-397(ces)-397(fonctions)-397(assure)]TJ /F9 1 Tf 10.807 1.3614 Td [(Z)]TJ 1 -0.40964 Td 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8.352 0 0 Tm 46.664 35.185 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.886 24.171 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.517 35.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.416 24.171 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.043 35.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.88 24.171 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.155 35.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.355 24.171 Td [(\).)-632(Comme)-397(les)]TJ -33.024 -1.7651 Td [(in)24(t)24(\266)475(egrales)-313(pr)24(\266)475(ec)24(\266)469(eden)24(tes)-313(son)24(t)-319(des)-313(fonctions)-313(mesurables)-313(p)-30(ositiv)24(es)-313(de)]TJ /F8 1 Tf 28.361 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.034 32.659 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.855 22.406 Td [(on)-313(en)-313(d)24(\266)475(eduit)-313(de)-313(m)24(^)475(eme)]TJ /F9 1 Tf -27.289 -0.29518 Td [(Z)]TJ 1 -0.24699 Td [(\263)]TJ 0.76506 0.24699 Td [(Z)]TJ 1 -0.40964 Td [(X)]TJ /F8 1 Tf 1.6145 -0.95181 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.793 29.478 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.765 20.75 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.647 30.288 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.012 20.75 Td [(;)-162(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.5 30.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.536 20.75 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.034 30.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.006 20.75 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.138 30.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.476 20.75 Td [(\))]TJ /F9 1 Tf 0.39157 1.1145 Td [(\264)]TJ /F8 1 Tf 0.75904 -1.1145 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.526 30.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.307 20.75 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.302 30.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.548 20.75 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ 1.6145 0.40964 Td [(Z)]TJ 1 -0.24699 Td [(\263)]TJ 0.76506 0.24699 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.129 29.478 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.476 20.75 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.983 30.288 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.717 20.75 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.836 30.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.241 20.75 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.371 30.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.711 20.75 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.474 30.288 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.181 20.75 Td [(\))]TJ /F9 1 Tf 0.39157 1.1145 Td [(\264)]TJ /F8 1 Tf 0.76506 -1.1145 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.862 30.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.012 20.75 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.638 30.288 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.259 20.75 Td [(\))-337(,)]TJ -35.928 -2.3434 Td [(c'est-)0(\265)500(a-dire)]TJ /F8 1 Tf 5.1205 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F9 1 Tf 0.39157 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.371 26.125 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.873 18.406 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 0.95181 Td [(X)]TJ /F8 1 Tf 1.6084 -0.95181 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.733 26.125 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.012 18.406 Td [(\).)]TJ -13.681 -1.7289 Td [(La)-427(prop)-30(osition)]TJ /F1 1 Tf 6.8916 0 Td [(IV.2.2.)]TJ /F2 1 Tf 3.8373 0 Td [(prouv)24(e)-427(alors)-427(que)-421(l'on)-427(p)-30(eut)-421(d)24(\266)475(e\257nir)-427(une)-421(mesure)]TJ /F8 1 Tf 20.904 0 Td [(\271)]TJ /F2 1 Tf 1.0241 0 Td [(sur)-427(\()]TJ /F8 1 Tf 2.1506 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))-427(par)]TJ /F8 1 Tf -36.904 -1.7108 Td [(\271)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F2 1 Tf 0.81325 0 Td [(\))-349(=)]TJ /F8 1 Tf 1.8554 0 Td [(\301)]TJ /F2 1 Tf 0.59639 0 Td [(\()-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.078 21.202 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.223 14.967 Td [(\))-379(et)-379(que,)-391(p)-24(our)]TJ /F8 1 Tf 6.5723 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(mesurable)-379(p)-30(ositiv)30(e,)-391(on)-373(a)]TJ /F8 1 Tf 11.114 0 Td [(\301)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\))-349(=)]TJ /F9 1 Tf 1.8614 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)-277(d\271)]TJ /F2 1 Tf 1.8916 0 Td [(.)-566(La)-379(premi)24(\265)469(ere)-373(par-)]TJ -30.88 -1.6566 Td [(tie)-415(de)-409(l')24(\266)469(egalit)24(\266)475(e)-421(de)-409(F)78(ubini)-415(est)-415(donc)-385(\266)469(etablie)-415(p)-30(our)-415(les)-415(fonctions)-415(mesurables)-415(p)-30(ositiv)24(es)-409(si)-421(l'on)]TJ 0 -1.2048 Td [(mon)24(tre)-319(que)]TJ /F8 1 Tf 5.253 0 Td [(\271)]TJ /F2 1 Tf 0.92169 0 Td [(co)108(\304)391(\263ncide)-325(a)30(v)24(ec)-325(la)-319(mesure)-319(pro)-30(duit)-319(tensoriel)]TJ /F8 1 Tf 18.048 0 Td [(m)]TJ /F2 1 Tf 0.8735 0 Td [(.)-445(Compte)-313(ten)24(u)-325(de)-319(l'unicit)30(\266)469(e)-325(de)-319(ce)]TJ -25.096 -1.1988 Td [(pro)-30(duit)-373(tensoriel,)-379(il)-373(su\261t)-379(de)-367(prouv)24(er)-373(la)-373(co)102(\304)391(\263ncidence)-373(sur)-373(les)-373(pa)24(v)48(\266)475(es)-373(mesurables,)-385(c'est-)-6(\265)500(a-dire)]TJ /F9 1 Tf 0 -0.38554 Td [(Z)]TJ 1 0.048192 Td [(\265)]TJ 0.73494 -0.048192 Td [(Z)]TJ /F2 1 Tf 1 -1.3614 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.345 12.892 Td [(A)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 18.458 18.427 Td [(0)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 13.595 12.892 Td [(\243)]TJ /F7 1 Tf 0.89655 0 Td [(A)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 21.458 18.427 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.271 9.1593 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.5 13.702 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.512 9.1593 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.362 13.702 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.036 9.1593 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.888 13.702 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.506 9.1593 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.991 13.702 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.976 9.1593 Td [(\))]TJ /F9 1 Tf 0.39157 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90361 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.578 13.702 Td 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8.352 0 0 Tm 16.638 9.4435 Td [(\243)]TJ /F7 1 Tf 0.89655 0 Td [(A)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 25.711 13.596 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.398 6.7497 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.543 10.245 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.639 6.7497 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.397 10.245 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.163 6.7497 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.931 10.245 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.633 6.7497 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.034 10.245 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.102 6.7497 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.121 10.245 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.355 6.7497 Td [(\))-6(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.216 9.4435 Td [(A)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 51.819 13.596 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.229 6.7497 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.905 10.245 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.47 6.7497 Td [(\).)]TJ -3.4759 -2.8614 Td [(115)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 381 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 380 0 R /ProcSet 2 0 R >> /Contents 379 0 R >> endobj 380 0 obj << /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F18 107 0 R >> endobj 382 0 obj << /Length 18260 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(En)-277(\266)475(ec)24(hangean)24(t)-307(les)-307(r)0(^)500(oles)-307(de)]TJ /F8 1 Tf 11.711 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.69 89.633 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.536 62.22 Td [(et)-301(de)]TJ /F8 1 Tf 2.4398 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.31 89.633 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.699 62.22 Td [(on)-301(prouv)24(e)-307(de)-301(m)24(^)469(eme)-301(la)-301(deuxi)24(\265)469(eme)-301(partie)-307(de)-301(l')24(\266)469(egalit)24(\266)475(e)]TJ -17.367 -1.1988 Td [(de)-331(F)78(ubini)-331(p)-30(our)-337(les)-337(fonctions)-331(mesurables)-343(p)-24(ositiv)24(es.)]TJ /F1 1 Tf 0.81325 -1.5602 Td [(2.)]TJ /F2 1 Tf 1.3795 0 Td [(Le)-246(r)30(\266)469(esultat)-246(pr)24(\266)475(ec)24(\266)469(eden)24(t)-246(mon)24(tre)-240(en)-246(particulier)-240(que,)-271(si)]TJ /F8 1 Tf 21.651 0 Td [(f)]TJ /F2 1 Tf 0.83735 0 Td [(est)-246(in)24(t)24(\266)469(egrable)-240(p)-30(ositiv)24(e,)-265(la)-240(fonction)]TJ /F8 1 Tf -24.681 -1.6566 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4483 83.306 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4578 57.804 Td [(7!)]TJ /F9 1 Tf 1.2831 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.733 83.306 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.578 57.804 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.595 83.306 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.108 57.804 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.121 83.306 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.572 57.804 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.233 83.306 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.042 57.804 Td [(\))-295(est)-295(in)24(t)24(\266)475(egrable)-295(et)-289(donc)-295(\257nie)-289(p)-30(our)-289(presque)-295(tout)]TJ /F8 1 Tf 20.44 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.026 83.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.337 57.804 Td [(,)-301(ce)-289(qui)-289(implique)]TJ -32.006 -1.7651 Td [(que,)-319(p)-30(our)-313(presque)-313(tout)]TJ /F8 1 Tf 10.241 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.103 80.788 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.422 56.039 Td [(,)-319(la)-313(fonction)-313(partielle)]TJ /F8 1 Tf 9.3976 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.181 80.788 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.657 56.039 Td [(;)-168(:)]TJ /F2 1 Tf 0.72289 0 Td [(\))-319(est)-319(in)30(t)24(\266)469(egrable.)-439(De)-313(m)24(^)469(eme)-313(on)-313(mon)24(tre)]TJ -23.048 -1.7108 Td [(que)]TJ /F8 1 Tf 1.9036 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.172 78.331 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6747 54.328 Td [(7!)]TJ /F9 1 Tf 1.3494 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.009 78.331 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.867 54.328 Td [(;)-162(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.862 78.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.392 54.328 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.397 78.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.627 54.328 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.172 78.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.867 54.328 Td [(\))-385(est)-379(in)24(t)24(\266)469(egrable,)-391(donc)-379(\257nie)-379(p)-30(our)-379(presque)-379(tout)]TJ /F8 1 Tf 20.319 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.466 78.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.265 54.328 Td [(,)-391(et)]TJ /F8 1 Tf 1.8795 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(:)-168(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.388 78.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 54.328 Td [(\))]TJ -38.777 -1.7651 Td [(est)-337(in)24(t)24(\266)475(egrable)-337(p)-30(our)-337(presque)-331(tout)]TJ /F8 1 Tf 14.386 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.034 75.814 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.801 52.563 Td [(.)]TJ -15.47 -1.8193 Td [(Lorsque)]TJ /F8 1 Tf 3.8253 0 Td [(f)]TJ /F2 1 Tf 0.98193 0 Td [(est)-391(in)24(t)24(\266)469(egrable)-385(de)-385(signe)-391(quelconque,)-403(on)-391(p)-30(eut)-385(l')24(\266)469(ecrire)-385(comme)-391(di\256)24(\266)475(erence)-391(de)-385(deux)]TJ -4.8072 -1.1988 Td [(fonctions)-349(in)24(t)24(\266)475(egrables)-355(p)-30(ositiv)30(es)-355(\(par)-349(exemple)]TJ /F8 1 Tf 19.5 0 Td [(f)]TJ /F2 1 Tf 0.90362 0 Td [(=)]TJ /F8 1 Tf 1.0783 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.078 70.685 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.205 49.545 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.509 70.685 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.38 49.545 Td [(\))-349(et)-355(le)-355(r)30(\266)469(esultat)-355(g)30(\266)469(en)30(\266)469(eral)-349(se)-349(d)30(\266)469(eduit)]TJ -25.048 -1.2048 Td [(du)-337(pr)30(\266)469(ec)24(\266)469(eden)30(t)-337(par)-337(additivit)24(\266)469(e)-337(de)-331(l'in)24(t)24(\266)469(egrale.)]TJ 0 -1.5602 Td [(En\257n,)-487(si)]TJ /F8 1 Tf 4.2349 0 Td [(f)]TJ /F2 1 Tf 1.0482 0 Td [(est)-463(\265)500(a)-451(v)54(aleurs)-463(v)24(ectorielles)-457(on)-451(se)-457(ram)24(\265)469(ene)-457(au)-451(cas)-457(pr)24(\266)475(ec)24(\266)469(eden)24(t)-451(en)-457(consid)30(\266)469(eran)30(t)-457(ses)]TJ -5.2831 -1.1988 Td [(co)-30(ordonn)18(\266)475(ees)-337(\(dans)-337(une)-331(base)-337(arbitraire\).)]TJ ET 512.64 543.56 m 519.84 543.56 l 519.84 551.98 l 512.64 551.98 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 43.238 Td [(Remarque.)]TJ /F2 1 Tf 6.253 0 Td [(Av)30(ec)-337(les)-337(notations)-337(pr)24(\266)475(ec)24(\266)469(eden)24(tes,)-331(la)-337(double)-337(in)30(t)24(\266)469(egrale)]TJ /F9 1 Tf 5.9217 -0.37952 Td [(Z)]TJ 1 0.048195 Td [(\265)]TJ 0.90361 -0.048195 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.44 59.969 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.247 41.497 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.293 59.969 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.777 41.497 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.828 59.969 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.241 41.497 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.931 59.969 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.717 41.497 Td [(\))]TJ /F9 1 Tf 0.38554 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90361 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.517 59.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.687 41.497 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.293 59.969 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.928 41.497 Td [(\))]TJ -26.596 -2.7771 Td [(se)-331(note)-337(aussi)-337(plus)-337(simplemen)24(t)]TJ /F9 1 Tf 13.283 1.3675 Td [(Z)-277(Z)]TJ /F8 1 Tf 1.8313 -1.3675 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.491 56.004 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.289 38.72 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.345 56.004 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.813 38.72 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.879 56.004 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.283 38.72 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.983 56.004 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.753 38.72 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.517 56.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.988 38.72 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.293 56.004 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.229 38.72 Td [(\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 29.359 Td [(VI)-35(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 35.195 Td [(4.)]TJ /F15 1 Tf 1.747 0 Td [(Cor)24(ollaire.)]TJ /F12 1 Tf 6.5422 0 Td [(Av)30(ec)-295(les)-289(h)30(yp)-36(oth)24(\265)469(eses)-289(et)-289(notations)-289(du)-289(th)24(\266)469(eor)24(\265)469(eme)-289(pr)30(\266)469(ec)24(\266)469(eden)30(t,)-301(p)-30(our)]TJ /F8 1 Tf -12.319 -1.2048 Td [(B)]TJ /F12 1 Tf 1.1747 0 Td [(partie)-367(mesurable)-367(de)]TJ /F8 1 Tf 9.0542 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-373(c)24(haque)-367(section)]TJ /F8 1 Tf 7.3855 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.224 48.435 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 50.145 68.09 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.392 33.991 Td [(,)-373(p)-30(our)]TJ /F8 1 Tf 3.0482 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.517 49.237 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 29.62 33.991 Td [(2)]TJ /F8 1 Tf 1 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.121 49.237 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 31.807 33.991 Td [(,)-373(est)-373(mesurable,)-373(l'application)]TJ /F8 1 Tf -26.476 -1.6506 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4483 46.866 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4578 32.34 Td [(7!)]TJ /F8 1 Tf 1.2831 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.328 46.866 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1265 32.34 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.776 46.866 Td [(0)]TJ /F7 1 Tf -0.068965 -0.94828 Td [(x)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.777 32.34 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-343(mesurable)-331(et)]TJ /F8 1 Tf 7.4639 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.267 46.866 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.157 32.34 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.638 46.064 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 53.518 64.776 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.078 32.34 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.543 46.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.313 32.34 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.319 46.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.554 32.34 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(.)]TJ -25.614 -2.0181 Td [(En)-301(particulier,)-313(une)-301(partie)-307(mesurable)-301(de)]TJ /F8 1 Tf 17.355 0 Td [(X)]TJ /F12 1 Tf 1.2048 0 Td [(est)-307(n)24(\266)469(egligeable)-301(si)-307(et)-301(seulemen)24(t)-301(si)-307(presque)-301(toutes)]TJ -18.56 -1.1988 Td [(ses)-337(sections)-337(son)24(t)-337(n)24(\266)475(egligeables.)]TJ 0 -1.5663 Td [(On)-337(a)-331(des)-307(\266)469(enonc)30(\266)469(es)-337(analogues)-331(en)-307(\266)469(ec)24(hangean)24(t)-337(les)-337(r)0(^)499(oles)-337(de)]TJ /F8 1 Tf 24.096 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.414 40.03 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 30.952 27.557 Td [(et)-331(de)]TJ /F8 1 Tf 2.5 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.164 40.03 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 34.867 27.557 Td [(.)]TJ /F1 1 Tf -29.536 -1.8193 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Il)-228(su\261t)-240(d'appliquer)-228(le)-240(th)24(\266)469(eor)24(\265)469(eme)-228(de)-234(F)84(ubini)-234(en)-234(prenan)24(t)-234(p)-30(our)]TJ /F8 1 Tf 24.976 0 Td [(f)]TJ /F2 1 Tf 0.83132 0 Td [(la)-228(fonction)]TJ -34.627 -1.1988 Td [(indicatrice)-331(de)]TJ /F8 1 Tf 6.2289 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(.)]TJ ET 512.64 292.06 m 519.84 292.06 l 519.84 300.49 l 512.64 300.49 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 22.72 Td [(Remarque.)]TJ /F2 1 Tf 6.6627 0 Td [(Si)]TJ /F8 1 Tf 1.2711 0 Td [(B)]TJ /F2 1 Tf 1.247 0 Td [(est)-439(une)-439(partie)-445(n)30(\266)469(egligeable)-439(de)]TJ /F8 1 Tf 13.307 0 Td [(X)]TJ /F2 1 Tf 1.3434 0 Td [(\(non)-439(supp)-30(os)24(\266)475(ee)-439(mesurable\),)-463(on)-439(sait)]TJ -23.831 -1.2048 Td [(qu'elle)-391(est)-397(con)30(ten)24(ue)-391(dans)-385(une)-391(partie)-397(mesurable)-385(n)24(\266)469(egligeable.)-608(P)24(ar)-385(suite)-391(presque)-391(toutes)-391(ses)]TJ 0 -1.1988 Td [(sections)-337(son)18(t)-337(n)24(\266)469(egligeables.)-457(Cep)-30(endan)24(t)-343(cette)-337(condition)-343(n'est)-337(plus)-343(su\261san)24(te)-337(p)-30(our)-343(assurer)-337(la)]TJ 0 -1.1988 Td [(n)30(\266)469(egligeabilit)24(\266)475(e)-337(d'une)-343(partie)-337(si)-337(on)-331(ne)-337(la)-331(supp)-30(ose)-337(pas)-337(mesurable.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 13.49 Td [(VI)-35(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 16.171 Td [(5.)-283(In)24(t)24(\266)548(egration)-385(sur)-385(une)-385(partie)-385(d'un)-385(pro)-36(duit.)]TJ /F2 1 Tf -4.0301 -1.5602 Td [(Si)]TJ /F8 1 Tf 1.1386 0 Td [(f)]TJ /F2 1 Tf 0.90361 0 Td [(est)-313(une)-313(fonction)-313(mesurable)-307(sur)-313(une)-307(partie)-313(mesurable)]TJ /F8 1 Tf 23.036 0 Td [(A)]TJ /F2 1 Tf 1.0542 0 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(X)]TJ /F2 1 Tf 1.1807 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.405 21.504 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 36.373 14.611 Td [(\243)]TJ /F8 1 Tf 0.95181 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.707 21.504 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.741 14.611 Td [(.)-439(On)-307(sait)-307(que)]TJ -33.41 -1.1988 Td [(le)-331(probl)24(\265)469(eme)-325(de)-331(l'in)24(t)24(\266)475(egration)-337(de)]TJ /F8 1 Tf 13.675 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(sur)]TJ /F8 1 Tf 1.6687 0 Td [(A)]TJ /F2 1 Tf 1.0783 0 Td [(se)-331(ram)24(\265)469(ene)-331(\265)499(a)-325(celui)-331(de)-325(l'in)24(t)24(\266)469(egration)-331(de)-331(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.328 18.978 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 40.223 13.412 Td [(\242)]TJ /F8 1 Tf 0.49398 0 Td [(f)]TJ /F2 1 Tf 0.92168 0 Td [(sur)]TJ /F8 1 Tf 1.6687 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -38.886 -1.1988 Td [(L'application)-337(du)-337(th)24(\266)469(eor)24(\265)475(eme)-337(de)-331(F)78(ubini)-331(donne)-337(alors)-337(:)]TJ /F15 1 Tf 0 -1.8253 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Av)24(ec)-337(les)-337(notations)-331(pr)24(\266)469(ec)24(\266)475(eden)24(tes,)-331(si)]TJ /F8 1 Tf 14.976 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(est)-337(p)-30(ositiv)30(e)-343(ou)-331(in)24(t)24(\266)469(egrable)-331(on)-337(a)]TJ /F9 1 Tf -15.283 -0.37952 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.147 11.064 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.036 8.6473 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.75 11.064 Td [(X)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 43 15.873 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 21.819 10.057 Td [(\265)]TJ 0.73494 -0.048193 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.069 11.064 Td [(A)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 47.422 15.102 Td [(x)]TJ /F18 1 Tf 0.79518 0.28916 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.476 8.6473 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.259 12.961 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 26.319 8.6473 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.121 12.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.843 8.6473 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.647 12.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.313 8.6473 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.75 12.961 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.783 8.6473 Td [(\))]TJ /F9 1 Tf 0.39157 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90361 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.336 12.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.759 8.6473 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.112 12.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36 8.6473 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(,)]TJ -31.392 -2.3614 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3855 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.69 8.7797 Td [(x)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 15.843 12.68 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5783 6.2858 Td [(est)-337(la)-337(section)-331(de)]TJ /F8 1 Tf 7.3494 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(d)30(\266)469(e\257nie)-331(par)]TJ /F8 1 Tf 4.9458 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.241 9.59 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 22.813 6.2858 Td [(.)]TJ /F5 1 Tf -17.482 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(116)]TJ ET endstream endobj 384 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 383 0 R /ProcSet 2 0 R >> /Contents 382 0 R >> endobj 383 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F18 107 0 R >> endobj 385 0 obj << /Length 16020 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(VI)-35(I)-35(I.3.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 62.22 Td [(6.)-283(Compl)18(\266)548(etion)-385(des)-391(mesures)-385(pro)-36(duits.)]TJ /F2 1 Tf -4.0301 -1.7169 Td [(M)30(^)469(eme)-301(si)-313(les)-301(mesures)]TJ /F8 1 Tf 9.1024 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.914 87.168 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.898 60.503 Td [(et)]TJ /F8 1 Tf 1.1386 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.638 87.168 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.729 60.503 Td [(son)24(t)-307(compl)24(\265)469(etes,)-313(la)-307(mesure)-307(pro)-30(duit)-307(tensoriel)-307(n'est)-301(en)-307(g)30(\266)469(en)24(\266)475(eral)]TJ -13.398 -1.2048 Td [(pas)-331(compl)24(\265)469(ete.)-445(Il)-331(est)-331(donc)-331(fr)18(\266)475(equen)24(t)-337(de)-325(la)-331(remplacer)-331(par)-331(la)-331(mesure)-337(compl)24(\266)475(et)24(\266)469(ee.)-445(Cela)-337(conduit)]TJ 0 -1.1988 Td [(\265)499(a)-421(consid)30(\266)469(erer)-421(encore)-427(comme)-421(mesurable)-427(une)-421(partie)-427(\(resp.)-710(une)-421(fonction\))]TJ /F6 1 Tf 31.681 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurable)-421(que)]TJ -32.482 -1.1988 Td [(l'on)-289(a)-283(mo)-30(di\257)24(\266)469(ee)-283(sur)-289(un)-289(ensem)24(ble)-289(n)24(\266)475(egligeable.)-433(Il)-283(s'ensuit)-283(que,)-301(p)-30(our)-289(presque)-283(tout)]TJ /F8 1 Tf 33.958 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.043 82.012 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.428 56.9 Td [(la)-283(section)]TJ -35.096 -1.1988 Td [(\(resp.)-427(la)-283(fonction)-289(partielle\))-283(d)24(\266)469(e\257nie)-283(par)]TJ /F8 1 Tf 16.831 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.534 80.297 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.301 55.701 Td [(n'a)-253(\266)469(et)24(\266)475(e)-289(mo)-30(di\257)24(\266)469(ee)-283(que)-289(sur)-289(un)-283(ensem)24(ble)-289(n)30(\266)469(egligeable.)]TJ -17.97 -1.2048 Td [(En)-499(particulier)-493(on)-500(p)-30(eut)-499(encore)-506(a\261rmer)-500(que)-500(cette)-500(section)-500(\(resp.)-939(fonction)-499(partielle\))-500(est)]TJ 0 -1.1988 Td [(mesurable)-349(p)-30(our)-349(presque)-355(tout)]TJ /F8 1 Tf 12.976 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.017 76.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.163 53.298 Td [(.)-493(Cep)-24(endan)24(t)-355(on)-349(ne)-349(p)-30(eut)-355(plus)-349(assurer)-355(la)-349(mesurabilit)30(\266)469(e)-355(p)-30(our)]TJ -13.831 -1.1988 Td [(tout)]TJ /F8 1 Tf 2.2892 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.724 75.142 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.9337 52.099 Td [(comme)-457(a)24(v)24(ec)-457(les)-463(parties)-457(\(resp.)-819(fonctions\))-463(qui)-457(son)24(t)-463(vraimen)24(t)]TJ /F6 1 Tf 27.301 0 Td [(T)]TJ /F2 1 Tf 0.8012 0 Td [(-mesurables.)-819(En)]TJ -31.705 -1.2048 Td [(dehors)-337(de)-331(cette)-337(mo)-36(di\257cation,)-337(il)-331(est)-337(clair)-337(que)-337(les)-337(autres)-337(r)30(\266)469(esultats)-331(resten)24(t)-337(v)54(alables.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 38.6 Td [(VI)-35(I)-35(I.4.)-236(G)25(\266)537(en)20(\266)537(eralisation)-381(aux)-376(pro)-35(duits)-376(\257nis.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 44.551 Td [(Dans)-271(un)-271(souci)-277(de)-271(simpli\257cation)-271(des)-271(notations)-277(nous)-265(n'a)24(v)24(ons)-277(jusqu'ici)-271(consid)30(\266)469(er)24(\266)469(e)-271(que)-277(le)-277(cas)-271(du)]TJ 0 -1.1988 Td [(pro)-30(duit)-301(de)-301(deux)-301(espaces)-301(mesurables)-307(ou)-301(mesur)30(\266)469(es.)-433(Il)-301(est)-307(facile)-307(de)-301(g)30(\266)469(en)30(\266)469(eraliser)-301(cela)-307(au)-301(pro)-30(duit)]TJ 0 -1.1988 Td [(d'une)-337(famille)-337(\257nie.)-445(On)-337(a)-331(en)-337(particulier)-331(les)-313(\266)475(enonc)24(\266)475(es)-337(suiv)54(an)24(ts)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 32.49 Td [(VI)-35(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 38.948 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(D)]TJ 0.83132 0.084339 Td [(\266)]TJ 0 -0.084339 Td [(efinition.)]TJ /F12 1 Tf 5.3012 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.0422 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.767 55.521 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.837 38.948 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.664 55.521 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.157 38.948 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.698 55.521 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.175 38.948 Td [(une)-343(famille)-349(\257nie)-349(d'espaces)-343(mesurables)-349(\()]TJ /F8 1 Tf 17.229 0 Td [(I)]TJ /F12 1 Tf 0.86145 0 Td [(est)]TJ -37.934 -1.1988 Td [(donc)-295(supp)-30(os)24(\266)469(e)-301(\257ni\).)-433(Dans)-301(le)-307(pro)-30(duit)]TJ /F8 1 Tf 15.651 0 Td [(X)]TJ /F2 1 Tf 1.1867 0 Td [(=)]TJ /F9 1 Tf 1.1988 0.75301 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.233 52.737 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.62 37.75 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.422 53.806 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 25.783 37.75 Td [(,)-307(on)-301(app)-30(elle)]TJ /F1 1 Tf 5.3193 0 Td [(pa)24(v)54(\266)548(e)-343(mesurable)]TJ /F12 1 Tf 8.006 0 Td [(tout)-301(pro)-30(duit)]TJ /F8 1 Tf -33.777 -1.8976 Td [(A)]TJ /F2 1 Tf 1.0241 0 Td [(=)]TJ /F9 1 Tf 1.2048 0.74699 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.612 50.021 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8133 35.852 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 13.681 51.09 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 10.229 35.852 Td [(o)-30(\265)530(u)-337(les)]TJ /F8 1 Tf 2.8434 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.776 51.09 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.488 35.852 Td [(son)24(t)-337(dans)]TJ /F6 1 Tf 4.4819 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.931 51.09 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 19.849 35.852 Td [(.)]TJ -14.518 -2.3614 Td [(La)-391(tribu)]TJ /F6 1 Tf 4.0663 0 Td [(T)]TJ /F12 1 Tf 1.1867 0 Td [(engendr)24(\266)469(ee)-391(par)-385(les)-391(pa)24(v)54(\266)469(es)-385(mesurables)-391(s'app)-30(elle)-391(la)-391(tribu)]TJ /F1 1 Tf 23.867 0 Td [(pro)-36(duit)]TJ /F12 1 Tf 4.1506 0 Td [(des)-391(tribus)]TJ /F6 1 Tf 4.7349 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.802 47.711 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 33.491 Td [(.)]TJ -38.886 -1.2048 Td [(On)-337(la)-331(note)]TJ /F9 1 Tf 5.1446 0.75301 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.784 44.926 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.729 32.286 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.569 45.987 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 12.608 32.286 Td [(.)]TJ -7.2771 -2.4157 Td [(L'espace)-331(mesurable)]TJ /F2 1 Tf 8.753 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-343(alors)-337(le)]TJ /F1 1 Tf 5.0181 0 Td [(pro)-36(duit)]TJ /F12 1 Tf 4.1024 0 Td [(des)-337(espaces)]TJ /F2 1 Tf 5.241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.034 42.53 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.199 29.87 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.922 42.53 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.524 29.87 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)]TJ /F1 1 Tf -29.578 -2.0783 Td [(Remarques.)]TJ /F2 1 Tf 6.6807 0 Td [(Dans)-325(ce)-319(cas)-319(encore)-319(l'ensem)30(ble)-319(des)-319(pa)24(v)54(\266)469(es)-319(mesurables)-319(est)-325(un)-319(semi-anneau)-319(de)]TJ -6.6807 -1.1988 Td [(parties)-337(de)]TJ /F8 1 Tf 4.6205 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(.)]TJ -5.5301 -1.7229 Td [(Lorsque)]TJ /F8 1 Tf 3.7711 0 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F6 1 Tf 1.0602 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(1)]TJ /F8 1 Tf 0.5 0 Td [(;)-168(:)-162(:)-168(:)-168(;)-168(n)]TJ /F6 1 Tf 2.8253 0 Td [(g)]TJ /F2 1 Tf 0.83132 0 Td [(on)-337(utilisera)-337(la)-331(notation)]TJ /F6 1 Tf 10.349 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.931 35.383 Td [(1)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 27.175 24.87 Td [(\243)-222(\242)-168(\242)-168(\242)-222(\243)-222(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.526 35.383 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.988 24.87 Td [(plut)-6(^)500(ot)-337(que)]TJ /F9 1 Tf 5.012 0.75301 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.741 34.314 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 38.253 24.87 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.517 35.383 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.133 24.87 Td [(.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 17.434 Td [(VI)-35(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 20.9 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.9518 0 Td [(Av)30(ec)-367(les)-361(notations)-361(pr)30(\266)469(ec)24(\266)469(eden)30(tes,)-367(p)-30(our)]TJ /F8 1 Tf 16.47 0 Td [(i)]TJ /F6 1 Tf 0.66265 0 Td [(2)]TJ /F8 1 Tf 0.98795 0 Td [(I)]TJ /F12 1 Tf 0.87952 0 Td [(notons)]TJ /F8 1 Tf 3.253 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.509 29.693 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 41.578 20.9 Td [(la)-355(pro-)]TJ -36.247 -1.1988 Td [(jection)-337(de)]TJ /F8 1 Tf 4.5843 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.552 27.978 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.325 19.701 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.879 27.978 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.62 19.701 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(est)-343(la)-331(comp)-30(osan)18(te)-331(d'indice)]TJ /F8 1 Tf 11.855 0 Td [(i)]TJ /F12 1 Tf 0.6747 0 Td [(de)]TJ /F8 1 Tf 1.3374 0 Td [(x)]TJ /F12 1 Tf 0.56626 0 Td [(\).)-451(Alors)-337(:)]TJ /F1 1 Tf -25.59 -1.7229 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(les)-337(applications)]TJ /F8 1 Tf 7.012 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.879 25.512 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 15.958 17.979 Td [(son)24(t)-337(mesurables)-337(;)]TJ /F1 1 Tf -9.8133 -1.7169 Td [(2.)]TJ /F12 1 Tf 1.6024 0 Td [(si)]TJ /F2 1 Tf 1.0301 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.302 23.056 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.44 16.262 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.353 23.056 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.88 16.262 Td [(\))]TJ /F12 1 Tf 0.74096 0 Td [(est)-361(un)-361(espace)-355(mesurable,)-361(une)-355(application)]TJ /F8 1 Tf 18.331 0 Td [(f)]TJ /F12 1 Tf 0.95181 0 Td [(de)]TJ /F8 1 Tf 1.3554 0 Td [(X)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.776 23.056 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 34.892 16.262 Td [(dans)]TJ /F8 1 Tf 2.3614 0 Td [(X)]TJ /F12 1 Tf 1.259 0 Td [(est)-361(mesurable)]TJ -33.181 -1.1988 Td [(si)-337(et)-337(seulemen)30(t)-337(si)-337(c)24(hacune)-337(de)-331(ses)-343(comp)-30(osan)24(tes)]TJ /F8 1 Tf 20.38 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.612 21.34 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.837 15.063 Td [(\261)]TJ /F8 1 Tf 0.72289 0 Td [(f)]TJ /F12 1 Tf 0.93374 0 Td [(est)-337(mesurable.)]TJ /F1 1 Tf -23.163 -2.0783 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Comme)-379(p)-30(our)-373(la)-379(prop)-30(osition)]TJ /F1 1 Tf 12.572 0 Td [(VI)-36(I)-36(I.2.2.)]TJ /F2 1 Tf 4.3554 0 Td [(,)-385(la)-379(mesurabilit)30(\266)469(e)-379(de)]TJ /F8 1 Tf 8.9157 0 Td [(\274)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.043 18.366 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.271 12.985 Td [(d)24(\266)475(ecoule)]TJ -35.94 -1.2048 Td [(du)-403(fait)-409(que,)-427(p)-30(our)]TJ /F8 1 Tf 8.0542 0 Td [(B)]TJ /F2 1 Tf 1.2169 0 Td [(appartenan)24(t)-409(\265)500(a)]TJ /F6 1 Tf 6.6205 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.147 16.642 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.102 11.78 Td [(,)-421(l'image)-409(r)30(\266)469(ecipro)-30(que)]TJ /F8 1 Tf 9.1807 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.638 17.469 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F7 1 Tf -0.94828 -1.0086 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.964 11.78 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-409(est)-409(le)-409(pa)24(v)54(\266)469(e)-409(mesurable)]TJ /F9 1 Tf -28.645 -0.44578 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 7.6293 13.857 Td [(j)]TJ /F10 1 Tf 0.52586 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8133 10.581 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.828 14.926 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3193 10.581 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.1988 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.121 14.926 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.97 10.581 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.819 14.926 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.608 10.581 Td [(sauf)-337(p)-30(our)]TJ /F8 1 Tf 4.4518 0 Td [(j)]TJ /F2 1 Tf 0.74699 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(i)]TJ /F2 1 Tf 0.6747 0 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.3916 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.888 14.926 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.295 10.581 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(.)]TJ -20.825 -2.4458 Td [(Le)-481(r)30(\266)469(esultat)]TJ /F1 1 Tf 5.3675 0 Td [(2.)]TJ /F2 1 Tf 1.7771 0 Td [(d)30(\266)469(ecoule)-481(du)-481(fait)-481(que)-481(les)-481(pa)30(v)48(\266)469(es)-475(mesurables)-481(engendren)24(t)]TJ /F6 1 Tf 24.247 0 Td [(T)]TJ /F2 1 Tf 1.2771 0 Td [(et)-481(que)-481(le)-487(pa)30(v)48(\266)475(e)]TJ -32.669 -1.1988 Td [(mesurable)]TJ /F9 1 Tf 4.8735 0.74699 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.397 8.6418 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 11.458 6.9364 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.474 9.7107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.88 6.9364 Td [(est)-337(l'in)24(tersection)-337(des)-337(images)-337(r)24(\266)475(ecipro)-30(ques)]TJ /F8 1 Tf 17.795 0 Td [(\274)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.759 10.538 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F7 1 Tf -0.94828 -1.0086 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.355 6.9364 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.931 9.7107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.825 6.9364 Td [(\).)]TJ ET 512.64 81.68 m 519.84 81.68 l 519.84 90.104 l 512.64 90.104 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.994 3.8882 Td [(117)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 387 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 386 0 R /ProcSet 2 0 R >> /Contents 385 0 R >> endobj 386 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 388 0 obj << /Length 24071 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 51.902 Td [(VI)-35(I)-35(I.4.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 62.22 Td [(3.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.0783 0 Td [(Soien)24(t)]TJ /F2 1 Tf 3.0964 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.198 88.823 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.831 62.22 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.095 88.823 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.157 62.22 Td [(;)-168(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.457 88.823 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.813 62.22 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.491 88.823 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 27.88 62.22 Td [(une)-403(famille)-403(\257nie)-403(d'espaces)-397(mesur)24(\266)475(es)-403(et)]TJ /F2 1 Tf -22.548 -1.1988 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.68675 0 Td [(l'espace)-301(mesurable)-301(pro)-30(duit)-301(des)]TJ /F2 1 Tf 13.584 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.345 87.107 Td [(i)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.639 61.021 Td [(;)]TJ /F6 1 Tf 0.43976 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.241 87.107 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.958 61.021 Td [(\))]TJ /F12 1 Tf 0.39157 0 Td [(,)-301(alors)-307(il)-301(existe)-301(sur)]TJ /F2 1 Tf 8.2169 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.69277 0 Td [(une)-295(mesure)]TJ /F8 1 Tf 5.2169 0 Td [(m)]TJ /F12 1 Tf 1.1747 0 Td [(qui)]TJ -37.801 -1.1988 Td [(est)-337(un)]TJ /F1 1 Tf 3.006 0 Td [(pro)-30(duit)-391(tensoriel)]TJ /F12 1 Tf 8.7651 0 Td [(des)]TJ /F8 1 Tf 1.7289 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.207 85.392 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.042 59.822 Td [(,)-337(c'est-)0(\265)500(a-dire)-337(telle)-337(que)-337(l'on)-337(ait)-337(:)]TJ -11.681 -1.5121 Td [(p)-30(our)-337(tout)-337(pa)30(v)48(\266)475(e)-337(mesurable)]TJ /F9 1 Tf 11.717 0.74699 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.526 82.159 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.331 58.31 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.603 83.228 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 22.747 58.31 Td [(a)24(v)24(ec)-337(\()]TJ /F8 1 Tf 2.5843 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.328 83.228 Td [(i)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.693 58.31 Td [(2)-283(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.336 83.228 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 28.518 58.31 Td [(\))]TJ /F8 1 Tf 2.3916 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F9 1 Tf 0.53012 0.74699 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.043 82.159 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.572 58.31 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.121 83.228 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.657 58.31 Td [(\))-283(=)]TJ /F9 1 Tf 1.8675 0.74699 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.06 82.159 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.777 58.31 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.319 83.228 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.988 58.31 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.422 83.228 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.464 58.31 Td [(\))]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -35.855 -2.3193 Td [(En)-391(outre)-397(si)-391(les)-397(mesures)]TJ /F8 1 Tf 10.723 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.233 79.909 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.657 55.991 Td [(son)24(t)]TJ /F8 1 Tf 2.2048 0 Td [(\276)]TJ /F12 1 Tf 0.60843 0 Td [(-\257nies,)-409(il)-391(n'y)-391(a)-391(qu'une)-391(mesure)-397(sur)]TJ /F2 1 Tf 15.337 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F2 1 Tf 0.79518 0 Td [(\))]TJ /F12 1 Tf 0.78313 0 Td [(qui)-391(ait)-397(cette)]TJ -33.741 -1.2048 Td [(propri)24(\266)469(et)30(\266)469(e.)-506(Elle)-355(est)]TJ /F8 1 Tf 8.3494 0 Td [(\276)]TJ /F12 1 Tf 0.60843 0 Td [(-\257nie)-355(et)-355(se)-355(note)-355(alors)]TJ /F9 1 Tf 9.6205 0.75301 Td [(N)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.129 77.125 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.247 54.786 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.379 78.193 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.458 54.786 Td [(,)-355(on)-355(dit)-361(que)-355(c'est)]TJ /F1 1 Tf 7.8072 0 Td [(le)]TJ /F12 1 Tf 1.1988 0 Td [(pro)-30(duit)-355(tensoriel)-355(des)]TJ -30.133 -1.5783 Td [(mesures)]TJ /F8 1 Tf 3.7892 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.31 75.926 Td [(i)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 10.337 53.207 Td [(.)]TJ /F2 1 Tf -5.006 -2.241 Td [(La)-277(d)24(\266)469(emonstration)-277(de)-277(cette)-283(prop)-30(osition)-283(ainsi)-277(que)-277(la)-277(g)24(\266)475(en)24(\266)475(eralisation)-283(\265)500(a)-277(ce)-277(con)24(texte)-277(des)-253(\266)469(enonc)30(\266)469(es)]TJ 0 -1.1988 Td [(obten)24(us)-307(p)-30(our)-307(le)-313(pro)-30(duit)-307(de)-301(deux)-307(espaces)-307(p)-30(eut)-307(se)-307(faire)-313(en)-301(adaptan)24(t)-307(les)-307(d)24(\266)469(emonstrations)-301(d)24(\266)469(ej)0(\265)500(a)]TJ 0 -1.1988 Td [(vues.)-445(Nous)-337(ne)-331(les)-337(reprendrons)-337(pas.)]TJ 0 -1.6205 Td [(On)-337(p)-30(eut)-313(\266)469(egalemen)24(t)-337(obtenir)-343(la)-337(plupart)-343(de)-337(ces)-337(r)24(\266)475(esultats)-343(en)-337(raisonnan)24(t)-343(par)-337(r)24(\266)475(ecurrence)-343(sur)-343(le)]TJ 0 -1.2048 Td [(nom)24(bre)-331(d')24(\266)469(el)24(\266)469(emen)30(ts)-337(de)]TJ /F8 1 Tf 9.8434 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-445(P)24(our)-331(cela)-343(on)-331(utilise)-337(la)-337(m)30(\266)469(etho)-30(de)-337(suiv)54(an)24(te)-337(:)]TJ -10.361 -1.6205 Td [(Supp)-24(osons)-373(que)-373(l'on)-343(\266)469(ecriv)24(e)]TJ /F8 1 Tf 11.717 0 Td [(I)]TJ /F2 1 Tf 0.88554 0 Td [(comme)-367(r)24(\266)469(eunion)-373(de)-367(deux)-367(parties)-373(disjoin)30(tes)]TJ /F8 1 Tf 18.657 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.103 63.737 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.759 44.123 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.5 63.737 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.994 44.123 Td [(.)-548(On)-373(p)-30(eut)]TJ -34.663 -1.1988 Td [(alors)-439(d)30(\266)469(e\257nir)-433(des)-439(pro)-30(duits)-439(partiels)-433(\()]TJ /F8 1 Tf 15.681 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.371 62.021 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.199 42.924 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.552 62.021 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.723 42.924 Td [(\))-439(et)-433(\()]TJ /F8 1 Tf 2.4819 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.802 62.021 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.62 42.924 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.31 62.021 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.38 42.924 Td [(\))-433(a)24(v)24(ec)]TJ /F8 1 Tf 3.1205 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.802 62.021 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.133 42.924 Td [(=)]TJ /F9 1 Tf 1.5 0.75301 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.595 60.142 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 73.169 84.463 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.024 42.924 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.164 61.211 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.187 42.924 Td [(,)]TJ /F6 1 Tf 0.73494 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.845 62.021 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.446 42.924 Td [(=)]TJ /F9 1 Tf 1.506 0.75301 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.629 60.142 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 85.807 84.463 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 43.337 42.924 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.802 61.211 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 42.924 Td [(,)]TJ /F8 1 Tf -38.886 -1.9518 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.9224 59.228 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.1627 40.973 Td [(=)]TJ /F9 1 Tf 1.5783 0.75301 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.948 57.349 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 19.169 80.56 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 10.241 40.973 Td [(X)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.845 58.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.819 40.973 Td [(et)]TJ /F6 1 Tf 1.247 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.845 59.228 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.789 40.973 Td [(=)]TJ /F9 1 Tf 1.5783 0.75301 Td [(Q)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.871 57.349 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 34.422 80.56 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.867 40.973 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.353 58.418 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.747 40.973 Td [(.)-686(T)78(out)-421(revien)24(t)-421(alors)-415(\265)500(a)-415(v)24(oir)-415(que)-421(l'espace)-415(mesurable)-415(\()]TJ /F8 1 Tf 23.265 0 Td [(X)-24(;)]TJ /F6 1 Tf 1.2952 0 Td [(T)]TJ /F2 1 Tf 0.80121 0 Td [(\))]TJ -38.777 -1.9518 Td [(s'iden)30(ti\257e)-337(naturellemen)24(t)-337(au)-337(pro)-30(duit)-337(de)-331(\()]TJ /F8 1 Tf 17.548 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.034 56.435 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.066 39.021 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.216 56.435 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.59 39.021 Td [(\))-337(a)30(v)24(ec)-337(\()]TJ /F8 1 Tf 3.3072 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.655 56.435 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.313 39.021 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.164 56.435 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.072 39.021 Td [(\).)]TJ -26.741 -1.6205 Td [(Plus)-361(pr)30(\266)469(ecis)24(\266)475(emen)24(t)-355(on)-355(d)24(\266)469(e\257nit)-355(une)-355(bijection)]TJ /F8 1 Tf 18.331 0 Td [(F)]TJ /F2 1 Tf 1.1325 0 Td [(de)]TJ /F8 1 Tf 1.3554 0 Td [(X)]TJ /F2 1 Tf 1.2651 0 Td [(sur)]TJ /F8 1 Tf 1.6928 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.957 54.116 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 30.536 37.4 Td [(\243)]TJ /F8 1 Tf 1.012 0 Td [(X)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.448 54.116 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.325 37.4 Td [(en)-355(p)-24(osan)18(t)]TJ /F8 1 Tf 4.6024 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))-319(=)-313(\()]TJ /F8 1 Tf 2.1867 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.595 54.116 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.62 37.4 Td [(;)-168(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.388 54.116 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 37.4 Td [(\))]TJ -38.777 -1.1988 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3916 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.379 52.4 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8193 36.201 Td [(=)-283(\()]TJ /F8 1 Tf 1.4518 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.017 51.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.133 36.201 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.052 51.59 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.77586 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 23.506 72.499 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.072 36.201 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(a)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.914 52.4 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.012 36.201 Td [(=)-283(\()]TJ /F8 1 Tf 1.4518 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.888 51.59 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.325 36.201 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.922 51.59 Td [(i)]TJ /F10 1 Tf 0.40517 0 Td [(2)]TJ /F7 1 Tf 0.76724 0 Td [(I)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 35.892 72.499 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.488 36.201 Td [(.)-457(Il)-337(reste)-337(alors)-343(\265)500(a)-337(v)24(oir)-337(que)-343(cette)-343(bijection)-343(en)30(v)24(oie)-343(les)-313(\266)475(el)18(\266)475(emen)24(ts)]TJ -13.157 -1.1988 Td [(de)]TJ /F6 1 Tf 1.3976 0 Td [(T)]TJ /F2 1 Tf 1.1928 0 Td [(exactemen)24(t)-397(sur)-403(les)-373(\266)475(el)18(\266)475(emen)24(ts)-397(de)]TJ /F6 1 Tf 14.151 0 Td [(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.733 50.676 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 23.416 35.003 Td [(\243)-265(T)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.147 50.676 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.771 35.003 Td [(.)-632(Autremen)24(t)-403(dit,)-409(il)-403(faut)-397(prouv)24(er)-397(que)]TJ /F8 1 Tf 16.711 0 Td [(F)]TJ /F2 1 Tf 1.1807 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.2048 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.75 48.961 Td [(\241)]TJ /F4 1 Tf 0.88793 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.494 33.798 Td [(son)24(t)-313(mesurables)-313(mais)-313(cela)-313(d)30(\266)469(ecoule)-313(de)-307(la)-313(caract)30(\266)469(erisation)-313(des)-313(applications)-307(mesurables)-313(\265)500(a)]TJ -2.1627 -1.1988 Td [(v)54(aleurs)-337(dans)-337(un)-337(espace)-331(pro)-30(duit)-337(donn)24(\266)469(ee)-337(par)-331(les)-337(prop)-36(ositions)]TJ /F1 1 Tf 26.139 0 Td [(VI)-36(I)-36(I.2.2.)]TJ /F2 1 Tf 4.6867 0 Td [(et)]TJ /F1 1 Tf 1.1687 0 Td [(VI)-36(I)-36(I.4.2.)]TJ /F2 1 Tf 4.3494 0 Td [(.)]TJ -36.343 -1.6205 Td [(On)-361(constate)-367(alors)-367(que,)-373(si)-367(l'on)-367(a)-361(une)-361(mesure)]TJ /F8 1 Tf 19.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.784 44.926 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.349 30.979 Td [(\(resp.)]TJ /F8 1 Tf 2.988 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.233 44.926 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.723 30.979 Td [(\))-367(pro)-30(duit)-361(tensoriel)-367(des)]TJ /F8 1 Tf 10.169 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.776 44.116 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.464 30.979 Td [(p)-30(our)]TJ /F8 1 Tf -37.133 -1.1988 Td [(i)]TJ /F6 1 Tf 0.62048 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.612 43.202 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.0241 29.78 Td [(\(resp.)]TJ /F8 1 Tf 2.8976 0 Td [(i)]TJ /F6 1 Tf 0.62048 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.612 43.202 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.518 29.78 Td [(\),)-331(tout)-331(pro)-30(duit)-331(tensoriel)-325(de)]TJ /F8 1 Tf 12.072 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.879 43.202 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.072 29.78 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.664 43.202 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.952 29.78 Td [(s'iden)24(ti\257e)-331(gr)0(^)500(ace)-331(\265)500(a)]TJ /F8 1 Tf 7.9096 0 Td [(F)]TJ /F2 1 Tf 1.1145 0 Td [(\265)500(a)-325(un)-331(pro)-30(duit)]TJ -33.645 -1.2048 Td [(tensoriel)-379(des)]TJ /F8 1 Tf 5.8253 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.216 40.676 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.741 28.575 Td [(p)-30(our)]TJ /F8 1 Tf 2.4096 0 Td [(i)]TJ /F6 1 Tf 0.69277 0 Td [(2)]TJ /F8 1 Tf 1.0181 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-572(La)-385(construction)-373(d'un)-385(pro)-30(duit)-373(tensoriel)-379(des)]TJ /F8 1 Tf 19.904 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.612 40.676 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.873 28.575 Td [(p)-30(eut)-379(donc)-379(se)]TJ -33.542 -1.1988 Td [(faire)-421(par)-415(r)30(\266)469(ecurrence)-421(\265)500(a)-415(partir)-415(de)-415(la)-415(construction)-415(d'un)-421(pro)-30(duit)-415(tensoriel)-421(de)-415(deux)-415(mesures.)]TJ 0 -1.1988 Td [(En)-331(particulier,)-337(lorsque)-337(les)-337(mesures)]TJ /F8 1 Tf 15.361 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.871 37.245 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.235 26.177 Td [(son)24(t)]TJ /F8 1 Tf 2.1446 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nies)-337(\(cas)-337(d'unicit)24(\266)469(e)-337(du)-337(pro)-30(duit)-337(tensoriel\),)-331(le)]TJ -19.657 -1.1988 Td [(pro)-30(duit)-331(tensoriel)-337(des)]TJ /F8 1 Tf 9.3193 0 Td [(m)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.224 35.53 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.199 24.979 Td [(s'iden)30(ti\257e)-337(au)-337(pro)-30(duit)-331(tensoriel)-331(des)-337(pro)-30(duits)-337(tensoriels)-337(partiels)]TJ /F8 1 Tf 27.139 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 36.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 23.774 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 10.552 34.616 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.8855 23.774 Td [(.)-445(C'est)-343(une)-331(propri)24(\266)469(et)24(\266)475(e)-343(d'asso)-30(ciativit)30(\266)469(e)-337(du)-337(pro)-30(duit)-331(tensoriel.)]TJ -2.5542 -1.6205 Td [(Ce)-439(p)-30(oin)24(t)-439(de)-433(vue)-439(p)-30(ermet)-439(d'utiliser)-433(le)-445(th)24(\266)469(eor)24(\265)469(eme)-433(de)-433(F)78(ubini)-433(p)-30(our)-439(calculer)-439(l'in)24(t)24(\266)475(egrale)-439(d'une)]TJ 0 -1.1988 Td [(fonction)-277(par)-271(rapp)-30(ort)-277(\265)500(a)-271(un)-277(pro)-30(duit)-271(tensoriel)-277(de)-271(mesures)]TJ /F8 1 Tf 23.735 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nies)-277(en)-271(in)24(t)24(\266)475(egran)24(t)-277(successiv)30(emen)24(t)]TJ -24.343 -1.1988 Td [(par)-451(rapp)-36(ort)-457(\265)500(a)-451(c)24(hacune)-457(des)-457(mesures)-457(et)-457(ceci)-457(dans)-451(un)-457(ordre)-451(arbitraire.)-813(Cela)-457(est)-457(l)24(\266)475(egitime)]TJ 0 -1.2048 Td [(lorsqu'il)-331(s'agit)-337(d'une)-337(fonction)-337(mesurable)-337(p)-30(ositiv)30(e)-337(ou)-337(d'une)-337(fonction)-343(in)30(t)24(\266)469(egrable.)]TJ /F1 1 Tf 0 -1.9277 Td [(Remarque.)]TJ /F2 1 Tf 0 -1.6205 Td [(Le)-367(th)18(\266)475(eor)24(\265)469(eme)-367(de)-367(F)78(ubini)-367(p)-30(eut)-343(^)475(etre)-367(utilis)24(\266)469(e)-373(comme)-367(on)-373(vien)24(t)-367(de)-367(le)-373(v)24(oir)-367(p)-30(our)-373(ramener)-367(un)-367(pro-)]TJ 0 -1.2048 Td [(bl)24(\265)475(eme)-367(d'in)18(t)24(\266)475(egrale)-373("m)24(ultiple",)-379(c'est-)0(\265)500(a-dire)-373(relatif)-373(\265)500(a)-367(un)-373(pro)-30(duit)-373(tensoriel)-367(de)-367(mesures,)-385(\265)500(a)-367(une)]TJ 0 -1.1988 Td [(succession)-271(de)-271(probl)24(\265)469(emes)-271(d'in)24(t)24(\266)469(egrales)-271("simples".)-427(Il)-271(est)-277(aussi)-271(utilis)24(\266)469(e)-277(p)-30(our)-240(\266)469(ec)24(hanger)-271(des)-271(ordres)]TJ 0 -1.1988 Td [(d'in)24(t)24(\266)469(egration.)-439(A)-337(ce)-337(sujet)-343(on)-331(remarquera)-337(que,)-337(p)-30(our)]TJ /F8 1 Tf 22.253 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(mesurable,)-337(une)-337(relation)-331(du)-337(t)24(yp)-36(e)]TJ /F9 1 Tf -19.217 -0.38554 Td [(Z)]TJ 1 0.048194 Td [(\265)]TJ 0.90361 -0.048194 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.69 14.409 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.036 9.6533 Td [(;)-168(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.543 14.409 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.566 9.6533 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.069 14.409 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.03 9.6533 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.181 14.409 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.506 9.6533 Td [(\))]TJ /F9 1 Tf 0.38554 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90361 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.759 14.409 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.476 9.6533 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.534 14.409 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.717 9.6533 Td [(\))-283(=)]TJ /F9 1 Tf 1.7229 1.3614 Td [(Z)]TJ 1 0.048194 Td [(\265)]TJ 0.90362 -0.048194 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.793 14.409 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.187 9.6533 Td [(;)-162(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.647 14.409 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.711 9.6533 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.181 14.409 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.946 9.6533 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.957 14.409 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.187 9.6533 Td [(\))]TJ /F9 1 Tf 0.39156 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90362 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.207 14.409 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.392 9.6533 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.319 14.409 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.861 9.6533 Td [(\))-337(,)]TJ -34.53 -2.1627 Td [(est)-277(automatique)-277(si)]TJ /F8 1 Tf 8.1928 0 Td [(f)]TJ /F2 1 Tf 0.86747 0 Td [(est)-277(p)-30(ositiv)30(e)-277(et)-277(se)-271(prouv)24(e)-277(dans)-271(le)-277(cas)-277(g)30(\266)469(en)30(\266)469(eral)-271(en)-277(justi\257an)24(t)-277(l'in)24(t)24(\266)475(egrabilit)24(\266)475(e)]TJ -9.0602 -1.2048 Td [(de)]TJ /F8 1 Tf 1.2289 0 Td [(f)]TJ /F2 1 Tf 0.8253 0 Td [(\(p)-30(our)]TJ /F8 1 Tf 2.6506 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.612 9.59 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 11.205 6.2858 Td [(\255)]TJ /F8 1 Tf 0.78916 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.422 9.59 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.386 6.2858 Td [(\).)-415(P)30(our)-228(cela)-234(il)-228(su\261t)-234(de)-228(remplacer)]TJ /F8 1 Tf 14.229 0 Td [(f)]TJ /F2 1 Tf 0.8253 0 Td [(par)]TJ /F6 1 Tf 1.6747 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.50602 0 Td [(dans)-228(les)-228(in)24(t)24(\266)475(egrales)-234(pr)30(\266)469(ec)24(\266)469(eden)30(tes)]TJ /F5 1 Tf -26.169 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(118)]TJ ET endstream endobj 390 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 389 0 R /ProcSet 2 0 R >> /Contents 388 0 R >> endobj 389 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R /F18 107 0 R >> endobj 391 0 obj << /Length 14277 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(et)-253(de)-259(mon)24(trer)-253(que)-259(l'une)-259(des)-259(in)24(t)24(\266)475(egrales)-259(doubles)-253(obten)18(ues)-259(est)-259(\257nie,)-271(sac)24(han)24(t)-259(que)-259(de)-253(toute)-253(fa)-6(\273)439(con)]TJ 0 -1.1988 Td [(elles)-337(son)24(t)-307(\266)469(egales)-331(\(puisque)]TJ /F6 1 Tf 11.247 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F2 1 Tf 0.60843 0 Td [(est)-337(p)-30(ositiv)24(e\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 47.163 Td [(VI)-35(I)-35(I.5.)-236(Le)-376(cas)-371(de)-381(la)-376(mesure)-376(de)-376(Leb)-35(esgue.)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 54.912 Td [(Soit)]TJ /F8 1 Tf 2.0783 0 Td [(E)]TJ /F2 1 Tf 1.1506 0 Td [(un)-361(espace)-355(v)24(ectoriel)-355(de)-355(dimension)-361(\257nie.)-518(On)-355(sait)-355(que)-361(les)-361(normes)-355(sur)]TJ /F8 1 Tf 29.5 0 Td [(E)]TJ /F2 1 Tf 1.1566 0 Td [(son)24(t)-331(\266)475(equiv)48(a-)]TJ -33.886 -1.1988 Td [(len)30(tes)-337(donc)-331(d)30(\266)469(e\257nissen)24(t)-331(toutes)-331(la)-331(m)24(^)475(eme)-331(top)-30(ologie)-331(et)-331(donc)-331(la)-331(m)30(^)469(eme)-325(tribu)-331(de)-331(b)-30(or)30(\266)469(eliens)]TJ /F6 1 Tf 36.922 0 Td [(B)]TJ /F2 1 Tf 0.68674 0 Td [(.)-445(Si)]TJ /F8 1 Tf -37.608 -1.2048 Td [(u)]TJ /F2 1 Tf 0.92771 0 Td [(est)-361(un)-361(v)24(ecteur)-355(de)]TJ /F8 1 Tf 7.9398 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(,)-361(notons)]TJ /F8 1 Tf 3.8916 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.655 74.935 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.205 52.509 Td [(la)-355(translation)-361(de)-355(v)24(ecteur)]TJ /F8 1 Tf 11.108 0 Td [(u)]TJ /F2 1 Tf 0.92771 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.319 74.935 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.59 52.509 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-319(=)]TJ /F8 1 Tf 1.8012 0 Td [(a)]TJ /F2 1 Tf 0.76506 0 Td [(+)]TJ /F8 1 Tf 1.0181 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\).)-518(Comme)]TJ /F8 1 Tf 4.8735 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.922 74.935 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 51.31 Td [(est)-337(con)24(tin)24(ue)-337(elle)-343(est)-343(b)-30(or)30(\266)469(elienne)-337(et)-337(donc,)-337(p)-30(our)]TJ /F8 1 Tf 19.934 0 Td [(B)]TJ /F2 1 Tf 1.1506 0 Td [(b)-24(or)24(\266)469(elien)-331(de)]TJ /F8 1 Tf 5.1446 0 Td [(E)]TJ /F2 1 Tf 0.8012 0 Td [(,)-337(l'image)-337(r)24(\266)475(ecipro)-30(que)]TJ /F8 1 Tf 8.9578 0 Td [(\277)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.759 73.211 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.91 51.31 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F2 1 Tf 0.81326 0 Td [(\))]TJ -38.777 -1.1988 Td [(est)-409(encore)-403(b)-30(or)24(\266)475(elienne.)-656(Si)]TJ /F8 1 Tf 11.458 0 Td [(m)]TJ /F2 1 Tf 1.2831 0 Td [(est)-409(une)-403(mesure)-403(sur)-409(\()]TJ /F8 1 Tf 9.1928 0 Td [(E)-60(;)]TJ /F6 1 Tf 1.241 0 Td [(B)]TJ /F2 1 Tf 0.68674 0 Td [(\))-403(on)-403(p)-30(eut)-409(alors)-403(d)24(\266)469(e\257nir)-403(une)-403(mesure)]TJ /F8 1 Tf -23.861 -1.1988 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.25 69.78 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.2892 48.912 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\))]TJ /F1 1 Tf 0.78313 0 Td [(translat)18(\266)548(ee)]TJ /F2 1 Tf 5.3554 0 Td [(de)]TJ /F8 1 Tf 1.3976 0 Td [(m)]TJ /F2 1 Tf 1.2771 0 Td [(par)]TJ /F8 1 Tf 1.8434 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.698 69.78 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.578 48.912 Td [(en)-397(p)-30(osan)24(t)]TJ /F8 1 Tf 4.6928 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.362 69.78 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.229 48.912 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\)\()]TJ /F8 1 Tf 0.77711 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-391(=)]TJ /F8 1 Tf 1.9398 0 Td [(m)]TJ /F9 1 Tf 0.87952 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(\277)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.509 69.78 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.946 48.912 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))]TJ /F9 1 Tf 0.39156 0.80723 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.80723 Td [(,)-415(c'est)-403(l'image)-403(directe)]TJ -29.663 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3916 0 Td [(m)]TJ /F2 1 Tf 1.2651 0 Td [(par)]TJ /F8 1 Tf 1.8373 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.69 68.056 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.175 47.707 Td [(\(cf.)]TJ /F1 1 Tf 2.0361 0 Td [(IV.9.3.)]TJ /F2 1 Tf 3.4157 0 Td [(\).)-614(On)-391(dira)-397(alors)-391(qu'une)-391(mesure)-397(sur)-391(\()]TJ /F8 1 Tf 16.458 0 Td [(E)-60(;)]TJ /F6 1 Tf 1.241 0 Td [(B)]TJ /F2 1 Tf 0.68674 0 Td [(\))-391(est)]TJ /F1 1 Tf 2.3976 0 Td [(in)30(v)60(arian)30(te)-457(par)]TJ -32.078 -1.1988 Td [(translation)]TJ /F2 1 Tf 5.7651 0 Td [(lorsque,)-337(p)-30(our)-337(tout)-337(v)24(ecteur)]TJ /F8 1 Tf 11.741 0 Td [(u)]TJ /F2 1 Tf 0.90964 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(,)-337(on)-337(a)]TJ /F8 1 Tf 2.8374 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.716 66.34 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.669 46.509 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(.)]TJ /F1 1 Tf -28.211 -2.247 Td [(Exemples.)]TJ /F2 1 Tf 5.5602 0 Td [(On)-337(v)54(\266)469(eri\257era)-337(que,)-337(si)]TJ /F8 1 Tf 8.6566 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.612 63.125 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.813 44.262 Td [(est)-343(la)-337(mesure)-337(de)-337(Dirac)-337(au)-337(p)-30(oin)24(t)]TJ /F8 1 Tf 14.133 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)-337(la)-337(translat)24(\266)475(ee)]TJ /F8 1 Tf 6.3494 0 Td [(\277)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.483 63.125 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.789 44.262 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.422 63.125 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 44.262 Td [(\))]TJ -38.777 -1.1988 Td [(est)-337(la)-337(mesure)-337(de)-331(Dirac)]TJ /F8 1 Tf 10.114 0 Td [(\261)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.741 61.409 Td [(a)]TJ /F4 1 Tf 0.62069 0 Td [(+)]TJ /F7 1 Tf 0.87931 0 Td [(u)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.464 43.063 Td [(.)]TJ -12.133 -1.6265 Td [(La)-337(mesure)-337(de)-337(d)30(\266)469(ecompte)-337(est)-337(in)24(v)54(arian)24(te)-337(par)-337(translation.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 32.012 Td [(VI)-35(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 38.376 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F1 1 Tf 6.8675 0 Td [(\(caract)18(\266)548(erisation)-385(de)-385(la)-391(mesure)-385(de)-385(Leb)-36(esgue\))]TJ /F12 1 Tf -12.645 -1.6265 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0843 0 Td [(E)]TJ /F12 1 Tf 1.1928 0 Td [(un)-391(espace)-397(v)24(ectoriel)-391(de)-391(dimension)-397(\257nie)-397(et)]TJ /F8 1 Tf 18.373 0 Td [(C)]TJ /F12 1 Tf 1.1807 0 Td [(une)-391(partie)-397(b)-30(or)24(\266)475(elienne)-397(de)]TJ /F8 1 Tf 11.163 0 Td [(E)]TJ /F12 1 Tf 1.1868 0 Td [(qui)-391(est)]TJ -36.181 -1.1988 Td [(b)-30(orn)30(\266)469(ee)-337(et)-337(d'in)24(t)24(\266)469(erieur)-331(non)-337(vide.)-445(Alors)-331(:)]TJ /F1 1 Tf 0.81325 -1.6265 Td [(1.)]TJ /F12 1 Tf 1.8675 0 Td [(Il)-487(y)-487(a)-487(des)-487(mesures)]TJ /F8 1 Tf 8.9518 0 Td [(m)]TJ /F12 1 Tf 0.87952 0 Td [(,)]TJ /F8 1 Tf 0.8012 0 Td [(\276)]TJ /F12 1 Tf 0.60843 0 Td [(-\257nies)-487(et)-487(non)-493(n)30(ulles)-493(sur)]TJ /F2 1 Tf 11.265 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(E)-60(;)]TJ /F6 1 Tf 1.241 0 Td [(B)]TJ /F2 1 Tf 0.68675 0 Td [(\))]TJ /F12 1 Tf 0.87952 0 Td [(qui)-481(son)18(t)-487(in)24(v)54(arian)24(tes)-493(par)]TJ -28.38 -1.1988 Td [(translation.)]TJ /F1 1 Tf 0.81325 -1.6205 Td [(2.)]TJ /F12 1 Tf 1.6988 0 Td [(Ces)-403(mesures)-403(son)18(t)-403(prop)-30(ortionnelles)-409(en)30(tre)-409(elles.)-656(En)-403(particulier,)-421(il)-403(y)-409(en)-397(a)-403(une)-409(et)-403(une)]TJ -2.512 -1.2048 Td [(seule)-337(qui)-331(v)54(\266)469(eri\257e)-337(en)-331(outre)]TJ /F8 1 Tf 11.042 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F2 1 Tf 0.78313 0 Td [(\))-283(=)-277(1)]TJ /F12 1 Tf 2.2229 0 Td [(.)]TJ /F1 1 Tf -15.319 -1.9337 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6265 Td [(Nous)-493(laissons)-500(au)-500(lecteur)-493(le)-506(soin)-500(de)-493(v)48(\266)475(eri\257er)-499(cet)-469(\266)475(enonc)24(\266)475(e)-506(lorsque)]TJ /F8 1 Tf 28.729 0 Td [(E)]TJ /F2 1 Tf 1.3494 0 Td [(=)]TJ /F6 1 Tf 1.3253 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(0)]TJ /F6 1 Tf 0.5 0 Td [(g)]TJ /F2 1 Tf 0.99398 0 Td [(et)-500(supp)-30(osons)]TJ -33.398 -1.1988 Td [(d)30(\266)469(esormais)-337(dim)]TJ /F8 1 Tf 6.4578 0 Td [(E)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(d)]TJ /F6 1 Tf 0.80121 0 Td [(6)]TJ /F2 1 Tf [(=)-277(0.)]TJ /F1 1 Tf -8.5783 -1.6265 Td [(1.)]TJ /F2 1 Tf 1.6265 0 Td [(Commen)-6(\273)439(cons)-373(par)-367(le)-373(cas)]TJ /F8 1 Tf 10.765 0 Td [(E)]TJ /F2 1 Tf 1.1325 0 Td [(=)]TJ /F3 1 Tf 1.1084 0 Td [(R)]TJ /F2 1 Tf 0.72289 0 Td [(.)-548(La)-373(mesure)-367(de)-367(Leb)-36(esgue)]TJ /F8 1 Tf 11.476 0 Td [(\270)]TJ /F2 1 Tf 0.95181 0 Td [(\(ou)-373(plut)0(^)500(ot)-373(sa)-367(restriction)]TJ -28.596 -1.1988 Td [(aux)-361(b)-30(or)24(\266)475(eliens\))-361(a)-355(clairemen)24(t)-361(les)-361(propri)24(\266)469(et)24(\266)475(es)-361(v)24(oulues.)-518(En)-361(e\256et,)-367(cette)-361(mesure)-361(est)-367(caract)30(\266)469(eris)24(\266)469(ee)]TJ 0 -1.1988 Td [(par)-397(le)-409(fait)-397(que)-403(sa)-403(v)48(aleur)-397(sur)-409(un)-397(in)24(terv)54(alle)-397(est)-379(\266)469(egale)-403(\265)500(a)-397(la)-403(longueur)-403(de)-397(cet)-403(in)30(terv)54(alle.)-644(Cela)]TJ 0 -1.3494 Td [(en)30(tra)108(^)385(\263ne)-283(imm)18(\266)475(ediatemen)24(t)-283(que)]TJ /F8 1 Tf 12.663 0 Td [(\270)]TJ /F2 1 Tf 0.86747 0 Td [(est)-289(non)-289(n)24(ulle)-283(et)]TJ /F8 1 Tf 6.8976 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nie)-289(puisque)]TJ /F3 1 Tf 6.0482 0 Td [(R)]TJ /F2 1 Tf 1 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [([)]TJ /F2 1 Tf 1.1084 -0.95181 Td [([)]TJ /F6 1 Tf 0.28313 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(].)-427(P)24(ar)-283(ailleurs,)]TJ -32.952 -1.3434 Td [(le)-385(fait)-385(que)-379(le)-391(translat)30(\266)469(e)-385(d'un)-385(in)24(terv)60(alle)-385(soit)-379(un)-385(in)24(terv)54(alle)-379(de)-379(m)24(^)475(eme)-385(longueur)-385(mon)24(tre)-379(que)]TJ /F8 1 Tf 38.578 0 Td [(\270)]TJ /F2 1 Tf -38.578 -1.1988 Td [(est)-337(in)24(v)54(arian)24(te)-337(par)-337(translation.)]TJ 0 -1.6265 Td [(L'existence)-307(dans)-301(le)-313(cas)]TJ /F8 1 Tf 10.114 0 Td [(E)]TJ /F2 1 Tf 1.0723 0 Td [(=)]TJ /F3 1 Tf 1.0542 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.181 22.918 Td [(d)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.066 15.599 Td [(en)-307(d)24(\266)475(ecoule)-313(en)-301(prenan)18(t)-307(p)-30(our)-307(mesure)-307(la)-307(puissance)-307(tensorielle)]TJ /F8 1 Tf -13.735 -1.8313 Td [(d)]TJ /F2 1 Tf 0.51807 0 Td [(-i)24(\265)469(eme)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.741 21.064 Td [(d)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5542 14.521 Td [(N)]TJ /F8 1 Tf 1.2771 -0.75301 Td [(\270)]TJ /F2 1 Tf 0.95181 0 Td [(de)-373(la)-367(mesure)-373(de)-373(Leb)-30(esgue.)-554(L'in)24(v)54(ariance)-373(par)-367(translation)-373(d)24(\266)469(ecoule)-373(alors)-373(du)-373(fait)]TJ -5.4518 -1.1988 Td [(que)-451(le)-457(translat)24(\266)475(e)-457(d'un)-457(pa)24(v)54(\266)469(e)-451(mesurable)-457(est)-451(un)-457(pa)30(v)48(\266)469(e)-451(mesurable)-451(don)24(t)-457(les)-451(facteurs)-457(son)24(t)-457(des)]TJ 0 -1.1988 Td [(translat)30(\266)469(es)-337(des)-337(facteurs)-337(du)-337(pa)30(v)48(\266)475(e)-343(de)-331(d)24(\266)475(epart.)]TJ 0 -1.6265 Td [(En\257n,)-283(dans)-265(le)-277(cas)-265(g)24(\266)475(en)24(\266)475(eral)-265(on)-271(sait)-265(qu'il)-271(existe)-271(une)-265(bijection)]TJ /F8 1 Tf 25.072 0 Td [(F)]TJ /F2 1 Tf 1.0482 0 Td [(lin)24(\266)469(eaire)-271(de)]TJ /F3 1 Tf 4.7048 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.776 14.53 Td [(d)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.614 9.7436 Td [(sur)]TJ /F8 1 Tf 1.6084 0 Td [(E)]TJ /F2 1 Tf 1.0602 0 Td [(et)-265(l'image)]TJ -34.952 -1.8313 Td [(directe)-337(par)]TJ /F8 1 Tf 5.0602 0 Td [(F)]TJ /F2 1 Tf 1.1205 0 Td [(de)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.879 12.685 Td [(d)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 12.843 8.6593 Td [(N)]TJ /F8 1 Tf 1.2771 -0.74699 Td [(\270)]TJ /F2 1 Tf 0.91566 0 Td [(a)-337(la)-331(propri)18(\266)475(et)24(\266)469(e)-337(v)24(oulue.)]TJ /F1 1 Tf -8.8916 -1.6265 Td [(2.)]TJ /F2 1 Tf 1.5602 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0301 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.621 9.59 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.229 6.2858 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.422 9.59 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.114 6.2858 Td [(des)-337(mesures)-337(a)24(y)24(an)30(t)-337(les)-337(propri)24(\266)469(et)24(\266)475(es)-337(du)]TJ /F1 1 Tf 15.825 0 Td [(1.)]TJ /F2 1 Tf -6.9458 -2.3976 Td [(119)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 393 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 392 0 R /ProcSet 2 0 R >> /Contents 391 0 R >> endobj 392 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F15 68 0 R >> endobj 394 0 obj << /Length 26839 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(F)84(ormons)-385(alors)-391(le)-385(pro)-30(duit)]TJ /F8 1 Tf 11.319 0 Td [(X)]TJ /F2 1 Tf 1.2892 0 Td [(des)-385(espaces)-391(mesur)30(\266)469(es)-385(\()]TJ /F8 1 Tf 9.5723 0 Td [(E)-60(;)]TJ /F6 1 Tf 1.241 0 Td [(B)]TJ /F8 1 Tf 0.68675 0 Td [(;)-168(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.026 89.633 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.048 62.22 Td [(\))-385(et)-385(\()]TJ /F8 1 Tf 2.3735 0 Td [(E)-66(;)]TJ /F6 1 Tf 1.241 0 Td [(B)]TJ /F8 1 Tf 0.69277 0 Td [(;)-162(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.483 89.633 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.187 62.22 Td [(\))-385(et)-385(notons)]TJ /F8 1 Tf 5.2711 0 Td [(m)]TJ /F2 1 Tf 1.259 0 Td [(la)]TJ -38.386 -1.1988 Td [(mesure)-337(pro)-30(duit)-337(tensoriel)]TJ /F8 1 Tf 11.006 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.629 87.909 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 17.717 61.021 Td [(\255)]TJ /F8 1 Tf 1.006 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.043 87.909 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.108 61.021 Td [(.)-451(P)24(our)]TJ /F8 1 Tf 3.1687 0 Td [(\256)]TJ /F2 1 Tf 0.98193 0 Td [(application)-337(mesurable)-337(de)]TJ /F8 1 Tf 11.241 0 Td [(E)]TJ /F2 1 Tf 1.1325 0 Td [(dans)]TJ /F8 1 Tf 2.3434 0 Td [(E)]TJ /F2 1 Tf 1.1325 0 Td [(notons)]TJ /F8 1 Tf 3.2289 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.862 87.107 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 59.822 Td [(l'application)-337(\()]TJ /F8 1 Tf 6.1145 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.198 86.193 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.295 59.822 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.052 86.193 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.825 59.822 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(7!)]TJ /F9 1 Tf 1.2771 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.043 86.193 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.084 59.822 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.905 86.193 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.837 59.822 Td [(+)]TJ /F8 1 Tf 1 0 Td [(\256)]TJ /F2 1 Tf 0.63855 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.681 86.193 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.723 59.822 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F2 1 Tf 0.79518 -0.80723 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(X)]TJ /F2 1 Tf 1.241 0 Td [(dans)]TJ /F8 1 Tf 2.3434 0 Td [(X)]TJ /F2 1 Tf 0.90361 0 Td [(.)]TJ -23.392 -2.1928 Td [(Si)]TJ /F8 1 Tf 1.2108 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(est)-385(mesurable)-379(p)-24(ositiv)24(e)-379(sur)]TJ /F8 1 Tf 11.843 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-391(l'in)24(t)24(\266)475(egrale)]TJ /F8 1 Tf 5.3554 0 Td [(I)]TJ /F2 1 Tf 0.86747 0 Td [(=)]TJ /F9 1 Tf 1.1265 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F6 1 Tf 0.8494 0 Td [(\261)]TJ /F8 1 Tf 0.74699 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.078 82.254 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.536 57.629 Td [(dm)]TJ /F2 1 Tf 1.7771 0 Td [(v)54(aut,)-385(d'apr)18(\265)475(es)-379(le)-379(th)18(\266)475(eor)18(\265)475(eme)]TJ -27.982 -2.4096 Td [(de)-475(F)84(ubini,)]TJ /F9 1 Tf 5.0602 1.3614 Td [(Z)]TJ 1 0.048191 Td [(\265)]TJ 0.73494 -0.048191 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F9 1 Tf 0.59639 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.112 79.607 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.036 55.22 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.974 79.607 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.789 55.22 Td [(+)]TJ /F8 1 Tf 1 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.75 79.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.675 55.22 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F8 1 Tf 0.62048 -0.80723 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.612 79.607 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.602 55.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.716 79.607 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.072 55.22 Td [(\))]TJ /F9 1 Tf 0.39157 1.4096 Td [(\266)]TJ /F8 1 Tf 0.90361 -1.4096 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.302 79.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.042 55.22 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.078 79.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.289 55.22 Td [(\).)-873(Compte)-481(ten)24(u)-475(de)-475(l'in)24(v)48(ariance)-475(par)]TJ -22.958 -2.4157 Td [(translation)-319(de)]TJ /F8 1 Tf 6.3675 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18 76.159 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.09 52.804 Td [(,)-319(l'in)24(t)24(\266)469(egrale)-319(par)-319(rapp)-30(ort)-325(\265)500(a)]TJ /F8 1 Tf 11.44 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.353 76.159 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.235 52.804 Td [(s')24(\266)469(ecrit)-319(encore)]TJ /F9 1 Tf 6.0361 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.845 76.159 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.114 52.804 Td [(;)-162(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.707 76.159 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.639 52.804 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.233 76.159 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.108 52.804 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.345 76.159 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.578 52.804 Td [(\))-325(et)-319(donc)]TJ /F8 1 Tf -35.247 -2.3253 Td [(I)]TJ /F2 1 Tf 0.79518 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3675 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)-277(dm)]TJ /F2 1 Tf 2.1627 0 Td [(.)-445(Autremen)18(t)-331(dit)]TJ /F8 1 Tf 7.3434 0 Td [(m)]TJ /F2 1 Tf 1.2108 0 Td [(est)-337(in)24(v)54(arian)24(te)-337(par)-337(l'action)-331(de)]TJ /F8 1 Tf 12.88 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.31 72.03 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.934 50.479 Td [(.)]TJ -27.602 -2.0301 Td [(En)-367(\266)469(ec)24(hangean)24(t)-397(les)-397(r)0(^)500(oles)-397(des)-403(facteurs)-397(de)]TJ /F8 1 Tf 17.789 0 Td [(X)]TJ /F2 1 Tf 0.90964 0 Td [(,)-409(on)-397(mon)24(tre)-397(de)-397(m)24(^)475(eme)-397(que,)-415(p)-30(our)]TJ /F8 1 Tf 14.614 0 Td [(\257)]TJ /F2 1 Tf 1.0121 0 Td [(application)]TJ -34.325 -1.1988 Td [(mesurable)-331(de)]TJ /F8 1 Tf 6.0663 0 Td [(E)]TJ /F2 1 Tf 1.1265 0 Td [(dans)]TJ /F8 1 Tf 2.3434 0 Td [(E)]TJ /F2 1 Tf 0.79518 0 Td [(,)]TJ /F8 1 Tf 0.60843 0 Td [(m)]TJ /F2 1 Tf 1.2169 0 Td [(est)-337(in)24(v)54(arian)24(te)-337(par)-337(l'action)-331(de)]TJ /F8 1 Tf 12.88 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.431 67.4 Td [(\257)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.867 47.25 Td [(:)-277(\()]TJ /F8 1 Tf 0.93976 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.776 68.202 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 33.663 47.25 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.629 68.202 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.193 47.25 Td [(\))]TJ /F6 1 Tf 0.66867 0 Td [(7!)]TJ /F2 1 Tf 1.2771 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.526 68.202 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.602 47.25 Td [(+)]TJ /F8 1 Tf 1 0 Td [(\257)]TJ /F2 1 Tf 0.62048 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.931 68.202 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.693 47.25 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.681 68.202 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.614 47.25 Td [(\).)]TJ -38.283 -1.5783 Td [(P)30(osons)-367(alors)]TJ /F8 1 Tf 5.7831 0 Td [(T)]TJ /F2 1 Tf 1.0542 0 Td [(=)]TJ /F8 1 Tf 1.1024 0 Td [(T)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.828 65.142 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.789 45.671 Td [(p)-24(our)]TJ /F8 1 Tf 2.3916 0 Td [(\256)]TJ /F2 1 Tf 0.96988 0 Td [(=)-325(Id)-361(et)]TJ /F8 1 Tf 3.5783 0 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(=)]TJ /F8 1 Tf 1.1024 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.25 65.142 Td [(\257)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.53 45.671 Td [(p)-30(our)]TJ /F8 1 Tf 2.3976 0 Td [(\257)]TJ /F2 1 Tf 0.94579 0 Td [(=)]TJ /F6 1 Tf 1.1024 0 Td [(\241)]TJ /F2 1 Tf 0.77711 0 Td [(Id)0(.)-530(P)24(ar)-361(comp)-30(osition,)-373(la)-361(mesure)]TJ /F8 1 Tf -25.422 -1.2048 Td [(m)]TJ /F2 1 Tf 1.2711 0 Td [(est)-397(in)24(v)54(arian)24(te)-397(par)]TJ /F8 1 Tf 8.1687 0 Td [(V)]TJ /F2 1 Tf 1.1807 0 Td [(=)]TJ /F8 1 Tf 1.1566 0 Td [(T)]TJ /F6 1 Tf 0.98795 0 Td [(\261)]TJ /F8 1 Tf 0.75904 0 Td [(U)]TJ /F6 1 Tf 1.0542 0 Td [(\261)]TJ /F8 1 Tf 0.76506 0 Td [(T)]TJ /F2 1 Tf 0.72289 0 Td [(.)-626(Or,)-409(on)-391(v)48(\266)469(eri\257e)-391(ais)24(\266)469(emen)30(t)]TJ /F8 1 Tf 11.422 0 Td [(V)]TJ /F2 1 Tf 0.80723 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.5 64.228 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 34.867 44.467 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.353 64.228 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.398 44.467 Td [(\))-379(=)-379(\()]TJ /F6 1 Tf 2.3133 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.328 64.228 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.572 44.467 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.517 64.228 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.867 44.467 Td [(\).)-632(On)]TJ -36.536 -1.7108 Td [(en)-433(conclut,)-457(p)-30(our)]TJ /F8 1 Tf 7.8012 0 Td [(f)]TJ /F2 1 Tf 1.0361 0 Td [(mesurable)-433(p)-30(ositiv)30(e)]TJ /F9 1 Tf 8.6325 1.3615 Td [(Z)]TJ /F8 1 Tf 1 -1.3615 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.293 61.78 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.645 42.756 Td [(;)-162(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.147 61.78 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.169 42.756 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.06 61.78 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.367 42.756 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.914 61.78 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.898 42.756 Td [(\))-445(=)]TJ /F9 1 Tf 2.0542 1.3615 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3615 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.371 61.78 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.801 42.756 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.552 61.78 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.102 42.756 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.129 61.78 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 42.301 42.756 Td [(;)-162(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.991 61.78 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.825 42.756 Td [(\).)]TJ -38.494 -1.7651 Td [(Appliquan)24(t)-373(cela)-379(p)-30(our)]TJ /F8 1 Tf 9.7349 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.793 59.254 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.91 40.991 Td [(;)-168(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.655 59.254 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.44 40.991 Td [(\))-343(=)-349(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.845 58.443 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.506 40.991 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.155 59.254 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.747 40.991 Td [(\))]TJ /F6 1 Tf 0.63855 0 Td [(\242)]TJ /F2 1 Tf 0.53012 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.026 58.443 Td [(B)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.163 40.991 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.388 59.254 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.639 40.991 Td [(\))-373(o)-30(\265)530(u)]TJ /F8 1 Tf 2.1928 0 Td [(A)]TJ /F2 1 Tf 1.1205 0 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(B)]TJ /F2 1 Tf 1.1867 0 Td [(son)24(t)-379(des)-373(parties)-373(b)-30(or)24(\266)475(eliennes)]TJ -27.012 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(E)]TJ /F2 1 Tf 1.1325 0 Td [(on)-331(obtien)24(t)]TJ /F8 1 Tf 4.8072 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.293 57.538 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.759 39.792 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.172 57.538 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.404 39.792 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F2 1 Tf 0.81325 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.336 57.538 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.711 39.792 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F6 1 Tf 0.60843 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.664 57.538 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.904 39.792 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\).)]TJ -21.765 -1.5783 Td [(Comme)]TJ /F8 1 Tf 3.6687 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.138 55.28 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.5 38.214 Td [(est)]TJ /F8 1 Tf 1.5663 0 Td [(\276)]TJ /F2 1 Tf 0.60843 0 Td [(-\257nie,)-343(on)-337(p)-30(eut)-343(prendre)]TJ /F8 1 Tf 10.169 0 Td [(A)]TJ /F2 1 Tf 1.0843 0 Td [(telle)-343(que)]TJ /F8 1 Tf 4.0422 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.284 55.28 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.127 38.214 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-343(soit)-337(\257ni)-343(non)-343(n)24(ul.)-457(Il)-337(vien)18(t)-337(alors,)]TJ -24.934 -1.7771 Td [(p)-30(our)-331(tout)-337(b)-30(or)24(\266)475(elien)]TJ /F8 1 Tf 8.3434 0 Td [(B)]TJ /F2 1 Tf 1.1386 0 Td [(:)]TJ /F8 1 Tf 0.72289 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.491 52.728 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.928 36.436 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(K)]TJ /F6 1 Tf 1.1446 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.017 52.728 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.651 36.436 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-337(a)30(v)18(ec)]TJ /F8 1 Tf 2.9217 0 Td [(K)]TJ /F2 1 Tf 1.1988 0 Td [(=)]TJ /F8 1 Tf 1.1747 0.6747 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 42.966 53.625 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.536 37.111 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(\241)]TJ /F8 1 Tf 0.77711 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ ET 0.504 w 348.34 438.48 m 392.54 438.48 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.651 35.75 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.681 51.573 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.807 35.75 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ 1.3494 0.68675 Td [(;)-331(on)-337(a)-331(donc)]TJ /F8 1 Tf 5.2229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.379 52.728 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.187 36.436 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(K)-72(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.595 52.728 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.325 36.436 Td [(.)]TJ -37.994 -2.2169 Td [(On)-337(remarquera)-331(d'ailleurs)-337(que)-337(le)-337(cas)]TJ /F8 1 Tf 15.861 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(B)]TJ /F2 1 Tf 1.1446 0 Td [(fournit)]TJ /F8 1 Tf 3.3072 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.94 49.556 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.12 34.22 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.138 49.556 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.151 34.22 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\).)]TJ -29.958 -1.8554 Td [(En)-337(particulier,)-343(si)-343(l'on)-343(prend)-343(p)-30(our)]TJ /F8 1 Tf 14.88 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 30.172 46.9 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.705 32.364 Td [(une)-343(des)-343(mesures)-337(construites)-343(en)]TJ /F1 1 Tf 13.849 0 Td [(1.)]TJ /F2 1 Tf 0.89157 0 Td [(,)-343(le)-343(fait)-343(que)]TJ /F8 1 Tf 5.3614 0 Td [(C)]TJ /F2 1 Tf 1.1265 0 Td [(soit)]TJ -37.602 -1.2048 Td [(b)-30(orn)30(\266)469(ee)-409(d'in)24(t)24(\266)469(erieur)-403(non)-409(vide)-409(assure)]TJ /F8 1 Tf 15.578 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.181 45.185 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.066 31.159 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(C)]TJ /F2 1 Tf 0.78313 0 Td [(\))-409(\257ni)-409(et)-409(non)-403(n)24(ul.)-662(La)-409(seule)-403(mesure)]TJ /F8 1 Tf 15.663 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.931 45.185 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.699 31.159 Td [(a)24(y)24(an)30(t)-409(en)]TJ -35.367 -1.6687 Td [(plus)-331(la)-337(propri)24(\266)469(et)30(\266)469(e)]TJ /F8 1 Tf 7.512 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.638 42.788 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.235 29.491 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(\))-283(=)-277(1)-337(est)-337(alors)]TJ /F8 1 Tf 6.5181 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.638 42.788 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.596 29.491 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(K)]TJ /F6 1 Tf 1.1446 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.888 42.788 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.789 29.491 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.1928 0 Td [(K)]TJ /F2 1 Tf 1.1988 0 Td [(=)]TJ 2.2892 0.6747 Td [(1)]TJ ET 386.71 355.46 m 419.33 355.46 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 32.355 28.804 Td [(m)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.56 41.633 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.518 28.804 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(\))]TJ /F6 1 Tf 0.84337 0.68674 Td [(\242)]TJ ET 512.64 351.25 m 519.84 351.25 l 519.84 359.67 l 512.64 359.67 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 26.822 Td [(De)-325(fa)-6(\273)439(con)-325(g)24(\266)475(en)24(\266)475(erale,)-337(toute)-325(mesure)-331(sur)-325(les)-331(b)-30(or)30(\266)469(eliens)-325(d'un)-331(espace)-325(v)24(ectoriel)-331(de)-325(dimension)-331(\257nie,)]TJ 0 -1.1988 Td [(in)24(v)54(arian)24(te)-409(par)-403(translation)-415(et)-403(prenan)24(t)-409(une)-409(v)48(aleur)-403(\257nie)-409(non)-409(n)24(ulle)-409(sur)-409(un)-409(b)-30(orn)30(\266)469(e)-409(d'in)24(t)24(\266)469(erieur)]TJ 0 -1.1988 Td [(non)-481(vide,)-518(s'app)-30(elle)-487(une)-481(mesure)-481(de)-481(Leb)-36(esgue.)-891(Ces)-481(mesures)-481(son)24(t)-487(donc)-481(d)30(\266)469(e\257nies)]TJ /F14 1 Tf 35.681 0 Td [(a)-493(priori)]TJ /F2 1 Tf -35.681 -1.1988 Td [(\265)499(a)-361(un)-367(co)-30(e\261cien)24(t)-361(de)-367(prop)-30(ortionnalit)30(\266)469(e)-367(pr)24(\265)475(es.)-536(Il)-361(est)-367(fr)24(\266)469(equen)24(t)-367(que)-367(l'on)-361(donne)-367(une)-361(condition)-367(de)]TJ 0 -1.2048 Td [(normalisation)-253(p)-30(our)-259(pr)30(\266)469(eciser)-253(la)-253(mesure)-259(de)-253(Leb)-36(esgue)-253(consid)24(\266)475(er)18(\266)475(ee.)-421(Ainsi,)-277(si)]TJ /F8 1 Tf 30.819 0 Td [(E)]TJ /F2 1 Tf 1.0783 0 Td [(=)]TJ /F3 1 Tf 1.0542 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.819 32.107 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.548 22.021 Td [(,)-271(on)-259(exigera)]TJ /F8 1 Tf -34.217 -1.1988 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(\))-277(=)-283(1)-331(p)-30(our)]TJ /F8 1 Tf 4.9157 0 Td [(C)]TJ /F2 1 Tf 1.0663 0 Td [(=)-277([0;)-168(1])]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.5 30.392 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.301 20.822 Td [(l'h)24(yp)-30(ercub)-30(e)-337(unit)24(\266)475(e.)-451(La)-337(mesure)-337(est)-337(alors)-337(le)-337(pro)-30(duit)-331(tensoriel)-337(des)]TJ -11.97 -1.1988 Td [(mesures)-331(de)-337(Leb)-30(esgue)-337(sur)-337(c)18(hacun)-337(de)-331(facteurs.)]TJ 0 -1.5783 Td [(En\257n)-319(il)-325(est)-319(fr)18(\266)475(equen)24(t)-325(que)-319(l'on)-325(remplace)-319(cette)-319(mesure)-325(d)24(\266)475(e\257nie)-325(sur)-319(les)-325(b)-24(or)24(\266)469(eliens)-319(par)-319(la)-319(mesure)]TJ 0 -1.2048 Td [(compl)24(\266)475(et)24(\266)469(ee)-337(en)-331(a)-54(joutan)18(t)-337(les)-337(parties)-331(n)24(\266)469(egligeables)-337(\265)499(a)-331(la)-337(tribu)-331(des)-337(mesurables.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 11.565 Td [(VI)-35(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 13.864 Td [(2.)-283(Le)-391(th)24(\266)548(eor)18(\265)548(eme)-385(de)-385(c)30(hangemen)24(t)-385(de)-385(v)60(ariables)]TJ /F2 1 Tf -4.0301 -1.5783 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0181 0 Td [(U)]TJ /F2 1 Tf 1.1205 0 Td [(un)-331(ouv)24(ert)-331(de)]TJ /F3 1 Tf 5.8735 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.991 18.168 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.934 12.286 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(F)]TJ /F2 1 Tf 1.1084 0 Td [(une)-325(application)-331(de)-331(classe)]TJ /F8 1 Tf 11.157 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.569 18.168 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.596 12.286 Td [(de)]TJ /F8 1 Tf 1.3253 0 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(dans)]TJ /F3 1 Tf 2.3373 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.086 18.168 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.639 12.286 Td [(.)-445(Un)-331(p)-24(oin)24(t)]TJ /F8 1 Tf 4.9578 0 Td [(x)]TJ /F2 1 Tf 0.89759 0 Td [(de)]TJ /F8 1 Tf -38.163 -1.1988 Td [(U)]TJ /F2 1 Tf 1.1446 0 Td [(de)-349(co)-30(ordonn)18(\266)475(ees)-355(\()]TJ /F8 1 Tf 7.4036 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 20.681 15.65 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.904 11.087 Td [(;)-162(:)-168(:)-168(:)-168(;)-168(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.328 15.65 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.241 11.087 Td [(\))-355(a)-349(donc)-355(p)-30(our)-349(image)-355(un)-355(p)-30(oin)24(t)]TJ /F8 1 Tf 13.38 0 Td [(y)]TJ /F2 1 Tf 0.87952 0 Td [(de)-349(co)-36(ordonn)24(\266)469(ees)-349(\()]TJ /F8 1 Tf 7.4096 0 Td [(y)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 57.81 15.65 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.849 11.087 Td [(;)-168(:)-162(:)-168(:)-168(;)-168(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.336 15.65 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.108 11.087 Td [(\))]TJ -38.777 -1.2048 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.3434 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.681 13.926 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0181 9.8822 Td [(=)]TJ /F8 1 Tf 1.3012 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.69 13.926 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.295 9.8822 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.543 13.926 Td [(1)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.705 9.8822 Td [(;)-168(:)-168(:)-168(:)-162(;)-168(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.181 13.926 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.042 9.8822 Td [(\).)-891(En)-481(tout)-481(p)-30(oin)24(t)]TJ /F8 1 Tf 8.3434 0 Td [(x)]TJ /F2 1 Tf 1.0542 0 Td [(de)]TJ /F8 1 Tf 1.4819 0 Td [(U)]TJ /F2 1 Tf 1.2711 0 Td [(la)-481(matrice)-487(jacobienne)-481(de)]TJ /F8 1 Tf 11.566 0 Td [(F)]TJ /F2 1 Tf 1.2651 0 Td [(est,)-518(par)]TJ -35.693 -1.7229 Td [(d)30(\266)469(e\257nition,)-295(la)-283(matrice)-283(de)-277(co)-36(e\261cien)24(ts)]TJ /F8 1 Tf 15.717 0.68072 Td [(@)-60(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.888 12.435 Td [(i)]TJ ET 251.5 100.58 m 270.43 100.58 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.042 7.4786 Td [(@)-54(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.767 10.487 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.741 8.1593 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\),)-295(c'est)-283(la)-283(matrice)-283(de)-283(l'application)-283(d)24(\266)475(eriv)48(\266)475(ee)]TJ /F8 1 Tf 18.096 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.94 12.271 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 42.867 8.1593 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\).)]TJ -38.494 -1.8735 Td [(Son)-403(d)24(\266)475(eterminan)24(t)-403(est)-409(le)-409(d)30(\266)469(eterminan)24(t)-403(jacobien)-403(det)]TJ /F8 1 Tf 21.735 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.853 9.59 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.133 6.2858 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)-656(Lorsque)-397(ce)-409(d)30(\266)469(eterminan)24(t)-403(est)-409(non)]TJ /F5 1 Tf -23.759 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(120)]TJ ET endstream endobj 396 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 395 0 R /ProcSet 2 0 R >> /Contents 394 0 R >> endobj 395 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F14 43 0 R >> endobj 397 0 obj << /Length 18136 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(n)30(ul)-367(en)-367(un)-367(p)-24(oin)24(t)]TJ /F8 1 Tf 7.2048 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-373(le)-367(th)18(\266)475(eor)24(\265)469(eme)-367(d'in)24(v)24(ersion)-367(lo)-30(cale)-367(a\261rme)-367(que)]TJ /F8 1 Tf 19.271 0 Td [(F)]TJ /F2 1 Tf 1.1446 0 Td [(est)-373(injectiv)30(e)-367(au)-367(v)24(oisinage)]TJ -28.193 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(x)]TJ /F2 1 Tf 0.90361 0 Td [(et)-337(admet)-337(une)-331(application)-337(r)24(\266)469(ecipro)-24(que)-313(\266)475(egalemen)24(t)-337(de)-331(classe)]TJ /F8 1 Tf 25.006 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.733 87.909 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.139 61.021 Td [(au)-337(v)24(oisinage)-337(de)]TJ /F8 1 Tf 7.0121 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)]TJ -37.56 -1.6868 Td [(En)-403(particulier,)-421(si)]TJ /F8 1 Tf 7.9157 0 Td [(F)]TJ /F2 1 Tf 1.1807 0 Td [(est)-409(injectiv)30(e)-409(et)-403(de)-403(d)30(\266)469(eterminan)24(t)-403(jacobien)-403(jamais)-403(n)24(ul,)-421(son)-403(image)]TJ /F8 1 Tf 28.09 0 Td [(V)]TJ /F2 1 Tf 1.1988 0 Td [(=)]TJ /F8 1 Tf -38.386 -1.1988 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-337(est)-349(un)-337(ouv)18(ert)-337(et)]TJ /F8 1 Tf 8.0361 0 Td [(F)]TJ /F2 1 Tf 1.1205 0 Td [(d)30(\266)469(e\257nit)-337(une)-343(bijection)-343(de)]TJ /F8 1 Tf 10.44 0 Td [(U)]TJ /F2 1 Tf 1.1325 0 Td [(sur)]TJ /F8 1 Tf 1.6807 0 Td [(V)]TJ /F2 1 Tf 1.1446 0 Td [(don)24(t)-337(la)-343(bijection)-343(r)30(\266)469(ecipro)-24(que)-343(est)]TJ -25.518 -1.2048 Td [(de)-331(classe)]TJ /F8 1 Tf 4.1205 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 14.655 82.064 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.687 56.93 Td [(.)-445(On)-337(dit)-343(alors)-337(que)]TJ /F8 1 Tf 8.2108 0 Td [(F)]TJ /F2 1 Tf 1.1145 0 Td [(est)-337(un)-337(di\256)24(\266)475(eomorphisme)-343(de)]TJ /F8 1 Tf 11.825 0 Td [(U)]TJ /F2 1 Tf 1.1265 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(V)]TJ /F2 1 Tf 0.80723 0 Td [(.)]TJ /F1 1 Tf -30.114 -2.0301 Td [(Remarque.)]TJ /F2 1 Tf 6.9096 0 Td [(Il)-493(n'y)-500(a)-493(pas)-506(de)-493(relation)-499(d'implication)-500(en)30(tre)-500(les)-475(\266)475(enonc)24(\266)469(es)-493(")]TJ /F8 1 Tf 26.145 0 Td [(F)]TJ /F2 1 Tf 1.2771 0 Td [(injectif")-500(et)]TJ -34.331 -1.1988 Td [(")0(det)]TJ /F8 1 Tf 2.0542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.69 77.443 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4518 53.701 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.66868 0 Td [(6)]TJ /F2 1 Tf [(=)-277(0".)-439(On)-307(p)-30(ourra,)-313(par)-313(exemple)-307(consid)24(\266)475(erer)-313(l'application)]TJ /F8 1 Tf 23.922 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 52.517 77.443 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.452 53.701 Td [(de)]TJ /F3 1 Tf 1.3072 0 Td [(R)]TJ /F2 1 Tf 1.0301 0 Td [(dans)]TJ /F3 1 Tf 2.3133 0 Td [(R)]TJ /F2 1 Tf 1.0301 0 Td [(qui)]TJ -37.801 -1.1988 Td [(v)54(\266)469(eri\257e)-373(le)-373(premier)-379(mais)-373(pas)-373(le)-379(second)-379(et)-373(l'application)]TJ /F8 1 Tf 23.331 0 Td [(z)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.3434 0 Td [(z)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.888 75.719 Td [(3)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.187 52.503 Td [(de)]TJ /F3 1 Tf 1.3735 0 Td [(C)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 49.06 75.719 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.114 52.503 Td [(dans)]TJ /F3 1 Tf 2.3795 0 Td [(C)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.69 75.719 Td [(\244)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.048 52.503 Td [(qui)-373(v)48(\266)475(eri\257e)-373(le)]TJ -33.717 -1.2048 Td [(second)-337(mais)-331(pas)-337(le)-343(premier.)]TJ /F15 1 Tf 0 -2.0241 Td [(Th)]TJ 1.4036 0.078316 Td [(\266)]TJ -0.0060239 -0.078316 Td [(eor)]TJ 1.8012 0.078316 Td [(\265)]TJ -0.0060244 -0.078316 Td [(eme.)]TJ /F12 1 Tf 2.7771 0 Td [(\()]TJ /F1 1 Tf 0.38554 0 Td [(de)-427(c)30(hangemen)24(t)-421(de)-421(v)60(ariables)]TJ /F12 1 Tf 13.946 0 Td [(\))-1006(Soien)24(t)]TJ /F8 1 Tf 4.4518 0 Td [(U)]TJ /F12 1 Tf 1.1566 0 Td [(et)]TJ /F8 1 Tf 1.1988 0 Td [(V)]TJ /F12 1 Tf 1.1687 0 Td [(des)-367(ouv)24(erts)-373(de)]TJ /F3 1 Tf 6.6627 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.664 71.099 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 41.904 49.274 Td [(m)24(unis)]TJ -36.572 -1.2048 Td [(de)-325(la)-325(mesure)-331(de)-325(Leb)-36(esgue,)]TJ /F8 1 Tf 11.747 0 Td [(F)]TJ /F12 1 Tf 1.1084 0 Td [(un)-325(di\256)24(\266)469(eomorphisme)-325(de)-331(classe)]TJ /F8 1 Tf 13.018 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 45.784 69.383 Td [(1)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 32.771 48.069 Td [(de)]TJ /F8 1 Tf 1.3253 0 Td [(U)]TJ /F12 1 Tf 1.1145 0 Td [(sur)]TJ /F8 1 Tf 1.6687 0 Td [(V)]TJ /F12 1 Tf 1.1325 0 Td [(et)]TJ /F8 1 Tf 1.1626 0 Td [(f)]TJ /F12 1 Tf 0.92169 0 Td [(mesurable)]TJ -34.765 -1.1988 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(V)]TJ /F12 1 Tf 1.1386 0 Td [(\265)499(a)-331(v)48(aleurs)-337(dans)]TJ ET 0.504 w 175.61 570.09 m 184.25 570.09 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 14.693 46.87 Td [(R)]TJ /F12 1 Tf 1.0542 0 Td [(ou)-337(dans)-331(un)-337(espace)-331(v)18(ectoriel)-331(de)-337(dimension)-337(\257nie.)-445(Alors)]TJ -9.4157 -1.6867 Td [(la)-343(fonction)]TJ /F2 1 Tf 4.994 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\))]TJ /F6 1 Tf 0.62048 0 Td [(\242)]TJ /F2 1 Tf 0.50602 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.414 65.254 Td [(0)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.687 45.183 Td [(est)-349(mesurable)-349(et)-343(elle)-355(est)-349(in)24(t)24(\266)475(egrable)-349(sur)]TJ /F8 1 Tf 17.205 0 Td [(U)]TJ /F12 1 Tf 1.1325 0 Td [(si)-349(et)-349(seulemen)30(t)-349(si)]TJ /F8 1 Tf 7.8735 0 Td [(f)]TJ /F12 1 Tf -38.566 -1.1988 Td [(est)-337(in)24(t)24(\266)475(egrable)-337(sur)]TJ /F8 1 Tf 7.8795 0 Td [(V)]TJ /F12 1 Tf 0.80723 0 Td [(.)]TJ -8.6867 -1.6867 Td [(En)-331(outre,)-343(si)]TJ /F8 1 Tf 5.4699 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)-337(in)24(t)24(\266)469(egrable)-331(ou)-337(\265)500(a)-331(v)48(aleurs)-337(p)-30(ositiv)30(es)-337(on)-337(a)]TJ /F9 1 Tf 4.5602 -0.29518 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.103 56.849 Td [(V)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.542 40.641 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.267 56.849 Td [(U)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.217 40.641 Td [(f)]TJ /F9 1 Tf 0.59639 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.80723 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.457 58.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.651 40.641 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F12 1 Tf -27.114 -2.1928 Td [(o)-24(\265)524(u)]TJ /F8 1 Tf 1.3855 0 Td [(dx)]TJ /F12 1 Tf 1.4277 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(dy)]TJ /F12 1 Tf 1.3795 0 Td [(d)24(\266)475(esignen)18(t)-337(la)-331(mesure)-337(de)-337(Leb)-30(esgue.)]TJ /F1 1 Tf -5.3614 -2.0301 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Comme)-415(dans)-415(le)-421(cas)-415(d'une)-421(v)54(ariable)-421(\(cf.)]TJ /F1 1 Tf 17.807 0 Td [(VI.2.6.)]TJ /F2 1 Tf 3.4157 0 Td [(\))-421(il)-415(su\261t)-421(de)-415(prouv)30(er)]TJ -30.042 -1.1988 Td [(le)-391(th)18(\266)475(eor)18(\265)475(eme)-391(lorsque)]TJ /F8 1 Tf 8.988 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(est)-397(b)-30(or)30(\266)469(elienne.)-614(En)-391(e\256et)-391(cela)-397(en)30(tra)108(^)385(\263ne)-391(imm)24(\266)469(ediatemen)30(t)-391(que)-397(si)]TJ /F8 1 Tf 26.97 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(est)]TJ -37.934 -1.2048 Td [(n)30(\266)469(egligeable)-265(\()]TJ /F8 1 Tf 5.4337 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\))]TJ /F6 1 Tf 0.46988 0 Td [(\242)]TJ /F8 1 Tf 0.36145 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.405 49.271 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.807 34.015 Td [(l'est)-265(aussi)-265(et,)-283(par)-265(addition,)-283(cela)-265(p)-30(ermet)-271(de)-265(conclure)-265(p)-30(our)]TJ /F8 1 Tf 24.428 0 Td [(f)]TJ /F2 1 Tf 0.86145 0 Td [(mesurable)]TJ -34.765 -1.1988 Td [(au)-331(sens)-337(de)-331(Leb)-36(esgue.)]TJ 0 -1.6867 Td [(T)78(out)-283(d'ab)-30(ord)-283(remarquons)-277(que)]TJ /F8 1 Tf 13.187 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(,)-265(\266)475(etan)18(t)-277(de)-283(classe)]TJ /F8 1 Tf 7.1145 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.922 45.142 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.651 31.129 Td [(,)-289(est)-283(con)24(tin)24(ue)-283(donc)-277(b)-30(or)24(\266)475(elienne)-283(et)-277(det)]TJ /F8 1 Tf 15.783 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 45.142 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 29.93 Td [(est)-415(\266)469(egalemen)30(t)-445(con)24(tin)24(ue)-439(donc)-445(b)-30(or)30(\266)469(elienne.)-765(Il)-445(s'ensuit)-439(que,)-469(p)-30(our)]TJ /F8 1 Tf 28.157 0 Td [(f)]TJ /F2 1 Tf 1.0361 0 Td [(b)-30(or)30(\266)469(elienne,)-469(la)-439(fonction)]TJ /F8 1 Tf -29.193 -1.2048 Td [(g)]TJ /F2 1 Tf 0.78916 0 Td [(=)-277(\()]TJ /F8 1 Tf 1.4458 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)-222(j)]TJ /F2 1 Tf 0.93976 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.095 41.702 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 13.62 28.726 Td [(j)]TJ /F2 1 Tf 0.61446 0 Td [(est)-307(\266)469(egalemen)24(t)-337(b)-24(or)24(\266)469(elienne)-331(donc)-337(mesurable)-337(au)-331(sens)-337(de)-331(Leb)-36(esgue.)]TJ -8.9036 -1.6867 Td [(Observ)24(ons)-355(\266)469(egalemen)30(t)-385(qu'il)-385(su\261t)-385(de)-379(prouv)24(er)-385(l')30(\266)469(egalit)24(\266)469(e)-385(p)-30(our)]TJ /F8 1 Tf 25.777 0 Td [(f)]TJ /F2 1 Tf 0.98193 0 Td [(\265)499(a)-379(v)54(aleurs)-385(p)-30(ositiv)24(es.)-590(En)-379(e\256et)]TJ -26.759 -1.1988 Td [(celle-ci,)-421(appliqu)24(\266)475(ee)-409(\265)500(a)]TJ /F6 1 Tf 9.0904 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.68072 0 Td [(ou)]TJ /F6 1 Tf 1.4639 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F2 1 Tf 0.90361 0 Td [(en)30(tra)102(^)391(\263ne)-403(l')24(\266)475(equiv)48(alence)-409(de)-403(l'in)24(t)24(\266)475(egrabilit)24(\266)475(e)-409(de)]TJ /F8 1 Tf 18.861 0 Td [(f)]TJ /F2 1 Tf 1 0 Td [(sur)]TJ /F8 1 Tf 1.747 0 Td [(V)]TJ /F2 1 Tf 1.2108 0 Td [(et)-403(de)]TJ -36.928 -1.1988 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\))-174(det)]TJ /F8 1 Tf 2.1145 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.017 35.849 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.084 24.641 Td [(sur)]TJ /F8 1 Tf 1.6566 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(.)-433(P)24(ar)-313(ailleurs)-313(cette)-283(\266)469(egalit)24(\266)469(e)-313(p)-30(our)]TJ /F8 1 Tf 13.916 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(p)-24(ositiv)24(e)-319(se)-313(g)30(\266)469(en)30(\266)469(eralise)-313(imm)24(\266)469(ediatemen)24(t)]TJ -23.036 -1.2048 Td [(au)-385(cas)-385(de)]TJ /F8 1 Tf 4.5482 0 Td [(f)]TJ /F2 1 Tf 0.98193 0 Td [(in)24(t)24(\266)469(egrable)-385(\265)499(a)-385(v)54(aleurs)-391(dans)]TJ ET 266.04 290.08 m 274.68 290.08 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 22.259 23.436 Td [(R)]TJ /F2 1 Tf 1.1084 0 Td [(en)-385(d)24(\266)475(ecomp)-30(osan)24(t)]TJ /F8 1 Tf 7.4458 0 Td [(f)]TJ /F2 1 Tf 0.98193 0 Td [(en)]TJ /F8 1 Tf 1.3855 0 Td [(f)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 48.181 33.331 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.59 23.436 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.681 33.331 Td [(\241)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.175 23.436 Td [(puis)-385(au)-391(cas)-385(de)]TJ /F8 1 Tf 6.7229 0 Td [(f)]TJ /F2 1 Tf -38.566 -1.1988 Td [(in)24(t)24(\266)475(egrable)-337(\265)500(a)-331(v)54(aleurs)-343(v)24(ectorielles)-337(en)-331(appliquan)24(t)-337(le)-337(r)30(\266)469(esultat)-337(obten)18(u)-331(aux)-343(co)-30(ordonn)24(\266)469(ees)-337(de)]TJ /F8 1 Tf 37.247 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(.)]TJ -37.843 -2.0301 Td [(La)-337(d)24(\266)475(emonstration)-337(v)48(a)-331(se)-337(faire)-337(par)-337(r)30(\266)469(ecurrence)-337(sur)]TJ /F8 1 Tf 21.114 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(.)]TJ -21.717 -1.6867 Td [(Le)-337(cas)]TJ /F8 1 Tf 3.0723 0 Td [(n)]TJ /F2 1 Tf 0.87952 0 Td [(=)-283(1)-331(a)-331(d)24(\266)469(ej)0(\265)500(a)-307(\266)469(et)30(\266)469(e)-337(vu)-337(\(cf.)]TJ /F1 1 Tf 9.759 0 Td [(VI.2.6.)]TJ /F2 1 Tf 3.4157 0 Td [(\).)]TJ -17.127 -1.6867 Td [(Supp)-24(osons)-385(donc)]TJ /F8 1 Tf 7.4157 0 Td [(n)]TJ /F11 1 Tf 0.95181 0 Td [(>)]TJ /F2 1 Tf 1.1386 0 Td [(2)-379(et)-379(le)-391(th)24(\266)475(eor)18(\265)475(eme)-355(\266)469(etabli)-385(p)-30(our)-379(des)-385(di\256)24(\266)475(eomorphismes)-385(en)30(tre)-385(ouv)24(erts)-385(de)]TJ /F3 1 Tf -9.506 -1.1988 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 22.969 Td [(n)]TJ /F10 1 Tf 0.7069 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6205 15.635 Td [(.)]TJ /F1 1 Tf -2.2892 -2.3735 Td [(1.)-512(Un)-385(cas)-385(sp)-6(\266)548(ecial.)]TJ /F2 1 Tf 0 -1.6867 Td [(Supp)-24(osons)-343(que,)-349(p)-30(our)-343(un)-343(couple)-343(\()]TJ /F8 1 Tf 14.416 0 Td [(i;)-168(j)]TJ /F2 1 Tf 1.259 0 Td [(\),)-349(on)-337(ait)]TJ /F8 1 Tf 3.9157 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.578 16.349 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.898 11.575 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-295(=)]TJ /F8 1 Tf 1.747 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.75 16.349 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.596 11.575 Td [(.)-463(P)24(our)]TJ /F8 1 Tf 3.1868 0 Td [(z)]TJ /F6 1 Tf 0.79518 0 Td [(2)]TJ /F3 1 Tf 0.95783 0 Td [(R)]TJ /F2 1 Tf 1.0602 0 Td [(on)-343(p)-30(eut)-343(alors)-343(d)24(\266)469(e\257nir)]TJ -30.265 -1.1988 Td [(une)-367(section)]TJ /F8 1 Tf 5.2952 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.181 14.633 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.133 10.376 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(U)]TJ /F2 1 Tf 1.1566 0 Td [(obten)18(ue)-367(en)-361(\257xan)24(t)-367(la)-367(co)-36(ordonn)24(\266)469(ee)]TJ /F8 1 Tf 14.452 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.474 14.633 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.44 10.376 Td [(\266)469(egale)-367(\265)500(a)]TJ /F8 1 Tf 3.4277 0 Td [(z)]TJ /F2 1 Tf 0.51205 0 Td [(.)-542(L'image)-367(de)]TJ /F8 1 Tf 6.0121 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.776 14.633 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.898 10.376 Td [(par)]TJ /F8 1 Tf 1.8133 0 Td [(F)]TJ /F2 1 Tf -38.38 -1.1988 Td [(est)-307(\266)469(evidemmen)24(t)-337(la)-331(section)]TJ /F8 1 Tf 11.627 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.103 12.918 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.337 9.1774 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(V)]TJ /F2 1 Tf 1.1386 0 Td [(obten)18(ue)-331(en)-337(\257xan)30(t)-337(la)-337(co)-30(ordonn)24(\266)469(ee)]TJ /F8 1 Tf 14.289 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.931 12.918 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.229 9.1774 Td [(\266)475(egale)-337(\265)500(a)]TJ /F8 1 Tf 3.3614 0 Td [(z)]TJ /F2 1 Tf 0.50602 0 Td [(.)]TJ -34.765 -1.6867 Td [(Ces)-439(sections)-439(s'iden)30(ti\257en)24(t)-439(naturellemen)24(t)-439(\265)500(a)-439(des)-439(ouv)18(erts)-439(de)]TJ /F3 1 Tf 25.633 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.345 11.306 Td [(n)]TJ /F10 1 Tf 0.70689 0 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.693 7.4906 Td [(et)-439(le)-445(di\256)30(\266)469(eomorphisme)]TJ /F8 1 Tf 10.018 0 Td [(F)]TJ /F2 1 Tf -38.38 -1.2048 Td [(induit)-337(alors)-337(un)-337(di\256)30(\266)469(eomorphisme)-337(partiel)]TJ /F8 1 Tf 17.446 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.509 8.7797 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.211 6.2858 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.534 8.7797 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.024 6.2858 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.905 8.7797 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.747 6.2858 Td [(.)]TJ -5.753 -2.3976 Td [(121)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 399 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 398 0 R /ProcSet 2 0 R >> /Contents 397 0 R >> endobj 398 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 400 0 obj << /Length 25505 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(Plus)-343(pr)24(\266)469(ecis)30(\266)469(emen)30(t,)-349(un)-319(\266)475(el)24(\266)469(emen)24(t)]TJ /F8 1 Tf 13.175 0 Td [(x)]TJ /F2 1 Tf 0.90964 0 Td [(dans)]TJ /F8 1 Tf 2.3494 0 Td [(U)]TJ /F2 1 Tf 1.1325 0 Td [(tel)-343(que)]TJ /F8 1 Tf 3.3193 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.336 88.823 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.5 62.22 Td [(=)]TJ /F8 1 Tf 1.0723 0 Td [(z)]TJ /F2 1 Tf 0.8494 0 Td [(s'iden)30(ti\257e)-349(\265)500(a)-337(un)-343(couple)-343(\()]TJ /F8 1 Tf 10.289 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.647 89.633 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 40.56 62.22 Td [(;)-168(z)]TJ /F2 1 Tf 0.95783 0 Td [(\))-343(o)-24(\265)524(u)]TJ /F8 1 Tf 2.1265 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 63.276 89.633 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 61.021 Td [(est)-403(form)24(\266)475(e)-409(par)-397(les)-403(co)-36(ordonn)24(\266)469(ees)]TJ /F8 1 Tf 13.584 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.888 87.107 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.386 61.021 Td [(p)-24(our)]TJ /F8 1 Tf 2.4277 0 Td [(k)]TJ /F6 1 Tf 0.94578 0 Td [(6)]TJ /F2 1 Tf [(=)]TJ /F8 1 Tf 1.1627 0 Td [(j)]TJ /F2 1 Tf 0.87349 0 Td [(et)]TJ /F8 1 Tf 1.2349 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.5 87.909 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.271 61.021 Td [(2)]TJ /F8 1 Tf 1.0542 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.948 87.107 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.873 61.021 Td [(;)-433(son)-403(image)]TJ /F8 1 Tf 5.5181 0 Td [(y)]TJ /F2 1 Tf 0.92771 0 Td [(par)]TJ /F8 1 Tf 1.8494 0 Td [(F)]TJ /F2 1 Tf 1.1807 0 Td [(s'iden)30(ti\257e)]TJ -35.048 -1.1988 Td [(\266)469(egalemen)30(t)-367(au)-361(couple)-361(\()]TJ /F8 1 Tf 9.6988 0 Td [(y)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.216 86.193 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.807 59.822 Td [(;)-168(z)]TJ /F2 1 Tf 0.95181 0 Td [(\))-361(o)-30(\265)524(u)]TJ /F8 1 Tf 2.1687 0 Td [(y)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.836 86.193 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.054 59.822 Td [(=)]TJ /F8 1 Tf 1.1024 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.198 85.392 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.259 59.822 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.233 86.193 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.5 59.822 Td [(\))-367(est)-367(form)30(\266)469(e)-361(par)-367(le)-361(co)-36(ordonn)24(\266)469(ees)]TJ /F8 1 Tf 13.747 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54 85.392 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.59 59.822 Td [(p)-30(our)]TJ /F8 1 Tf 2.3916 0 Td [(k)]TJ /F6 1 Tf 0.87349 0 Td [(6)]TJ /F2 1 Tf [(=)]TJ /F8 1 Tf 1.1024 0 Td [(i)]TJ /F2 1 Tf 0.70482 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.2048 Td [(y)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.3793 84.478 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5422 58.617 Td [(2)]TJ /F8 1 Tf 1.0783 0 Td [(V)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.741 83.668 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6627 58.617 Td [(.)-680(La)-415(matrice)-415(jacobienne)-415(de)]TJ /F8 1 Tf 12.596 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.345 83.668 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.777 58.617 Td [(en)]TJ /F8 1 Tf 1.4096 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.431 84.478 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.452 58.617 Td [(est)-415(clairemen)24(t)-415(obten)24(ue)-409(en)-415(enlev)54(an)24(t)-415(dans)-409(la)]TJ -20.12 -1.1988 Td [(matrice)-307(jacobienne)-307(de)]TJ /F8 1 Tf 9.7771 0 Td [(F)]TJ /F2 1 Tf 1.0843 0 Td [(en)]TJ /F8 1 Tf 1.3072 0 Td [(x)]TJ /F2 1 Tf 0.8735 0 Td [(la)-307(colonne)-307(d'indice)]TJ /F8 1 Tf 8.3614 0 Td [(j)]TJ /F2 1 Tf 0.77108 0 Td [(et)-307(la)-307(ligne)-307(d'indice)]TJ /F8 1 Tf 8.2771 0 Td [(i)]TJ /F2 1 Tf 0.6506 0 Td [(laquelle)-307(est)-307(form)24(\266)469(ee)]TJ -31.102 -1.1988 Td [(de)-391(0)-397(sauf)-397(le)-397(terme)-397(de)-397(la)]TJ /F8 1 Tf 11.018 0 Td [(j)]TJ /F2 1 Tf 0.46988 0 Td [(-i)24(\265)469(eme)-391(colonne)-397(qui)-397(v)54(aut)-397(1.)-632(On)-391(conclut)-403(donc,)-409(en)-397(d)24(\266)475(ev)24(eloppan)24(t)-397(le)]TJ -11.488 -1.2048 Td [(d)30(\266)469(eterminan)24(t)-337(par)-331(rapp)-36(ort)-337(\265)500(a)-331(cette)-337(ligne,)]TJ /F6 1 Tf 17.018 0 Td [(j)]TJ /F2 1 Tf 0.44578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.966 79.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.416 55.015 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F2 1 Tf 0.55422 0 Td [(=)]TJ /F6 1 Tf 1.0542 0 Td [(j)]TJ /F2 1 Tf 0.44578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.586 79.323 Td [(0)]TJ /F7 1 Tf -0.19828 -0.94828 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.482 55.015 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.422 79.323 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.723 55.015 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F2 1 Tf 0.28313 0 Td [(.)]TJ -28.06 -1.5964 Td [(On)-337(a)-331(alors)-337(les)-307(\266)469(egalit)24(\266)469(es)-331(:)]TJ /F9 1 Tf 0.16265 -0.78313 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 72.064 Td [(V)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.747 51.268 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3675 Td [(Z)]TJ /F16 1 Tf 8.352 0 0 8.352 0 0 Tm 16.819 72.064 Td [(R)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 12.307 52.683 Td [(\265)]TJ 0.74096 -0.048191 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.466 72.064 Td [(V)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 28.145 100.52 Td [(z)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.536 51.268 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.966 73.961 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.331 51.268 Td [(;)-168(z)]TJ /F2 1 Tf 0.95181 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.026 73.961 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 19.169 52.683 Td [(\266)]TJ /F8 1 Tf 0.90362 -1.4157 Td [(dz)]TJ /F2 1 Tf 3.0301 0 Td [(\(d'apr)24(\265)475(es)-337(le)-337(th)18(\266)475(eor)24(\265)469(eme)-331(de)-337(F)84(ubini\))]TJ -12.964 -2.8614 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F16 1 Tf 8.352 0 0 8.352 0 0 Tm 16.819 67.969 Td [(R)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 12.307 49.816 Td [(\265)]TJ 0.74096 -0.048199 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.466 67.969 Td [(U)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 28.289 94.788 Td [(z)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.608 48.406 Td [(f)]TJ /F9 1 Tf 0.59638 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.336 69.056 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.771 48.406 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.371 69.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.012 48.406 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(;)-168(z)]TJ /F9 1 Tf 0.95783 0.81325 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.81325 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.371 69.866 Td [(0)]TJ /F7 1 Tf -0.19828 -0.94828 Td [(z)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.639 48.406 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.207 69.866 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.88 48.406 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.362 69.866 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 27.09 49.816 Td [(\266)]TJ /F8 1 Tf 0.90362 -1.4096 Td [(dz)]TJ /F2 1 Tf 3.0301 0 Td [(\(par)-337(h)24(yp)-30(oth)18(\265)475(ese)-337(de)-331(r)24(\266)475(ecurrence\))]TJ -20.886 -2.8614 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F16 1 Tf 8.352 0 0 8.352 0 0 Tm 16.819 63.875 Td [(R)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 12.307 46.954 Td [(\265)]TJ 0.74096 -0.048195 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.466 63.875 Td [(U)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 28.289 89.066 Td [(z)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 14.608 45.545 Td [(f)]TJ /F9 1 Tf 0.59638 0.81326 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 24.914 65.771 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.687 45.545 Td [(;)-168(z)]TJ /F2 1 Tf 0.95783 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.81326 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.81326 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 32.905 65.771 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.277 45.545 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.69 65.771 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.518 45.545 Td [(;)-168(z)]TJ /F2 1 Tf 0.95181 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 39.207 65.771 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 27.681 46.954 Td [(\266)]TJ /F8 1 Tf 0.90362 -1.4096 Td [(dz)]TJ /F2 1 Tf -18.446 -2.7711 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.819 59.9 Td [(U)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.422 42.774 Td [(f)]TJ /F9 1 Tf 0.59639 0.80723 Td [(\241)]TJ /F8 1 Tf 0.46386 -0.80723 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.80723 Td [(\242)-222(j)]TJ /F2 1 Tf 0.93976 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.017 61.797 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.855 42.774 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 3.0964 0 Td [(\(d'apr)30(\265)469(es)-337(le)-337(th)24(\266)469(eor)24(\265)469(eme)-331(de)-337(F)84(ubini\))]TJ /F8 1 Tf 13.753 0 Td [(:)]TJ /F2 1 Tf -33.169 -2.1084 Td [(Le)-337(th)24(\266)469(eor)24(\265)475(eme)-337(est)-337(donc)-337(prouv)54(\266)469(e)-337(dans)-331(ce)-337(cas)-337(sp)-6(\266)475(ecial.)]TJ /F1 1 Tf 0 -2.1988 Td [(2.)-512(Comp)-36(osition)-391(de)-385(di\256)18(\266)548(eomorphismes.)]TJ /F2 1 Tf 0 -1.6024 Td [(Supp)-24(osons)-337(que)]TJ /F8 1 Tf 6.7892 0 Td [(F)]TJ /F2 1 Tf 1.1084 0 Td [(soit)-337(le)-337(comp)-30(os)24(\266)475(e)-337(d'un)-337(di\256)24(\266)475(eomorphisme)]TJ /F8 1 Tf 16.741 0 Td [(G)]TJ /F2 1 Tf 1.1205 0 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(U)]TJ /F2 1 Tf 1.1205 0 Td [(sur)]TJ /F8 1 Tf 1.6747 0 Td [(W)]TJ /F2 1 Tf 1.4157 0 Td [(et)-331(d'un)-337(di\256)24(\266)475(eomor-)]TJ -31.301 -1.1988 Td [(phisme)]TJ /F8 1 Tf 3.4337 0 Td [(H)]TJ /F2 1 Tf 1.2831 0 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(W)]TJ /F2 1 Tf 1.4518 0 Td [(sur)]TJ /F8 1 Tf 1.7169 0 Td [(V)]TJ /F2 1 Tf 0.8012 0 Td [(.)-560(Si)-367(le)-379(th)24(\266)469(eor)24(\265)469(eme)-367(a)-343(\266)469(et)30(\266)469(e)-349(\266)475(etabli)-379(p)-30(our)-367(c)18(hacun)-373(des)-373(di\256)30(\266)469(eomorphismes)]TJ /F8 1 Tf 28.325 0 Td [(G)]TJ /F2 1 Tf -38.38 -1.2048 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(H)]TJ /F2 1 Tf 1.241 0 Td [(alors)-343(on)-331(a)]TJ /F9 1 Tf -2.5241 -0.39759 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.2586 45.487 Td [(V)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.4699 32.701 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.422 45.487 Td [(W)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.386 32.701 Td [(f)]TJ /F9 1 Tf 0.59639 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(H)]TJ /F2 1 Tf 0.91566 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(z)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.80723 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(H)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.241 47.383 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.018 32.701 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(z)]TJ /F2 1 Tf 0.51205 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dz)]TJ /F2 1 Tf 1.3072 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.302 45.487 Td [(U)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.337 32.701 Td [(f)]TJ /F9 1 Tf 0.59639 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(H)]TJ /F6 1 Tf 0.91566 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(G)]TJ /F2 1 Tf 0.78916 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F6 1 Tf 0.68073 -0.80723 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(H)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.707 47.383 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 34.319 33.509 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(G)]TJ /F2 1 Tf 0.78916 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.80723 Td [(j)-222(\242)-222(j)]TJ /F2 1 Tf 1.4458 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(G)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.905 47.383 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.44 32.701 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dx)]TJ /F2 1 Tf 1.4277 0 Td [(.)]TJ -39.331 -2.1084 Td [(Cep)-24(endan)24(t)-548(la)-542(r)30(\265)469(egle)-548(de)-542(d)30(\266)469(eriv)54(ation)-542(des)-548(applications)-542(comp)-30(os)24(\266)469(ees)-542(p)-30(our)]TJ /F8 1 Tf 31 0 Td [(F)]TJ /F2 1 Tf 1.4096 0 Td [(=)]TJ /F8 1 Tf 1.3976 0 Td [(H)]TJ /F6 1 Tf 0.91566 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(G)]TJ /F2 1 Tf 1.3313 0 Td [(donne)]TJ /F8 1 Tf -36.554 -1.1988 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 8.75 42.65 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.3916 29.394 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-277(=)]TJ /F8 1 Tf 1.7229 0 Td [(H)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.293 42.65 Td [(0)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 10.271 30.201 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(G)]TJ /F2 1 Tf 0.78916 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80723 Td [(\242)]TJ /F6 1 Tf 0.45783 -0.80723 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(G)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 20.905 42.65 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.892 29.394 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-240(et)-246(donc)-246(en)-240(prenan)24(t)-246(la)-240(v)48(aleur)-240(absolue)-240(des)-246(d)24(\266)475(eterminan)24(ts,)-265(la)-240(derni)24(\265)469(ere)]TJ -10.524 -1.8133 Td [(in)24(t)24(\266)475(egrale)-253(de)-246(la)-246(relation)-246(pr)30(\266)469(ec)24(\266)469(eden)30(te)-246(se)-253(transforme)-246(en)]TJ /F9 1 Tf 22.03 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.948 38.168 Td [(U)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.584 27.581 Td [(f)]TJ /F9 1 Tf 0.60241 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F6 1 Tf 0.50602 -0.81325 Td [(\242)-48(j)]TJ /F2 1 Tf 0.76506 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.647 40.064 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.669 27.581 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.3373 0 Td [(ce)-253(qui)-246(prouv)30(e)]TJ -33.47 -1.7048 Td [(le)-337(th)24(\266)469(eor)24(\265)469(eme)-331(p)-30(our)]TJ /F8 1 Tf 7.759 0 Td [(F)]TJ /F2 1 Tf 1.0602 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(H)]TJ /F6 1 Tf 0.91566 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(G)]TJ /F2 1 Tf 0.78313 0 Td [(.)]TJ /F1 1 Tf -12.072 -2.2048 Td [(3.)-512(Preuv)24(e)-385(lo)-36(cale.)]TJ /F2 1 Tf 0 -1.5964 Td [(Fixons)]TJ /F8 1 Tf 3.3373 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 13.224 31.375 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.114 22.075 Td [(dans)]TJ /F8 1 Tf 2.4337 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(.)-722(La)-433(matrice)-433(jacobienne)-427(de)]TJ /F8 1 Tf 12.705 0 Td [(F)]TJ /F2 1 Tf 1.2108 0 Td [(en)]TJ /F8 1 Tf 1.4277 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 41.871 31.375 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.133 22.075 Td [(n'est)-427(pas)-433(iden)24(tiquemen)24(t)-433(n)24(ulle,)-451(il)]TJ -24.801 -1.6687 Td [(existe)-361(donc)]TJ /F8 1 Tf 5.253 0 Td [(i)]TJ /F2 1 Tf 0.6988 0 Td [(et)]TJ /F8 1 Tf 1.1928 0 Td [(j)]TJ /F2 1 Tf 0.8253 0 Td [(tels)-361(que)]TJ /F8 1 Tf 3.8795 0.6747 Td [(@)-60(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.353 29.952 Td [(i)]TJ ET 0.504 w 205.27 246.88 m 224.21 246.88 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.175 19.72 Td [(@)-54(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.233 28.004 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.873 20.406 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 28.388 28.987 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.283 20.406 Td [(\))]TJ /F6 1 Tf 0.71084 0 Td [(6)]TJ /F2 1 Tf [(=)-319(0.)-518(In)24(tro)-30(duisons)-361(alors)-367(une)-355(application)]TJ /F8 1 Tf 17.717 0 Td [(H)]TJ /F2 1 Tf 1.2711 0 Td [(de)]TJ /F8 1 Tf 1.3554 0 Td [(U)]TJ /F2 1 Tf 1.1506 0 Td [(dans)]TJ /F3 1 Tf -37.157 -1.8253 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 27.185 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8735 18.581 Td [(d)30(\266)469(e\257nie)-277(par)]TJ /F8 1 Tf 4.8313 0 Td [(H)]TJ /F2 1 Tf 0.91566 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(z)]TJ /F2 1 Tf 0.78313 0 Td [(o)-30(\265)530(u)]TJ /F8 1 Tf 1.3313 0 Td [(z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.586 26.375 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.651 18.581 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.026 26.375 Td [(k)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.048 18.581 Td [(si)]TJ /F8 1 Tf 0.94578 0 Td [(k)]TJ /F6 1 Tf 0.83133 0 Td [(6)]TJ /F2 1 Tf [(=)]TJ /F8 1 Tf 1.0542 0 Td [(j)]TJ /F2 1 Tf 0.74699 0 Td [(et)]TJ /F8 1 Tf 1.1084 0 Td [(z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.491 26.375 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.898 18.581 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.922 26.375 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.934 18.581 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\).)-427(La)-283(matrice)-277(jacobienne)-277(de)]TJ /F8 1 Tf 12.187 0 Td [(H)]TJ /F2 1 Tf 1.1867 0 Td [(est)]TJ -37.934 -1.1988 Td [(form)30(\266)469(ee)-271(des)-265(d)24(\266)469(eriv)54(\266)469(ees)-265(partielles)-271(de)]TJ /F8 1 Tf 14.054 0 Td [(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.664 24.659 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.627 17.382 Td [(plac)24(\266)475(ee)-271(en)]TJ /F8 1 Tf 4.1988 0 Td [(j)]TJ /F2 1 Tf 0.46988 0 Td [(-i)24(\265)469(eme)-265(ligne)-265(et)-271(d'une)-271(diagonale)-271(de)-265(1)-265(en)-265(dehors)]TJ -19.964 -1.6626 Td [(de)-403(cette)-409(ligne.)-662(On)-409(a)-403(donc)-403(det)]TJ /F8 1 Tf 13.578 0 Td [(H)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.362 23.081 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.5 15.72 Td [(=)]TJ /F8 1 Tf 1.3012 0.6747 Td [(@)-60(F)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.957 23.245 Td [(i)]TJ ET 260.5 190.87 m 279.43 190.87 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.795 15.033 Td [(@)-54(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.845 21.297 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.898 15.72 Td [(et)-409(en)-403(particulier)-409(det)]TJ /F8 1 Tf 9.1145 0 Td [(H)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.552 23.081 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.205 15.72 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 50.328 22.28 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.614 15.72 Td [(\))]TJ /F6 1 Tf 0.78916 0 Td [(6)]TJ /F2 1 Tf [(=)-397(0.)-662(Le)-403(th)18(\266)475(eor)18(\265)475(eme)]TJ -31.072 -1.7711 Td [(d'in)24(v)24(ersion)-319(lo)-30(cale)-319(a\261rme)-319(alors)-325(que)]TJ /F8 1 Tf 15.355 0 Td [(H)]TJ /F2 1 Tf 1.2289 0 Td [(induit)-319(un)-319(di\256)30(\266)469(eomorphisme)-319(d'un)-325(v)24(oisinage)-319(ouv)24(ert)]TJ /F8 1 Tf 21.446 0 Td [(U)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 63.034 19.745 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 12.75 Td [(de)]TJ /F8 1 Tf 1.4096 0 Td [(x)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 10.466 18.03 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.1747 12.75 Td [(sur)-415(son)-421(image)]TJ /F8 1 Tf 6.5904 0 Td [(W)]TJ /F2 1 Tf 1.0783 0 Td [(.)-686(On)-415(p)-24(eut)-421(donc)-409(d)24(\266)469(e\257nir)-415(sur)]TJ /F8 1 Tf 12.512 0 Td [(W)]TJ /F2 1 Tf 1.494 0 Td [(une)-415(application)]TJ /F8 1 Tf 7.2169 0 Td [(G)]TJ /F2 1 Tf 1.1988 0 Td [(=)]TJ /F8 1 Tf 1.1867 0 Td [(F)]TJ /F6 1 Tf 0.78313 0 Td [(\261)]TJ /F8 1 Tf 0.5 0 Td [(H)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.595 18.831 Td [(\241)]TJ /F4 1 Tf 0.89655 0 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.133 12.75 Td [(qui)]TJ -37.801 -1.1988 Td [(en)30(v)24(oie)]TJ /F8 1 Tf 3.0422 0 Td [(z)]TJ /F2 1 Tf 0.85542 0 Td [(sur)]TJ /F8 1 Tf 1.6928 0 Td [(y)]TJ /F2 1 Tf 0.87349 0 Td [(a)24(v)24(ec)]TJ /F8 1 Tf 2.2108 0 Td [(y)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 20.741 16.314 Td [(i)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.133 11.551 Td [(=)]TJ /F8 1 Tf 1.0783 0 Td [(z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.862 16.314 Td [(j)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.096 11.551 Td [(.)-493(En)-349(p)-30(osan)24(t)]TJ /F8 1 Tf 5.5964 0 Td [(V)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 33.31 16.314 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.03 11.551 Td [(=)]TJ /F8 1 Tf 1.0783 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 38.586 16.314 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.41 11.551 Td [(\))-355(on)-349(v)24(oit)-349(que)-355(le)-355(di\256)30(\266)469(eomorphisme)-349(de)]TJ /F8 1 Tf 15.952 0 Td [(U)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 63.034 16.314 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 10.346 Td [(sur)]TJ /F8 1 Tf 1.6988 0 Td [(V)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 10.897 14.59 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4217 10.346 Td [(induit)-367(par)]TJ /F8 1 Tf 4.7831 0 Td [(F)]TJ /F2 1 Tf 1.1386 0 Td [(est)-367(le)-367(pro)-30(duit)-355(des)-367(di\256)30(\266)469(eomorphismes)-361(induits)-361(par)]TJ /F8 1 Tf 21.12 0 Td [(G)]TJ /F2 1 Tf 1.1446 0 Td [(et)]TJ /F8 1 Tf 1.1928 0 Td [(H)]TJ /F2 1 Tf 0.91566 0 Td [(.)-524(Comme)-361(ces)]TJ -33.386 -1.1988 Td [(di\256)30(\266)469(eomorphismes)-283(rel)24(\265)469(ev)24(en)30(t)-283(du)-283(cas)-283(sp)-6(\266)469(ecial)-259(\266)475(etudi)24(\266)475(e)-289(en)]TJ /F1 1 Tf 21.976 0 Td [(1.)]TJ /F2 1 Tf 0.89157 0 Td [(,)-295(le)-283(th)24(\266)469(eor)24(\265)469(eme)-277(est)-289(prouv)54(\266)469(e)-283(p)-30(our)]TJ /F8 1 Tf 12.91 0 Td [(G)]TJ /F2 1 Tf 0.78313 0 Td [(,)-295(p)-30(our)]TJ /F8 1 Tf -36.56 -1.1988 Td [(H)]TJ /F2 1 Tf 1.247 0 Td [(et)-331(donc)-337(p)-30(our)-337(la)-331(restriction)-343(de)]TJ /F8 1 Tf 13.157 0 Td [(F)]TJ /F2 1 Tf 1.1145 0 Td [(\265)500(a)]TJ /F8 1 Tf 0.83735 0 Td [(U)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 32.009 11.159 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.819 7.9485 Td [(.)]TJ /F5 1 Tf -17.488 -4.0602 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(122)]TJ ET endstream endobj 402 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 401 0 R /ProcSet 2 0 R >> /Contents 400 0 R >> endobj 401 0 obj << /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F13 39 0 R /F16 81 0 R >> endobj 403 0 obj << /Length 25681 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ 0 -2.4036 Td [(4.)-512(Fin)-385(de)-385(la)-391(preuv)30(e.)]TJ /F2 1 Tf 0 -1.6084 Td [(App)-24(elons)-319("b)-30(ons)-313(ouv)24(erts")-319(les)-313(ouv)24(erts)]TJ /F8 1 Tf 16.12 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.828 87.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.831 60.611 Td [(de)]TJ /F8 1 Tf 1.3133 0 Td [(U)]TJ /F2 1 Tf 1.1024 0 Td [(sur)-313(lesquels)-319(le)-313(th)24(\266)469(eor)24(\265)469(eme)-313(est)-313(vrai,)-313(c'est-)-6(\265)500(a-dire)]TJ -19.916 -1.6566 Td [(tels)-331(que)]TJ /F9 1 Tf 4.6988 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 15.155 83.064 Td [(F)]TJ /F4 1 Tf 0.87931 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 24.277 116.49 Td [(0)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.75 83.064 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.771 58.954 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 1.3253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.44 83.064 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 36.795 116.49 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.717 58.954 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.81325 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.017 84.961 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.151 58.954 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.4217 0 Td [(.)]TJ -24.036 -2.2892 Td [(Nous)-385(v)24(enons)-391(de)-391(v)24(oir)-385(que)-391(l'on)-391(p)-30(ouv)54(ait)-391(recouvrir)]TJ /F8 1 Tf 21.259 0 Td [(U)]TJ /F2 1 Tf 1.1807 0 Td [(par)-391(des)-385(b)-30(ons)-391(ouv)18(erts.)-602(Mon)24(trons)-385(alors)]TJ -22.44 -1.1988 Td [(que)-349(l'ensem)30(ble)-349(des)-343(b)-30(ons)-349(ouv)18(erts)-349(est)-349(stable)-349(par)-343(r)24(\266)475(eunion)-349(\(\257nie\).)-487(Quitte)-349(\265)499(a)-343(compl)24(\266)469(eter)-343(par)-349(un)]TJ 0 -1.1988 Td [(raisonnemen)24(t)-337(par)-331(r)24(\266)475(ecurrence,)-343(il)-331(su\261t)-337(de)-337(le)-337(faire)-337(p)-30(our)-337(la)-331(r)24(\266)475(eunion)-343(de)-331(deux)-331(ouv)18(erts.)]TJ 0 -1.6084 Td [(Supp)-24(osons)-403(le)-397(th)24(\266)469(eor)24(\265)469(eme)-397(vrai)-397(sur)-403(les)-397(ouv)24(erts)]TJ /F8 1 Tf 19.428 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.569 75.952 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.229 52.659 Td [(et)]TJ /F8 1 Tf 1.2289 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.431 75.952 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.759 52.659 Td [(.)-638(Alors,)-409(en)-397(notan)24(t)-397(\()]TJ /F8 1 Tf 8.9639 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 55.112 75.952 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.795 52.659 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.069 75.952 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.988 52.659 Td [(le)-397(compl)24(\266)469(e-)]TJ -34.657 -1.1988 Td [(men)24(taire)-391(de)]TJ /F8 1 Tf 5.5843 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 16.759 74.228 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.988 51.461 Td [(,)]TJ /F8 1 Tf 0.68072 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 19.267 74.228 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 14.006 51.461 Td [([)]TJ /F8 1 Tf 0.92169 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.5 74.228 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.62 51.461 Td [(est)-397(la)-385(r)24(\266)475(eunion)-397(disjoin)30(te)-391(de)]TJ /F8 1 Tf 11.91 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.957 74.228 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.988 51.461 Td [(et)-391(de)]TJ /F8 1 Tf 2.6145 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 47.784 74.228 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 34.163 51.461 Td [(\\)]TJ /F2 1 Tf 0.92771 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.905 74.228 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.554 51.461 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.862 74.228 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.735 51.461 Td [(qui)-391(est)-391(con)24(ten)24(u)]TJ -32.404 -1.2048 Td [(dans)]TJ /F8 1 Tf 2.3795 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 12.172 72.512 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0181 50.256 Td [(.)-572(Si)]TJ /F8 1 Tf 2.0602 0 Td [(\256)]TJ /F2 1 Tf 1.0241 0 Td [(est)-379(la)-379(fonction)-385(indicatrice)-373(de)]TJ /F8 1 Tf 12.982 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 37.026 72.512 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 26.639 50.256 Td [(\\)]TJ /F2 1 Tf 0.92169 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 41.129 72.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.018 50.256 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.086 72.512 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.813 50.256 Td [(,)-391(on)-379(a)-373(donc)-379(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.216 71.702 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 72.819 100.62 Td [(0)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.491 71.702 Td [([)]TJ /F7 1 Tf 0.76724 0 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 75.675 100.62 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.729 50.256 Td [(=)-349(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.845 71.702 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 82.072 100.62 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.608 50.256 Td [(+)]TJ /F8 1 Tf 1.0301 0 Td [(\256)]TJ /F2 1 Tf 1.0241 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.1988 Td [(\256)]TJ /F2 1 Tf 0.92169 0 Td [(=)-277(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.259 69.987 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 16.976 98.21 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 9.253 49.057 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(.)-445(On)-337(en)-331(d)24(\266)475(eduit)]TJ /F9 1 Tf -5.0663 -0.65663 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4224 66.012 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 13.012 92.656 Td [(0)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 9.6983 66.012 Td [([)]TJ /F7 1 Tf 0.76724 0 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 15.867 92.656 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.4759 47.039 Td [(f)]TJ /F9 1 Tf 0.59639 0.80723 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F6 1 Tf 0.68072 -0.80723 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.371 67.909 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.91 47.039 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf -2.6566 -1.012 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.293 62.616 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 35.193 87.909 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.922 44.665 Td [(f)]TJ /F9 1 Tf 0.59639 0.80722 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80722 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.80722 Td [(\242)]TJ /F6 1 Tf 0.68073 -0.80722 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.879 64.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.355 44.665 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.3133 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.957 62.616 Td [(U)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 61.277 87.909 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.181 44.665 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F9 1 Tf 0.59639 0.80722 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80722 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80722 Td [(\242)]TJ /F6 1 Tf 0.68073 -0.80722 Td [(\242)-222(j)]TJ /F2 1 Tf 0.94578 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.741 64.512 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.331 44.665 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F8 1 Tf 0.44579 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)]TJ -38.886 -2.2229 Td [(P)30(osons)-373(alors)]TJ /F8 1 Tf 5.8012 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.078 61.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.554 42.442 Td [(=)]TJ /F8 1 Tf 1.1145 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.371 61.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.91 42.442 Td [(\),)]TJ /F8 1 Tf 1.0422 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.414 61.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.608 42.442 Td [(=)]TJ /F8 1 Tf 1.1145 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 31.034 61.331 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.199 42.442 Td [(\))-367(et)-373(notons)]TJ /F8 1 Tf 5.2229 0 Td [(\257)]TJ /F2 1 Tf 0.98193 0 Td [(la)-373(fonction)-373(indicatrice)-367(de)]TJ /F8 1 Tf 11.343 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.034 61.331 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.313 42.442 Td [(\\)]TJ /F2 1 Tf 0.90964 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.129 61.331 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.699 42.442 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.095 61.331 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 41.244 Td [(o)-24(\265)524(u)-361(\()]TJ /F8 1 Tf 1.8012 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.362 59.616 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2229 41.244 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.319 59.616 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3795 41.244 Td [(d)24(\266)475(esigne)-367(le)-361(compl)24(\266)469(emen)24(taire)-361(de)]TJ /F8 1 Tf 13.205 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.474 59.616 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.675 41.244 Td [(.)-524(On)-361(a)-361(clairemen)24(t)]TJ /F8 1 Tf 8.247 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.836 59.616 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.482 41.244 Td [(\\)]TJ /F2 1 Tf 0.90964 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 50.922 59.616 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.867 41.244 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.879 59.616 Td [(c)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.982 41.244 Td [(=)]TJ /F8 1 Tf 1.1024 0 Td [(F)]TJ /F9 1 Tf 0.78313 0.81326 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.414 59.616 Td [(00)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 40.873 41.244 Td [(\\)]TJ /F2 1 Tf 0.90361 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.474 59.616 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.241 41.244 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.431 59.616 Td [(c)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 44.036 42.057 Td [(\242)]TJ /F2 1 Tf -38.705 -2.1145 Td [(et)-307(donc)]TJ /F8 1 Tf 3.5 0 Td [(\256)]TJ /F2 1 Tf 0.64458 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-283(=)]TJ /F8 1 Tf 1.7229 0 Td [(\257)]TJ /F9 1 Tf 0.62048 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(.)-433(Le)-313(fait)-307(que)-307(le)-313(th)24(\266)469(eor)24(\265)475(eme)-307(soit)-307(vrai)-307(sur)-313(les)-307(ouv)24(erts)]TJ /F8 1 Tf 21.458 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.483 57.754 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.663 39.942 Td [(et)]TJ /F8 1 Tf 1.1386 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 58.086 57.754 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.41 39.942 Td [(mon)24(tre)]TJ -36.078 -1.1988 Td [(donc)-331(que)-343(les)-331(in)24(t)24(\266)469(egrales)-337(pr)30(\266)469(ec)24(\266)475(eden)24(tes)-337(son)18(t)-331(encore)-313(\266)469(egales)-331(\265)500(a)]TJ /F9 1 Tf 5.994 -0.29518 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17 51.771 Td [(V)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 25.048 72.752 Td [(0)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.843 37.087 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dy)]TJ /F2 1 Tf 1.2651 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.388 51.771 Td [(V)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 36.771 72.752 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.922 37.087 Td [(\257)]TJ /F2 1 Tf 0.62048 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F6 1 Tf 0.61446 0 Td [(\242)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 3.3795 0 Td [(donc)-337(\265)500(a)]TJ /F9 1 Tf 5.2289 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.94 51.771 Td [(V)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 66.892 72.752 Td [(0)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.25 51.771 Td [([)]TJ /F7 1 Tf 0.76724 0 Td [(V)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 69.807 72.752 Td [(00)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.44 37.087 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(dy)]TJ /F2 1 Tf 1.0482 0 Td [(.)]TJ -33.229 -2.2229 Td [(Comme)]TJ /F8 1 Tf 3.6627 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 14.026 50.487 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 10.295 34.864 Td [([)]TJ /F8 1 Tf 0.87952 0 Td [(V)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.147 50.487 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 12.771 34.864 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.595 50.487 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.283 34.864 Td [([)]TJ /F8 1 Tf 0.88554 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.698 50.487 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.47 34.864 Td [(\))-331(le)-337(th)24(\266)469(eor)24(\265)469(eme)-325(est)-337(vrai)-337(sur)]TJ /F8 1 Tf 11.361 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 43.828 50.487 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.12 34.864 Td [([)]TJ /F8 1 Tf 0.87952 0 Td [(U)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 46.931 50.487 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.301 34.864 Td [(,)-331(ce)-337(qui)-325(prouv)24(e)-337(le)-331(r)24(\266)475(esultat)]TJ -27.97 -1.1988 Td [(annonc)24(\266)475(e.)]TJ 0 -1.9096 Td [(En)-295(particulier,)-307(si)]TJ /F8 1 Tf 7.5904 0 Td [(K)]TJ /F2 1 Tf 1.2169 0 Td [(est)-301(un)-301(compact,)-301(comme)]TJ /F8 1 Tf 10.536 0 Td [(U)]TJ /F2 1 Tf 1.0843 0 Td [(est)-307(recouv)24(ert)-295(par)-301(des)-295(b)-30(ons)-301(ouv)18(erts,)-301(on)-301(p)-30(eut)]TJ -20.428 -1.2048 Td [(recouvrir)]TJ /F8 1 Tf 4.2349 0 Td [(K)]TJ /F2 1 Tf 1.2289 0 Td [(par)-313(un)-313(nom)24(bre)-313(\257ni)-313(de)-313(ces)-307(b)-30(ons)-313(ouv)18(erts,)-313(et)-313(en)-307(prenan)24(t)-319(leur)-307(r)24(\266)475(eunion)-319(on)-307(conclut)]TJ -5.4639 -1.1988 Td [(que)-337(tout)-337(compact)-337(de)]TJ /F8 1 Tf 9.3614 0 Td [(U)]TJ /F2 1 Tf 1.1265 0 Td [(est)-343(con)30(ten)24(u)-337(dans)-331(un)-337(b)-30(on)-337(ouv)24(ert.)]TJ -10.488 -1.9096 Td [(Main)24(tenan)24(t)-331(on)-325(sait)-325(que)]TJ /F8 1 Tf 10.476 0 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(p)-30(eut-)30(^)469(etre)-325(recouv)24(ert)-325(par)-331(une)-325(suite)-325(de)-325(compacts)]TJ /F8 1 Tf 20.44 0 Td [(K)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.681 39.056 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39 27.442 Td [(\(il)-331(su\261t,)-331(par)]TJ -33.669 -1.2048 Td [(exemple,)-385(de)-373(prendre)-379(p)-30(our)]TJ /F8 1 Tf 11.687 0 Td [(K)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.569 37.34 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.705 26.238 Td [(l'ensem)24(ble)-373(des)]TJ /F8 1 Tf 6.6265 0 Td [(x)]TJ /F2 1 Tf 0.94578 0 Td [(qui)-373(son)18(t)-373(de)-379(norme)-373(inf)24(\266)469(erieure)-379(ou)-349(\266)475(egale)-379(\265)500(a)]TJ /F8 1 Tf 17.717 0 Td [(p)]TJ /F2 1 Tf -38.663 -1.1988 Td [(et)-379(don)24(t)-379(la)-379(distance)-385(au)-379(compl)24(\266)475(emen)24(taire)-385(de)]TJ /F8 1 Tf 18.717 0 Td [(U)]TJ /F2 1 Tf 1.1747 0 Td [(est)-385(sup)0(\266)475(erieure)-385(ou)-349(\266)469(egale)-385(\265)500(a)-379(2)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.026 36.426 Td [(\241)]TJ /F7 1 Tf 0.88793 0 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.139 25.039 Td [(\).)-584(Chacun)-385(des)]TJ /F8 1 Tf -32.807 -1.1988 Td [(K)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.8448 33.9 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 7.0361 23.84 Td [(est)-397(donc)-397(con)24(ten)24(u)-397(dans)-391(un)-397(b)-30(on)-397(ouv)24(ert)]TJ /F8 1 Tf 16.976 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.345 33.9 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.157 23.84 Td [(.)-632(Quitte)-397(\265)500(a)-391(remplacer)]TJ /F8 1 Tf 9.7108 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.879 33.9 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.41 23.84 Td [(par)-397(la)-391(r)24(\266)475(eunion)-403(des)]TJ /F8 1 Tf -31.078 -1.1988 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6034 32.185 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8072 22.641 Td [(p)-24(our)]TJ /F8 1 Tf 2.3795 0 Td [(q)]TJ /F11 1 Tf 0.78916 0 Td [(6)]TJ /F8 1 Tf 1.0843 0 Td [(p)]TJ /F2 1 Tf 0.8494 0 Td [(on)-355(p)-24(eut)-355(m)24(^)469(eme)-349(supp)-30(oser)-349(que)-355(la)-349(suite)-355(des)]TJ /F8 1 Tf 17.97 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.741 32.185 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.373 22.641 Td [(est)-355(croissan)24(te.)-500(La)-355(r)30(\266)469(eunion)-355(des)]TJ /F8 1 Tf -26.042 -1.2048 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6034 30.461 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8193 21.436 Td [(est)-355(alors)]TJ /F8 1 Tf 3.988 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(,)-349(celle)-349(des)]TJ /F8 1 Tf 4.6024 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.836 30.461 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.518 21.436 Td [(\))-349(est)]TJ /F8 1 Tf 2.3072 0 Td [(V)]TJ /F2 1 Tf 1.1506 0 Td [(et)-349(la)-343(propri)18(\266)475(et)24(\266)469(e)-349(de)-343(Bepp)-36(o)-343(Levi)-349(assure)-349(que)-349(l'in)24(t)24(\266)475(egrale)]TJ /F9 1 Tf -16.645 -0.38554 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 8.4224 26.875 Td [(V)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 6.5783 19.689 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(y)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 1.3133 0 Td [(est)-277(la)-271(limite)-271(des)]TJ /F9 1 Tf 6.9819 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.052 26.875 Td [(F)]TJ /F4 1 Tf 0.87931 0 Td [(\()]TJ /F7 1 Tf 0.44827 0 Td [(U)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 37.952 37.355 Td [(p)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 27.759 26.875 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 19.759 19.689 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(y)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dy)]TJ /F2 1 Tf 1.3193 0 Td [(tandis)-277(que)]TJ /F9 1 Tf 4.741 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.698 26.875 Td [(U)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.114 19.689 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F6 1 Tf 0.55421 -0.81325 Td [(\242)-96(j)]TJ /F2 1 Tf 0.81326 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 51.534 28.771 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.289 19.689 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.3614 0 Td [(est)-277(la)-271(limite)]TJ -34.114 -2.5723 Td [(des)]TJ /F9 1 Tf 1.7892 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.983 23.193 Td [(U)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 16.434 32.21 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 8.6928 17.117 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F6 1 Tf 0.72289 -0.81325 Td [(\242)-259(j)]TJ /F2 1 Tf 0.98193 0 Td [(det)]TJ /F8 1 Tf 1.5542 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 22.784 25.09 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.205 17.117 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dx)]TJ /F2 1 Tf 1.0904 0 Td [(.)-626(Le)-397(th)18(\266)475(eor)24(\265)469(eme)-367(\266)475(etan)18(t)-397(vrai)-397(sur)-397(les)]TJ /F8 1 Tf 14.753 0 Td [(U)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.405 24.28 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.988 17.117 Td [(,)-409(ces)-397(deux)-391(suites)-397(son)18(t)]TJ -29.687 -1.9036 Td [(\266)469(egales)-331(ainsi)-337(donc)-337(que)-337(leur)-331(limite,)-337(ce)-337(qui)-331(prouv)24(e)-337(le)-343(th)24(\266)475(eor)18(\265)475(eme)-337(sur)]TJ /F8 1 Tf 27.904 0 Td [(U)]TJ /F2 1 Tf 0.78916 0 Td [(.)]TJ ET 512.64 180.68 m 519.84 180.68 l 519.84 189.1 l 512.64 189.1 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 12.695 Td [(Remarque.)]TJ /F2 1 Tf 6.5964 0 Td [(Dans)-427(l')30(\266)469(enonc)30(\266)469(e)-427(pr)30(\266)469(ec)24(\266)469(eden)30(t)-421(on)-421(prendra)-427(garde)-421(\265)499(a)-421(ne)-415(pas)-427(oublier)-421(la)-421(v)48(aleur)-421(ab-)]TJ -6.5964 -1.1988 Td [(solue)-367(p)-30(our)-361(le)-367(d)24(\266)469(eterminan)30(t)-367(jacobien.)-536(En)-361(e\256et)-367(les)-361(in)24(t)24(\266)469(egrales)-361(utilis)24(\266)469(ees)-361(ne)-367(son)24(t)-367(pas)-367(orien)30(t)24(\266)469(ees.)]TJ 0 -1.1988 Td [(Elles)-319(prennen)24(t)-325(des)-319(v)48(aleurs)-319(p)-30(ositiv)24(es)-319(lorsque)-319(l'on)-325(in)24(t)24(\265)475(egre)-325(des)-319(fonctions)-325(p)-30(ositiv)30(es,)-325(il)-319(est)-325(donc)]TJ 0 -1.2048 Td [(imp)-30(ortan)18(t)-331(que)-343(le)-337(facteur)-337(m)30(ultiplicatif)]TJ /F6 1 Tf 16.681 0 Td [(j)]TJ /F2 1 Tf 0.43976 0 Td [(det)]TJ /F8 1 Tf 1.5602 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 35.483 13.607 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 25.072 9.093 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F2 1 Tf 0.61446 0 Td [(soit)-337(p)-24(ositif.)]TJ -21.705 -1.6024 Td [(En)-499(fait,)-542(on)-500(p)-24(eut)-506(comme)-500(en)-493(dimension)-506(1)-493(in)24(tro)-30(duire)-506(une)-493(notion)-500(d'in)18(t)24(\266)475(egrale)-506(orien)30(t)24(\266)469(ee)-500(en)]TJ 0 -1.2048 Td [(rempla)0(\273)439(can)24(t)-325(les)-331(ouv)24(erts)]TJ /F8 1 Tf 10.133 0 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(et)]TJ /F8 1 Tf 1.1626 0 Td [(V)]TJ /F2 1 Tf 1.1265 0 Td [(par)-331(des)-325(ouv)24(erts)-331(orien)24(t)24(\266)469(es)-325(\(i.e.)-445(m)24(unis)-325(d'une)-325(orien)24(tation\).)-445(On)]TJ 5.1265 -2.3976 Td [(123)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 405 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 404 0 R /ProcSet 2 0 R >> /Contents 403 0 R >> endobj 404 0 obj << /F1 12 0 R /F2 13 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F13 39 0 R /F18 107 0 R >> endobj 406 0 obj << /Length 14985 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(con)30(vien)18(t)-385(alors)-385(que)-385(l'in)24(t)24(\266)469(egrale)-385(sur)]TJ /F8 1 Tf 14.801 0 Td [(U)]TJ /F2 1 Tf 1.1747 0 Td [(\(resp.)]TJ /F8 1 Tf 3.0482 0 Td [(V)]TJ /F2 1 Tf 0.8012 0 Td [(\))-391(est)-385(l'in)24(t)24(\266)469(egrale)-385(usuelle)-391(si)-385(l'orien)24(tation)-385(de)]TJ /F8 1 Tf 18.548 0 Td [(U)]TJ /F2 1 Tf -38.373 -1.1988 Td [(\(resp.)]TJ /F8 1 Tf 3.0542 0 Td [(V)]TJ /F2 1 Tf 0.80723 0 Td [(\))-385(co)102(\304)391(\263ncide)-391(a)30(v)24(ec)-391(l'orien)24(tation)-385(de)-385(l'espace)-391(am)24(bian)24(t)]TJ /F3 1 Tf 21.663 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.19 87.909 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.12 61.021 Td [(\(\257x)24(\266)475(ee)-391(au)-385(pr)24(\266)469(ealable\),)-403(et)-385(c'est)]TJ -26.789 -1.1988 Td [(l'opp)-30(os)24(\266)469(ee)-385(de)-385(l'in)24(t)24(\266)475(egrale)-391(usuelle)-385(dans)-385(le)-385(cas)-385(con)24(traire.)-596(Alors,)-397(dans)-385(ce)-385(con)30(texte,)-403(la)-379(form)24(ule)]TJ 0 -1.2048 Td [(de)-439(c)18(hangemen)24(t)-445(de)-445(v)54(ariables)-451(s')24(\266)475(ecrit)-445(sans)-445(mettre)-445(la)-445(v)54(aleur)-445(absolue)-445(mais)-439(elle)-451(supp)-30(ose)-439(que)]TJ 0 -1.1988 Td [(l'ouv)30(ert)-307(image)]TJ /F8 1 Tf 6.5 0 Td [(V)]TJ /F2 1 Tf 1.1145 0 Td [(est)-307(m)24(uni)-307(de)-307(l'orien)24(tation)-307(obten)24(ue)-307(en)-301(transp)-36(ortan)24(t)-307(celle)-307(de)]TJ /F8 1 Tf 25.295 0 Td [(U)]TJ /F2 1 Tf 1.0964 0 Td [(par)]TJ /F8 1 Tf 1.753 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(.)-433(Plus)]TJ -36.542 -1.1988 Td [(pr)30(\266)469(ecis)24(\266)475(emen)24(t,)-265(comme)-246(det)]TJ /F8 1 Tf 10.548 0 Td [(F)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.836 81.038 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.94 56.22 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-246(ne)-240(s'ann)24(ule)-246(pas,)-265(il)-246(garde)-240(un)-253(signe)-246(constan)24(t)-246(et)-246(on)-246(con)30(vien)24(t)-246(que)]TJ -12.572 -1.2048 Td [(l'orien)24(tation)-331(de)]TJ /F8 1 Tf 6.9699 0 Td [(V)]TJ /F2 1 Tf 1.1386 0 Td [(est)-337(la)-331(m)24(^)469(eme)-331(que)-331(celle)-337(de)]TJ /F8 1 Tf 10.97 0 Td [(U)]TJ /F2 1 Tf 1.1205 0 Td [(si)-337(ce)-337(d)30(\266)469(eterminan)24(t)-331(est)-337(p)-30(ositif)-331(et)-337(l'orien)24(tation)]TJ -20.199 -1.1988 Td [(con)30(traire)-319(si)-319(le)-325(d)30(\266)469(eterminan)24(t)-319(est)-319(n)24(\266)475(egatif.)-439(Ce)-325(p)-24(oin)24(t)-319(de)-319(vue)-319(est)-319(celui)-319(adopt)30(\266)469(e)-319(dans)-313(la)-319(th)24(\266)469(eorie)-319(de)]TJ 0 -1.1988 Td [(l'in)24(t)24(\266)475(egration)-343(des)-337(formes)-337(di\256)30(\266)469(eren)24(tielles.)]TJ 0 -2.2711 Td [(Le)-475(th)24(\266)469(eor)24(\265)475(eme)-475(de)-469(c)24(hangemen)18(t)-475(de)-469(v)48(ariables)-475(est)-475(souv)24(en)30(t)-475(utilis)18(\266)475(e)-475(dans)-475(des)-475(situations)-469(o)-30(\265)530(u)-475(les)]TJ 0 -1.1988 Td [(h)30(yp)-36(oth)24(\265)469(eses)-319(ne)-319(son)24(t)-319(pas)-319(v)48(\266)475(eri\257)24(\266)469(ees.)-439(P)30(our)-319(cela)-319(on)-319(se)-319(ram)24(\265)475(ene)-325(aux)-319(h)24(yp)-30(oth)18(\265)475(eses)-319(du)-325(th)24(\266)475(eor)18(\265)475(eme)-319(cit)24(\266)475(e)]TJ 0 -1.1988 Td [(en)-307(\266)475(eliminan)24(t)-337(des)-337(parties)-331(n)24(\266)469(egligeables.)-445(En)-337(v)24(oici)-337(un)-331(exemple.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 37.429 Td [(VI)-35(I)-35(I.5.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 44.87 Td [(3.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.4458 0 Td [(Soit)]TJ /F8 1 Tf 2.2349 0 Td [(F)]TJ /F12 1 Tf 1.3012 0 Td [(l'application)-518(de)]TJ /F3 1 Tf 7.4217 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 43.267 64.004 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 31.241 44.87 Td [(\243)]TJ /F2 1 Tf 1.1265 0 Td [([0;)-168(2)]TJ /F8 1 Tf 1.7229 0 Td [(\274)]TJ /F2 1 Tf 0.60241 0 Td [(])]TJ /F12 1 Tf 0.79518 0 Td [(dans)]TJ /F3 1 Tf 2.5241 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 55.431 64.806 Td [(2)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 39.699 44.87 Td [(d)30(\266)469(e\257nie)-518(par)]TJ /F8 1 Tf -34.367 -1.1988 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(\275;)-174(\265)]TJ /F2 1 Tf 1.4578 0 Td [(\))-283(=)-277(\()]TJ /F8 1 Tf 2.1145 0 Td [(\275)]TJ /F2 1 Tf 0.68675 0 Td [(cos)]TJ /F8 1 Tf 1.506 0 Td [(\265)-30(;)-168(\275)]TJ /F2 1 Tf 1.6265 0 Td [(sin)]TJ /F8 1 Tf 1.3916 0 Td [(\265)]TJ /F2 1 Tf 0.5 0 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)-445(Alors,)-337(p)-30(our)]TJ /F8 1 Tf 6.0181 0 Td [(A)]TJ /F12 1 Tf 1.0843 0 Td [(partie)-337(mesurable)-337(de)]TJ /F3 1 Tf 8.9578 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 47.155 62.28 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 33.837 43.671 Td [(\243)]TJ /F2 1 Tf 1 0 Td [([0;)-168(2)]TJ /F8 1 Tf 1.7229 0 Td [(\274)]TJ /F2 1 Tf 0.60843 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(,)-331(on)-337(a)]TJ /F9 1 Tf -21.38 -0.3494 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.793 58.745 Td [(F)]TJ /F4 1 Tf 0.87931 0 Td [(\()]TJ /F7 1 Tf 0.44828 0 Td [(A)]TJ /F4 1 Tf 0.86207 0 Td [(\))]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.518 41.961 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F2 1 Tf 1.3675 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.793 58.745 Td [(A)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.265 41.961 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(\275)]TJ /F2 1 Tf 0.68072 0 Td [(cos)]TJ /F8 1 Tf 1.506 0 Td [(\265)-36(;)-168(\275)]TJ /F2 1 Tf 1.6265 0 Td [(sin)]TJ /F8 1 Tf 1.3976 0 Td [(\265)]TJ /F2 1 Tf 0.49397 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(\275)-174(d\275d\265)]TJ /F12 1 Tf -26.018 -2.3735 Td [(p)-30(our)]TJ /F8 1 Tf 2.3614 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(mesurable)-337(p)-24(ositiv)24(e)-337(ou)-337(in)24(t)30(\266)469(egrable)-337(sur)]TJ /F8 1 Tf 16.145 0 Td [(F)]TJ /F2 1 Tf 0.78313 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F12 1 Tf 0.38554 0 Td [(.)]TJ /F1 1 Tf -21.747 -1.9458 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6386 Td [(P)30(our)-409(se)-415(ramener)-409(aux)-415(h)24(yp)-30(oth)24(\265)469(eses)-415(du)-409(th)24(\266)469(eor)24(\265)469(eme)-409(de)-409(c)24(hangemen)24(t)-415(de)-409(v)54(ariables)-415(on)-415(doit)-409(com-)]TJ 0 -1.1988 Td [(mencer)-445(par)-445(restreindre)]TJ /F8 1 Tf 10.578 0 Td [(F)]TJ /F2 1 Tf 1.2289 0 Td [(\265)500(a)-445(un)-445(ouv)18(ert)]TJ /F8 1 Tf 5.7289 0 Td [(U)]TJ /F2 1 Tf 1.241 0 Td [(sur)-445(lequel)]TJ /F8 1 Tf 4.759 0 Td [(F)]TJ /F2 1 Tf 1.2289 0 Td [(est)-451(une)-445(injection)-445(de)-445(classe)]TJ /F8 1 Tf 12.217 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 61.681 50.4 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 43.994 34.804 Td [(\265)500(a)]TJ -38.663 -1.1988 Td [(d)30(\266)469(eterminan)24(t)-319(jacobien)-319(non)-319(n)24(ul.)-439(On)-319(v)48(a)-313(donc)-319(prendre)-325(p)-24(our)]TJ /F8 1 Tf 24.855 0 Td [(U)]TJ /F2 1 Tf 1.1145 0 Td [(l'in)24(t)24(\266)475(erieur)-319(du)-319(domaine)-325(de)-313(d)24(\266)469(e\257-)]TJ -25.97 -1.1988 Td [(nition)-349(de)]TJ /F8 1 Tf 4.253 0 Td [(F)]TJ /F2 1 Tf 1.1265 0 Td [(\265)500(a)-349(sa)24(v)24(oir)]TJ /F3 1 Tf 3.7349 0 Td [(R)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.707 46.961 Td [(\244)]TJ /F4 1 Tf 0 -0.94828 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.831 32.406 Td [(\243)]TJ /F2 1 Tf 0.77711 0 Td [(]0;)-168(2)]TJ /F8 1 Tf 1.7229 0 Td [(\274)]TJ /F2 1 Tf 0.60843 0 Td [([.)-487(La)-355(restriction)-349(est)-355(alors)-349(injectiv)24(e,)-355(de)-349(classe)]TJ /F8 1 Tf 19.729 0 Td [(C)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 56.466 46.961 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 40.253 32.406 Td [(a)24(v)24(ec)-349(p)-30(our)]TJ -34.922 -1.8855 Td [(matrice)-307(jacobienne)]TJ /F9 1 Tf 8.4699 1.4096 Td [(\265)]TJ /F2 1 Tf 0.90361 -0.80723 Td [(cos)]TJ /F8 1 Tf 1.506 0 Td [(\265)]TJ /F6 1 Tf 1.494 0 Td [(\241)]TJ /F8 1 Tf 0.78313 0 Td [(\275)]TJ /F2 1 Tf 0.68072 0 Td [(sin)]TJ /F8 1 Tf 1.3976 0 Td [(\265)]TJ /F2 1 Tf -5.8072 -1.1988 Td [(sin)]TJ /F8 1 Tf 1.3976 0 Td [(\265)-1421(\275)]TJ /F2 1 Tf 2.5663 0 Td [(cos)]TJ /F8 1 Tf 1.506 0 Td [(\265)]TJ /F9 1 Tf 1 2.006 Td [(\266)]TJ /F2 1 Tf 1.0361 -1.4096 Td [(et)-307(le)-307(d)30(\266)469(eterminan)24(t)-301(jacobien)-301(v)48(aut)]TJ /F8 1 Tf 13.88 0 Td [(\275)]TJ /F2 1 Tf 0.81928 0 Td [(donc)-301(n'est)-301(jamais)]TJ -31.633 -1.747 Td [(n)30(ul.)-457(Ainsi)-337(le)-343(th)24(\266)475(eor)18(\265)475(eme)-337(s'applique)-337(si)]TJ /F8 1 Tf 15.789 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F8 1 Tf 1.0602 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(.)-451(Plus)-337(g)24(\266)475(en)24(\266)475(eralemen)24(t,)-343(en)-331(m)24(ultiplian)24(t)-313(\266)469(ev)24(en)30(tuelle-)]TJ -18.675 -1.1988 Td [(men)24(t)-331(par)-337(une)-331(fonction)-337(caract)24(\266)475(eristique)-337(con)24(v)24(enable,)-337(on)-331(v)24(oit)-337(qu'il)-331(s'applique)-331(si)]TJ /F8 1 Tf 33.464 0 Td [(A)]TJ /F6 1 Tf 1.0301 0 Td [(\275)]TJ /F8 1 Tf 1.0542 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(.)-439(P)24(our)]TJ -36.343 -1.1988 Td [(justi\257er)-391(la)-397(form)24(ule)-391(g)30(\266)469(en)24(\266)475(erale)-397(il)-397(su\261t)-397(alors)-397(de)-391(constater)-397(que)-397(le)-403(compl)24(\266)475(emen)24(taire)-397(de)]TJ /F8 1 Tf 35.97 0 Td [(U)]TJ /F2 1 Tf 1.1867 0 Td [(dans)]TJ /F3 1 Tf -37.157 -1.1988 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 8.6638 35.814 Td [(+)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 6.8253 25.177 Td [(\243)]TJ /F2 1 Tf 0.89157 0 Td [([0;)-168(2)]TJ /F8 1 Tf 1.7229 0 Td [(\274)]TJ /F2 1 Tf 0.60241 0 Td [(])-283(\(r)24(\266)475(eunion)-283(de)-277(deux)-277(demi-droites)-283(et)-277(d'un)-283(segmen)24(t)-283(de)-277(droite\))-283(et)-277(le)-283(compl)24(\266)469(emen)24(taire)]TJ -4.7108 -1.2048 Td [(de)]TJ /F8 1 Tf 1.3253 0 Td [(F)]TJ /F2 1 Tf 0.77711 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(U)]TJ /F2 1 Tf 0.79518 0 Td [(\))-325(dans)]TJ /F3 1 Tf 3.0422 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 17.724 34.9 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.157 23.973 Td [(\(une)-331(demi-droite)-325(ferm)24(\266)475(ee\))-325(son)18(t)-325(n)24(\266)475(egligeables)-331(et)-325(on)24(t)-325(donc)-325(une)-325(con)24(tribution)]TJ -7.8253 -1.1988 Td [(n)30(ulle)-337(dans)-337(les)-331(deux)-337(in)24(t)30(\266)469(egrales.)]TJ ET 512.64 270.97 m 519.84 270.97 l 519.84 279.39 l 512.64 279.39 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 15.243 Td [(VI)-35(I)-35(I.6.)-236(Le)-376(pro)-30(duit)-376(de)-381(con)30(v)30(olution.)]TJ 0 -2.5628 Td [(VI)-35(I)-35(I.6.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 15.201 Td [(1.)]TJ /F15 1 Tf 1.747 0 Td [(D)]TJ 0.83132 0.084336 Td [(\266)]TJ 0 -0.084336 Td [(efinition.)]TJ /F12 1 Tf 5.259 0 Td [(Soien)24(t)]TJ /F8 1 Tf 2.994 0 Td [(f)]TJ /F12 1 Tf 0.90362 0 Td [(et)]TJ /F8 1 Tf 1.1325 0 Td [(g)]TJ /F12 1 Tf 0.81325 0 Td [(des)-307(fonctions)-307(mesurables)-307(d)30(\266)469(e\257nies)-307(presque)-301(partout)]TJ -17.711 -1.1988 Td [(sur)]TJ /F3 1 Tf 1.7711 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.198 20.625 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 8.3675 14.003 Td [(.)-734(On)-433(app)-30(elle)]TJ /F1 1 Tf 6.2952 0 Td [(pro)-30(duit)-500(de)-493(con)24(v)30(olution)]TJ /F12 1 Tf 12.133 0 Td [(de)]TJ /F8 1 Tf 1.4277 0 Td [(f)]TJ /F12 1 Tf 1.0301 0 Td [(par)]TJ /F8 1 Tf 1.8795 0 Td [(g)]TJ /F12 1 Tf 0.93976 0 Td [(et)-433(on)-433(note)]TJ /F8 1 Tf 5.0723 0 Td [(f)]TJ /F6 1 Tf 0.88554 0 Td [(\244)]TJ /F8 1 Tf 0.78313 0 Td [(g)]TJ /F12 1 Tf 0.94579 0 Td [(la)-427(fonction)]TJ /F8 1 Tf -34.428 -1.6566 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F16 1 Tf 8.352 0 0 8.352 0 0 Tm 11.466 16.366 Td [(R)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 17.036 23.27 Td [(n)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.0602 12.346 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dt)]TJ /F12 1 Tf 2.2169 0 Td [(\(o)-30(\265)524(u)]TJ /F8 1 Tf 1.7771 0 Td [(dt)]TJ /F12 1 Tf 1.2169 0 Td [(d)24(\266)475(esigne)-337(la)-337(mesure)-337(de)-337(Leb)-30(esgue)-337(sur)]TJ /F3 1 Tf 15.355 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.086 18.254 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.241 12.346 Td [(\).)]TJ /F2 1 Tf -30.91 -2.4578 Td [(Bien)-500(s)-30(^)530(ur)-493(la)-493(fonction)]TJ /F8 1 Tf 9.6084 0 Td [(f)]TJ /F6 1 Tf 0.92771 0 Td [(\244)]TJ /F8 1 Tf 0.83133 0 Td [(g)]TJ /F2 1 Tf 1.006 0 Td [(n'est)-493(d)24(\266)475(e\257nie)-493(que)-499(p)-30(our)-493(les)]TJ /F8 1 Tf 12.042 0 Td [(x)]TJ /F2 1 Tf 1.0602 0 Td [(tels)-493(que)-500(l'in)24(t)24(\266)469(egrale)-500(du)-493(pro)-30(duit)]TJ /F8 1 Tf -25.476 -1.1988 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.75301 0 Td [(\241)]TJ /F8 1 Tf 0.95783 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-319(a)-313(un)-313(sens.)-439(C'est)-319(le)-319(cas)-313(en)-313(particulier)-313(lorsque)]TJ /F8 1 Tf 20.217 0 Td [(f)]TJ /F2 1 Tf 0.90964 0 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(g)]TJ /F2 1 Tf 0.8253 0 Td [(son)18(t)-313(p)-30(ositiv)24(es,)-319(\265)500(a)-313(condition)]TJ -27.807 -1.1988 Td [(d'admettre)-469(des)-469(v)48(aleurs)-469(in\257nies.)-849(Le)-469(th)24(\266)469(eor)24(\265)469(eme)-463(de)-469(F)84(ubini)-469(assure)-469(alors)-469(que)]TJ /F8 1 Tf 33.211 0 Td [(f)]TJ /F6 1 Tf 0.90964 0 Td [(\244)]TJ /F8 1 Tf 0.81325 0 Td [(g)]TJ /F2 1 Tf 0.9759 0 Td [(est)-475(une)]TJ -35.91 -1.2048 Td [(fonction)-337(mesurable)-337(p)-30(ositiv)30(e.)]TJ /F5 1 Tf 0 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(124)]TJ ET endstream endobj 408 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 407 0 R /ProcSet 2 0 R >> /Contents 406 0 R >> endobj 407 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F16 81 0 R >> endobj 409 0 obj << /Length 19139 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ /F2 1 Tf 0 -2.4036 Td [(Dans)-253(le)-253(cas)-253(g)30(\266)469(en)30(\266)469(eral,)]TJ /F8 1 Tf 8.6807 0 Td [(f)]TJ /F6 1 Tf 0.65663 0 Td [(\244)]TJ /F8 1 Tf 0.55422 0 Td [(g)]TJ /F2 1 Tf 0.75904 0 Td [(est)-259(d)30(\266)469(e\257nie)-253(aux)-253(p)-30(oin)24(ts)-246(o)-30(\265)530(u)-253(la)-246(fonction)]TJ /F8 1 Tf 15.434 0 Td [(t)]TJ /F6 1 Tf 0.63856 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F6 1 Tf 0.62651 0 Td [(\241)]TJ /F8 1 Tf 0.83133 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-253(est)-259(in)30(t)24(\266)469(egrable,)]TJ -32.458 -1.1988 Td [(c'est-)0(\265)500(a-dire)-391(aux)-391(p)-30(oin)24(ts)-385(o)-30(\265)530(u)]TJ /F6 1 Tf 11.651 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)-259(\244)-259(j)]TJ /F8 1 Tf 1.5723 0 Td [(g)]TJ /F6 1 Tf 0.51807 0 Td [(j)]TJ /F2 1 Tf 0.66265 0 Td [(est)-391(\257ni.)-614(En)-361(\266)475(ecriv)48(an)30(t)]TJ /F8 1 Tf 9.3494 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(g)]TJ /F2 1 Tf 0.89759 0 Td [(comme)-391(di\256)30(\266)469(erence)-391(de)-385(leurs)]TJ -27.735 -1.1988 Td [(parties)-271(p)-30(ositiv)30(es)-271(et)-277(n)30(\266)469(egativ)30(es)-277(on)-271(v)24(oit)-271(que)]TJ /F8 1 Tf 17.699 0 Td [(f)]TJ /F6 1 Tf 0.69277 0 Td [(\244)]TJ /F8 1 Tf 0.59036 0 Td [(g)]TJ /F2 1 Tf 0.78313 0 Td [(est)-277(d)24(\266)475(e\257nie)-271(et)-277(\257nie)-271(sur)-271(un)-277(ensem)24(ble)-271(mesurable)]TJ -19.765 -1.2048 Td [(sur)-337(lequel)-337(elle)-337(est)-343(une)-331(fonction)-343(mesurable.)]TJ 0 -1.6988 Td [(En)-259(faisan)24(t)-259(le)-265(c)24(hangemen)18(t)-259(de)-259(v)54(ariables)]TJ /F8 1 Tf 16.386 0 Td [(t)]TJ /F6 1 Tf 0.63855 0 Td [(7!)]TJ /F8 1 Tf 1.2831 0 Td [(x)]TJ /F6 1 Tf 0.63855 0 Td [(\241)]TJ /F8 1 Tf 0.8494 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(,)-277(qui)-259(est)-259(un)-265(di\256)30(\266)469(eomorphisme)-265(de)-259(d)30(\266)469(eterminan)24(t)]TJ -20.157 -1.1988 Td [(jacobien)]TJ /F6 1 Tf 3.8735 0 Td [(\247)]TJ /F2 1 Tf 0.77711 0 Td [(1,)-301(on)-289(p)-30(eut)-295(en)-289(outre)-289(remarquer)-295(que)-289(l'in)24(t)24(\266)469(egrabilit)24(\266)475(e)-295(de)]TJ /F8 1 Tf 22.524 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.70482 0 Td [(\241)]TJ /F8 1 Tf 0.90964 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-265(\266)475(equiv)48(aut)-289(\265)499(a)-289(celle)]TJ -31.795 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))-337(et)-337(les)-337(in)24(t)24(\266)475(egrales)-307(\266)469(ev)24(en)30(tuelles)-337(de)-331(ces)-337(fonctions)-337(son)24(t)-307(\266)469(egales.)-445(Autremen)24(t)-337(dit)-343(:)]TJ /F15 1 Tf -6.5482 -2.0542 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8614 0 Td [(Les)-313(pro)-30(duits)-313(de)-313(con)24(v)24(olution)]TJ /F8 1 Tf 12.392 0 Td [(f)]TJ /F6 1 Tf 0.77711 0 Td [(\244)]TJ /F8 1 Tf 0.68073 0 Td [(g)]TJ /F12 1 Tf 0.8253 0 Td [(et)]TJ /F8 1 Tf 1.1446 0 Td [(g)]TJ /F6 1 Tf 0.69277 0 Td [(\244)]TJ /F8 1 Tf 0.68072 0 Td [(f)]TJ /F12 1 Tf 0.90964 0 Td [(son)24(t)-319(d)30(\266)469(e\257nis)-319(aux)-313(m)24(^)469(emes)-313(p)-30(oin)24(ts)-313(et)]TJ -24.964 -1.1988 Td [(son)24(t)-307(\266)469(egaux.)]TJ /F2 1 Tf 0 -2.3976 Td [(V)84(oici)-337(quelques)-331(cas)-337(imp)-36(ortan)24(ts)-331(o)-30(\265)530(u)-337(le)-337(pro)-30(duit)-337(de)-331(con)24(v)24(olution)-337(est)-337(d)24(\266)475(e\257ni.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 38.118 Td [(VI)-35(I)-35(I.6.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 45.695 Td [(2.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0301 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(g)]TJ /F12 1 Tf 0.84337 0 Td [(des)-337(fonctions)-337(mesurables)-337(sur)]TJ /F3 1 Tf 12.789 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.612 65.987 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 38.012 45.695 Td [(.)-439(Alors)-337(:)]TJ /F1 1 Tf -31.867 -1.6988 Td [(1.)]TJ /F12 1 Tf 1.5542 0 Td [(si)-337(l'une)-331(des)-337(fonctions)-331(est)-337(b)-30(orn)24(\266)475(ee)-337(et)-331(l'autre)-331(est)-337(in)24(t)24(\266)469(egrable,)-331(la)-331(fonction)]TJ /F8 1 Tf 30.024 0 Td [(f)]TJ /F6 1 Tf 0.81325 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F12 1 Tf 0.84337 0 Td [(est)-337(d)30(\266)469(e\257nie)]TJ -34.771 -1.1988 Td [(et)-337(\257nie)-331(sur)-343(tout)]TJ /F3 1 Tf 7.1747 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.931 61.831 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 14.108 42.798 Td [(et)-337(est)-337(b)-30(orn)24(\266)475(ee)-337(;)]TJ /F1 1 Tf -7.9639 -1.6988 Td [(2.)]TJ /F12 1 Tf 1.759 0 Td [(si)]TJ /F8 1 Tf 1.1084 0 Td [(f)]TJ /F12 1 Tf 1.0301 0 Td [(et)]TJ /F8 1 Tf 1.2651 0 Td [(g)]TJ /F12 1 Tf 0.94578 0 Td [(son)18(t)-433(in)24(t)24(\266)469(egrables,)-463(le)-433(pro)-30(duit)]TJ /F8 1 Tf 12.536 0 Td [(f)]TJ /F6 1 Tf 0.88554 0 Td [(\244)]TJ /F8 1 Tf 0.78915 0 Td [(g)]TJ /F12 1 Tf 0.94579 0 Td [(est)-439(d)30(\266)469(e\257ni)-439(et)-433(\257ni)-439(presque)-433(partout)-439(et)-433(la)]TJ -22.078 -1.2048 Td [(fonction)]TJ /F8 1 Tf 3.8614 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F12 1 Tf 0.8494 0 Td [(est)-337(in)24(t)24(\266)475(egrable.)-445(En)-337(outre)-337(on)-337(a)-331(:)]TJ /F9 1 Tf 4.994 -0.29518 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.3735 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F6 1 Tf 1.3133 0 Td [(\242)]TJ /F9 1 Tf 0.33735 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dx)]TJ /F12 1 Tf 1.4217 0 Td [(.)]TJ /F1 1 Tf -27.633 -2.5 Td [(D)24(\266)548(emonstration.)]TJ 0.81325 -1.7048 Td [(1.)]TJ /F2 1 Tf 1.5602 0 Td [(Supp)-30(osons,)-337(par)-331(exemple,)]TJ /F8 1 Tf 11.133 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(b)-30(orn)24(\266)475(ee)-337(et)]TJ /F8 1 Tf 4.4217 0 Td [(g)]TJ /F2 1 Tf 0.8494 0 Td [(in)24(t)30(\266)469(egrable.)]TJ -19.705 -1.6988 Td [(On)-415(a)-409(donc)-415(une)-415(ma)-54(joration)]TJ /F6 1 Tf 12.313 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(j)]TJ /F11 1 Tf 0.68675 0 Td [(6)]TJ /F8 1 Tf 1.1867 0 Td [(M)]TJ /F2 1 Tf 1.488 0 Td [(qui)-415(donne)]TJ /F6 1 Tf 4.8012 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.84337 0 Td [(\241)]TJ /F8 1 Tf 1.0542 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.68675 0 Td [(6)]TJ /F8 1 Tf 1.1867 0 Td [(M)]TJ /F6 1 Tf 1.0783 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F2 1 Tf 0.68675 0 Td [(et)-415(assure,)-433(p)-30(our)]TJ -32.494 -1.7108 Td [(tout)]TJ /F8 1 Tf 2.1988 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)]TJ /F9 1 Tf 0.6506 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.81325 0 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dt)]TJ /F11 1 Tf 1.2108 0 Td [(6)]TJ /F8 1 Tf 1.1084 0 Td [(M)]TJ /F9 1 Tf 1.247 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dt)-331(<)]TJ /F2 1 Tf 2.3193 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.3675 0 Td [(et)-367(l'in)24(t)24(\266)475(egrabilit)24(\266)469(e)-367(de)]TJ /F8 1 Tf 8.741 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.81928 0 Td [(\241)]TJ /F8 1 Tf 1.0181 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-367(comme)]TJ -35.355 -1.6747 Td [(fonction)-337(de)]TJ /F8 1 Tf 5.1928 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(.)-445(La)-343(fonction)]TJ /F8 1 Tf 6.0482 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F2 1 Tf 0.84337 0 Td [(est)-343(donc)-331(d)24(\266)469(e\257nie)-331(et)-337(\257nie)-337(sur)-337(tout)]TJ /F3 1 Tf 14.301 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.147 42.012 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 35.223 28.948 Td [(et)-331(v)48(\266)475(eri\257e)]TJ /F6 1 Tf -20.494 -1.6566 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F2 1 Tf 0.68072 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dt)]TJ /F11 1 Tf 1.1627 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F9 1 Tf 1.247 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(dt)]TJ /F2 1 Tf 0.88554 0 Td [(.)]TJ /F1 1 Tf -28.675 -2.2108 Td [(2.)]TJ /F2 1 Tf 1.6687 0 Td [(Supp)-30(osons)]TJ /F8 1 Tf 4.9819 0 Td [(f)]TJ /F2 1 Tf 0.98795 0 Td [(et)]TJ /F8 1 Tf 1.2229 0 Td [(g)]TJ /F2 1 Tf 0.89759 0 Td [(in)24(t)24(\266)469(egrables)-391(et)-385(consid)24(\266)475(erons)]TJ /F8 1 Tf 11.717 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.83133 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))-391(comme)-391(fonction)-391(des)-391(deux)]TJ -27.157 -1.1988 Td [(v)54(ariables)]TJ /F8 1 Tf 4.1867 0 Td [(x)]TJ /F2 1 Tf 0.93976 0 Td [(et)]TJ /F8 1 Tf 1.2108 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(.)-560(Le)-373(c)24(hangemen)24(t)-379(de)-367(v)48(ariables)-379(\()]TJ /F8 1 Tf 13.765 0 Td [(x;)-168(t)]TJ /F2 1 Tf 1.3795 0 Td [(\))]TJ /F6 1 Tf 0.72892 0 Td [(7!)]TJ /F2 1 Tf 1.3434 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u;)-168(v)]TJ /F2 1 Tf 1.5422 0 Td [(\))-373(a)24(v)24(ec)]TJ /F8 1 Tf 2.994 0 Td [(u)]TJ /F2 1 Tf 0.91566 0 Td [(=)]TJ /F8 1 Tf 1.1205 0 Td [(x)]TJ /F6 1 Tf 0.81928 0 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(,)]TJ /F8 1 Tf 0.65663 0 Td [(v)]TJ /F2 1 Tf 0.86145 0 Td [(=)]TJ /F8 1 Tf 1.1205 0 Td [(t)]TJ /F2 1 Tf 0.73494 0 Td [(est)-379(un)]TJ -36.452 -1.2048 Td [(di\256)30(\266)469(eomorphisme)-337(de)]TJ /F3 1 Tf 8.8193 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.284 33.047 Td [(n)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.639 22.677 Td [(\243)]TJ /F3 1 Tf 1 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.845 33.047 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.241 22.677 Td [(sur)-337(lui-m)30(^)469(eme)-331(de)-337(d)24(\266)475(eterminan)24(t)-337(jacobien)-331(1.)-445(Il)-337(s'ensuit)]TJ /F9 1 Tf -9.8072 -0.29518 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dxdt)]TJ /F2 1 Tf 2.253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(v)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dudv)]TJ /F2 1 Tf 2.4157 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.43976 0 Td [(du)]TJ /F6 1 Tf 1.3193 0 Td [(\242)]TJ /F9 1 Tf 0.5 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(v)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.44578 0 Td [(dv)-319(<)]TJ /F2 1 Tf 2.3735 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.3373 0 Td [(.)]TJ -35.783 -2.2108 Td [(Cela)-325(prouv)24(e)-331(que)]TJ /F8 1 Tf 7.3916 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.77108 0 Td [(\241)]TJ /F8 1 Tf 0.98193 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))-331(est)-325(in)24(t)24(\266)469(egrable)-325(en)-319(tan)18(t)-325(que)-331(fonction)-325(des)-325(deux)-325(v)48(ariables)]TJ /F8 1 Tf 24.325 0 Td [(x)]TJ /F2 1 Tf 0.89759 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(.)]TJ -38.886 -1.1988 Td [(On)-343(sait)-343(alors)-343(que,)-349(p)-30(our)-343(presque)-343(tout)]TJ /F8 1 Tf 16.337 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(,)-343(il)-343(s'agit)-343(d'une)-343(fonction)-349(in)24(t)24(\266)475(egrable)-343(de)]TJ /F8 1 Tf 16.795 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(,)-343(autremen)24(t)]TJ -34.066 -1.1988 Td [(dit)]TJ /F8 1 Tf 1.5542 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F2 1 Tf 0.68072 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))-337(est)-343(d)30(\266)469(e\257ni)-337(et)-337(\257ni.)-445(En)-337(outre,)-337(on)-337(a)]TJ /F9 1 Tf 6.7952 -0.3494 Td [(Z)]TJ /F8 1 Tf 1 -1.3675 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dxdt)]TJ /F2 1 Tf 2.253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3675 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3675 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F2 1 Tf 0.6747 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dx)]TJ /F2 1 Tf 1.4277 0 Td [(.)]TJ -27.355 -2.1506 Td [(Cep)-24(endan)24(t,)-349(la)-343(fonction)-349(consid)30(\266)469(er)24(\266)469(ee)-319(\266)475(etan)24(t)-349(in)24(t)30(\266)469(egrable,)-349(le)-343(m)24(^)469(eme)-337(c)18(hangemen)24(t)-343(de)-343(v)48(ariables)-343(que)]TJ 0 -1.2048 Td [(pr)30(\266)469(ec)24(\266)469(edemmen)24(t)-337(donne)]TJ /F9 1 Tf 5.7711 -0.29518 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dxdt)]TJ /F2 1 Tf 2.253 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1.006 -1.3614 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(v)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(dudv)]TJ /F2 1 Tf 2.4157 0 Td [(=)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(du)]TJ /F6 1 Tf 1.3133 0 Td [(\242)]TJ /F9 1 Tf 0.5 1.3614 Td [(Z)]TJ /F8 1 Tf 1 -1.3614 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(v)]TJ /F2 1 Tf 0.51807 0 Td [(\))]TJ /F8 1 Tf 0.56024 0 Td [(dv)]TJ /F2 1 Tf 1.3735 0 Td [(,)]TJ -33.114 -2.1506 Td [(ce)-337(qui)-331(prouv)24(e)-337(le)-337(r)24(\266)475(esultat)-343(annonc)24(\266)475(e.)]TJ ET 512.64 88.808 m 519.84 88.808 l 519.84 97.232 l 512.64 97.232 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 23.94 3.8882 Td [(125)]TJ /F5 1 Tf 6.4458 0 Td [(U.P.M.C.)-530(Cours)-530(de)-530(P.)-530(MAZET)]TJ ET endstream endobj 411 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 410 0 R /ProcSet 2 0 R >> /Contents 409 0 R >> endobj 410 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R >> endobj 412 0 obj << /Length 18496 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.623 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-11867(VI)-36(I)-36(I.)-385(Les)-385(mesures)-391(pro)-30(duits)]TJ 0 -2.4036 Td [(Remarque.)]TJ /F2 1 Tf 6.1566 0 Td [(La)-265(partie)]TJ /F1 1 Tf 4.2048 0 Td [(1.)]TJ /F2 1 Tf 1.3133 0 Td [(de)-265(l')30(\266)469(enonc)30(\266)469(e)-265(pr)24(\266)475(ec)24(\266)469(eden)24(t)-259(est)-271(un)-259(exemple)-271(de)-259(l'e\256et)-259("r)24(\266)469(egularisan)30(t")-265(de)]TJ -11.675 -1.1988 Td [(la)-343(con)30(v)24(olution.)-475(Le)-349(fait)-343(que)-349(l'un)-349(des)-343(facteurs)-349(soit)-343(b)-30(orn)24(\266)475(e)-349(p)-30(ermet)-343(de)-343(conclure)-343(que)-349(le)-349(pro)-30(duit)]TJ 0 -1.1988 Td [(l'est)-222(\266)469(egalemen)24(t)-259(\(on)-259(mon)24(trera)-253(m)24(^)469(eme)-253(que,)-277(dans)-253(ce)-259(cas,)-271(le)-259(pro)-30(duit)-259(de)-253(con)24(v)24(olution)-253(est)-259(con)24(tin)24(u\).)]TJ 0 -1.2048 Td [(V)84(oici)-337(un)-337(autre)-337(exemple)-337(imp)-36(ortan)24(t)-337(de)-331(cet)-337(e\256et)-331(r)24(\266)475(egularisan)24(t.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 46.374 Td [(VI)-35(I)-35(I.6.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 55.593 Td [(3.)]TJ /F15 1 Tf 1.747 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0301 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(g)]TJ /F12 1 Tf 0.84337 0 Td [(des)-331(fonctions)-337(d)30(\266)469(e\257nies)-337(sur)]TJ /F3 1 Tf 11.211 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.336 80.15 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.422 55.593 Td [(.)-445(On)-331(supp)-30(ose)]TJ /F8 1 Tf 6.1446 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(de)]TJ -38.163 -1.1988 Td [(classe)]TJ /F8 1 Tf 2.7892 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.75 78.435 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7048 54.394 Td [(\265)500(a)-331(supp)-30(ort)-337(compact)-337(et)]TJ /F8 1 Tf 9.7048 0 Td [(g)]TJ /F12 1 Tf 0.8494 0 Td [(lo)-24(calemen)24(t)-337(in)24(t)24(\266)475(egrable.)]TJ -14.928 -1.6024 Td [(Alors)-331(le)-343(pro)-30(duit)-331(de)-337(con)30(v)24(olution)]TJ /F8 1 Tf 13.988 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F12 1 Tf 0.8494 0 Td [(est)-337(d)24(\266)475(e\257ni)-337(partout)-337(et)-337(est)-337(de)-337(classe)]TJ /F8 1 Tf 14.753 0 Td [(C)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.31 76.142 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 37.711 52.792 Td [(.)]TJ -32.38 -1.6024 Td [(En)-427(outre,)-457(p)-24(our)-433(tout)]TJ /F8 1 Tf 9.3916 0 Td [(n)]TJ /F12 1 Tf 0.60241 0 Td [(-uplet)]TJ /F8 1 Tf 2.9819 0 Td [(\256)]TJ /F12 1 Tf 1.0723 0 Td [(tel)-427(que)]TJ /F6 1 Tf 3.494 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F11 1 Tf 0.71084 0 Td [(6)]TJ /F8 1 Tf 1.2108 0 Td [(p)]TJ /F12 1 Tf 0.5 0 Td [(,)-451(en)-427(notan)24(t)]TJ /F8 1 Tf 5.4458 0 Td [(D)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.534 73.849 Td [(\256)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 33.518 51.189 Td [(l'op)0(\266)469(erateur)-427(de)-427(d)24(\266)469(eriv)54(ation)]TJ -28.187 -1.1988 Td [(d'ordre)]TJ /F8 1 Tf 3.4518 0 Td [(\256)]TJ /F12 1 Tf 0.64458 0 Td [(,)-331(on)-337(a)]TJ /F8 1 Tf 4.5 0 Td [(D)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.155 72.133 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.355 49.991 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F6 1 Tf 0.81928 0 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\))-277(=)]TJ /F9 1 Tf 1.7229 0.81326 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(D)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.819 72.133 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 21.404 49.991 Td [(f)]TJ /F9 1 Tf 0.60241 0.81326 Td [(\242)]TJ /F6 1 Tf 0.68073 -0.81326 Td [(\244)]TJ /F8 1 Tf 0.72289 0 Td [(g)]TJ /F12 1 Tf 0.51205 0 Td [(.)]TJ /F1 1 Tf -18.59 -1.8976 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(Soit)]TJ /F8 1 Tf 2.1145 0 Td [(K)]TJ /F2 1 Tf 1.3193 0 Td [(le)-397(supp)-30(ort)-397(de)]TJ /F8 1 Tf 6.2771 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(,)]TJ /F8 1 Tf 0.68675 0 Td [(f)]TJ /F2 1 Tf 0.96386 0 Td [(\266)469(etan)24(t)-397(con)30(tin)18(ue)-391(sur)]TJ /F8 1 Tf 8.4699 0 Td [(K)]TJ /F2 1 Tf 1.3193 0 Td [(y)-397(est)-397(ma)-60(jor)24(\266)469(ee,)-409(d'o)-36(\265)530(u)]TJ -30.566 -1.2048 Td [(l'existence)-391(de)]TJ /F8 1 Tf 6.253 0 Td [(M)]TJ /F2 1 Tf 1.4639 0 Td [(tel)-391(que)]TJ /F6 1 Tf 3.4157 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.64458 0 Td [(6)]TJ /F8 1 Tf 1.1446 0 Td [(M)]TJ /F2 1 Tf 1.0843 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.724 66.883 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.542 46.888 Td [(.)-602(Alors,)-403(p)-30(our)]TJ /F8 1 Tf 6.2892 0 Td [(x)]TJ /F6 1 Tf 0.93976 0 Td [(2)]TJ /F3 1 Tf 1.0361 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.69 67.693 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.072 46.888 Td [(,)-403(notons)]TJ /F8 1 Tf 3.9639 0 Td [(K)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 51.353 66.883 Td [(x)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.771 46.888 Td [(le)-397(compact)-391(image)]TJ -31.44 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3855 0 Td [(K)]TJ /F2 1 Tf 1.3072 0 Td [(par)-385(l'application)]TJ /F8 1 Tf 7.6145 0 Td [(u)]TJ /F6 1 Tf 0.93374 0 Td [(7!)]TJ /F8 1 Tf 1.3675 0 Td [(x)]TJ /F6 1 Tf 0.83133 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(u)]TJ /F2 1 Tf 0.57229 0 Td [(,)-397(on)-391(a)-385(donc)]TJ /F6 1 Tf 5.4458 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.83133 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.64458 0 Td [(6)]TJ /F8 1 Tf 1.1446 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.422 65.168 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.91 45.689 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.83133 0 Td [(\241)]TJ /F8 1 Tf 1.0361 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-367(=)]TJ /F8 1 Tf 1.8976 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.328 65.168 Td [(K)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 81.47 90.885 Td [(x)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.229 45.689 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\),)-403(par)]TJ -36.651 -1.1988 Td [(suite)]TJ /F6 1 Tf 2.4036 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.8012 0 Td [(\241)]TJ /F8 1 Tf 1.006 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.57229 0 Td [(6)]TJ /F8 1 Tf 1.0663 0 Td [(M)]TJ /F6 1 Tf 1.0843 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24 63.452 Td [(K)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 34.892 88.475 Td [(x)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.94 44.491 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F2 1 Tf 0.28313 0 Td [(.)-469(La)-349(fonction)-349(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.586 63.452 Td [(K)]TJ /F13 1 Tf 5.976 0 0 5.976 0 0 Tm 56.675 88.475 Td [(x)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 28.831 44.491 Td [(g)]TJ /F2 1 Tf 0.8253 0 Td [(\266)469(etan)24(t)-349(in)30(t)24(\266)469(egrable)-343(par)-343(h)24(yp)-36(oth)24(\265)475(ese,)-349(la)]TJ -24.325 -1.2048 Td [(fonction)]TJ /F8 1 Tf 3.8253 0 Td [(t)]TJ /F6 1 Tf 0.63855 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.72289 0 Td [(\241)]TJ /F8 1 Tf 0.93373 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-301(l'est)-271(\266)469(egalemen)30(t.)-439(Le)-301(pro)-30(duit)-301(de)-295(con)24(v)24(olution)]TJ /F8 1 Tf 19.392 0 Td [(f)]TJ /F6 1 Tf 0.74699 0 Td [(\244)]TJ /F8 1 Tf 0.6506 0 Td [(g)]TJ /F2 1 Tf 0.81326 0 Td [(est)-301(donc)-301(partout)]TJ -32 -1.1988 Td [(d)30(\266)469(e\257ni)-337(et)-337(\257ni.)]TJ 0 -1.8976 Td [(Alors,)-313(p)-30(our)-307(tout)]TJ /F8 1 Tf 7.3795 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(-uplet)]TJ /F8 1 Tf 2.8614 0 Td [(\256)]TJ /F2 1 Tf 0.94578 0 Td [(a)24(v)24(ec)]TJ /F6 1 Tf 2.1687 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(p)]TJ /F2 1 Tf 0.81325 0 Td [(et)-307(tout)]TJ /F8 1 Tf 3.2771 0 Td [(t)]TJ /F6 1 Tf 0.63855 0 Td [(2)]TJ /F3 1 Tf 0.94579 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 40.379 58.099 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.759 40.189 Td [(,)-313(la)-307(fonction)]TJ /F8 1 Tf 5.506 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2831 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.74096 0 Td [(\241)]TJ /F8 1 Tf 0.94578 0 Td [(t)]TJ /F2 1 Tf 0.35542 0 Td [(\))]TJ /F8 1 Tf 0.39156 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))-307(admet)]TJ -35.747 -1.9819 Td [(une)-331(d)24(\266)469(eriv)54(\266)469(ee)-337(d'ordre)]TJ /F8 1 Tf 8.7349 0 Td [(\256)]TJ /F2 1 Tf 0.9759 0 Td [(qui)-331(v)48(aut)]TJ /F8 1 Tf 4.0663 0.74096 Td [(@)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 28.181 56.254 Td [(j)]TJ /F7 1 Tf 0.34483 0 Td [(\256)]TJ /F10 1 Tf 0.74138 0 Td [(j)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 20.741 39.762 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81326 Td [(\242)]TJ ET 0.504 w 228.38 459.64 m 320.83 459.64 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.108 37.521 Td [(@)-60(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.302 54.107 Td [(\256)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.241 38.207 Td [(=)]TJ /F9 1 Tf 1.0542 0.81326 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81326 Td [(D)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.371 55.271 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 30.181 38.207 Td [(f)]TJ /F9 1 Tf 0.59638 0.81326 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81326 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F6 1 Tf 0.79518 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.55421 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\).)]TJ -30.271 -1.8795 Td [(P)30(ar)-367(cons)24(\266)469(equen)24(t,)-379(p)-30(our)-373(justi\257er)-367(le)-373(reste)-373(de)-373(la)-367(prop)-30(osition)-373(il)-367(su\261t)-373(de)-367(prouv)24(er)-373(que)-373(l'on)-373(p)-30(eut)]TJ 0 -1.2048 Td [(appliquer)-331(le)-331(th)18(\266)475(eor)18(\265)475(eme)-331(de)-325(d)24(\266)469(eriv)54(ation)-331(sous)-331(l'in)24(t)24(\266)475(egrale.)-445(Nous)-331(dev)24(ons)-331(donc,)-337(p)-30(our)-325(tout)-337(p)-24(oin)24(t)]TJ /F8 1 Tf 38.633 0 Td [(a)]TJ /F2 1 Tf -38.633 -1.1988 Td [(de)]TJ /F3 1 Tf 1.3373 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 10.578 49.142 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2711 33.924 Td [(et)-337(p)-30(our)-337(tout)]TJ /F8 1 Tf 5.7108 0 Td [(n)]TJ /F2 1 Tf 0.59639 0 Td [(-uplet)]TJ /F8 1 Tf 2.8976 0 Td [(\256)]TJ /F2 1 Tf 0.9759 0 Td [(a)24(v)24(ec)]TJ /F6 1 Tf 2.1988 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(\256)]TJ /F6 1 Tf 0.64458 0 Td [(j)]TJ /F11 1 Tf 0.56024 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(p)]TJ /F2 1 Tf 0.50602 0 Td [(,)-337(trouv)24(er)-343(un)-337(v)24(oisinage)]TJ /F8 1 Tf 9.8614 0 Td [(!)]TJ /F2 1 Tf 0.99398 0 Td [(de)]TJ /F8 1 Tf 1.3373 0 Td [(a)]TJ /F2 1 Tf 0.86747 0 Td [(et)-337(une)-337(relation)-337(de)]TJ -31.428 -1.1988 Td [(domination)]TJ /F9 1 Tf 5.3133 0.8494 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ 0.33735 0.56024 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.80723 Td [(D)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.595 47.418 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 12.861 32.726 Td [(f)]TJ /F9 1 Tf 0.60241 0.80723 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.80723 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F6 1 Tf 0.81928 0 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.8494 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ /F11 1 Tf 0.66868 -0.24699 Td [(6)]TJ /F8 1 Tf 1.1145 0 Td [(h)]TJ /F2 1 Tf 0.57831 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\),)-385(v)54(alable)-373(p)-24(our)-373(tout)]TJ /F8 1 Tf 9.0482 0 Td [(t)]TJ /F2 1 Tf 0.72892 0 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(R)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.293 47.418 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.657 32.726 Td [(et)-373(tout)]TJ /F8 1 Tf 3.4096 0 Td [(x)]TJ /F2 1 Tf 0.93976 0 Td [(de)]TJ /F8 1 Tf 1.3675 0 Td [(!)]TJ /F2 1 Tf 0.65663 0 Td [(,)-379(a)24(v)24(ec)]TJ /F8 1 Tf 2.8855 0 Td [(h)]TJ /F2 1 Tf -38.584 -1.1988 Td [(in)24(t)24(\266)475(egrable.)]TJ 0 -1.6024 Td [(P)30(our)-349(cela)-343(prenons)-349(p)-30(our)]TJ /F8 1 Tf 10.572 0 Td [(!)]TJ /F2 1 Tf 1 0 Td [(n'imp)-30(orte)-349(quel)-349(v)24(oisinage)-349(compact)-349(de)]TJ /F8 1 Tf 16.404 0 Td [(a)]TJ /F2 1 Tf 0.87349 0 Td [(\(par)-349(exemple)-349(une)-343(b)-30(oule)]TJ -28.849 -1.1988 Td [(ferm)30(\266)469(ee)-277(cen)24(tr)24(\266)469(ee)-277(en)]TJ /F8 1 Tf 7.7711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\).)-427(La)-277(d)24(\266)469(eriv)54(\266)469(ee)-277(d'ordre)]TJ /F8 1 Tf 9.2169 0 Td [(\256)]TJ /F2 1 Tf 0.91566 0 Td [(de)]TJ /F8 1 Tf 1.2711 0 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(est)-277(encore)-277(\265)499(a)-277(supp)-30(ort)-277(compact)-277(con)30(ten)24(u)-277(dans)]TJ /F8 1 Tf -20.578 -1.2048 Td [(K)]TJ /F2 1 Tf 0.92169 0 Td [(,)-277(il)-265(existe)-265(donc)-265(une)-259(ma)-60(joration)-271(du)-265(t)24(yp)-30(e)]TJ /F9 1 Tf 16.88 0.8494 Td [(\257)]TJ 0 -0.59638 Td [(\257)]TJ /F8 1 Tf 0.33133 -0.25301 Td [(D)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 34.802 39.978 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.892 27.521 Td [(f)]TJ /F9 1 Tf 0.59639 0.8494 Td [(\257)]TJ 0 -0.59638 Td [(\257)]TJ /F11 1 Tf 0.60843 -0.25301 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(M)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 40.405 39.978 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.512 27.521 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.612 39.168 Td [(K)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.849 27.521 Td [(.)-421(Alors,)-277(si)-265(l'on)-265(note)]TJ /F8 1 Tf 8.5301 0 Td [(K)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.241 39.978 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.843 27.521 Td [(le)-265(compact)]TJ -34.512 -1.3012 Td [(image)-319(de)]TJ /F8 1 Tf 4.1928 0 Td [(K)]TJ /F6 1 Tf 1.1205 0 Td [(\243)]TJ /F8 1 Tf 0.96988 0 Td [(!)]TJ /F2 1 Tf 0.9759 0 Td [(par)-325(l'application)-325(\()]TJ /F8 1 Tf 7.8675 0 Td [(u;)-168(x)]TJ /F2 1 Tf 1.5904 0 Td [(\))]TJ /F6 1 Tf 0.66868 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(x)]TJ /F6 1 Tf 0.76506 0 Td [(\241)]TJ /F8 1 Tf 0.9759 0 Td [(u)]TJ /F2 1 Tf 0.89156 0 Td [(on)-319(a)]TJ /F9 1 Tf 2.1928 0.85542 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ 0.33735 0.56024 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(D)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 43.603 38.116 Td [(\256)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.042 26.22 Td [(f)]TJ /F9 1 Tf 0.59639 0.81325 Td [(\242)]TJ /F2 1 Tf 0.45783 -0.81325 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(x)]TJ /F6 1 Tf 0.76506 0 Td [(\241)]TJ /F8 1 Tf 0.96988 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F8 1 Tf 0.55422 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.85542 Td [(\257)]TJ 0 -0.60241 Td [(\257)]TJ /F11 1 Tf 0.60843 -0.25301 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(M)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 56.586 38.116 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.819 26.22 Td [(1)295(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 57.793 37.306 Td [(K)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 82.205 52.548 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.428 26.22 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36144 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\).)]TJ -38.494 -1.1988 Td [(Comme)-391(la)-397(fonction)-397(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.052 35.59 Td [(K)]TJ /F18 1 Tf 5.976 0 0 5.976 0 0 Tm 30.867 50.138 Td [(0)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.753 25.021 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(j)]TJ /F2 1 Tf 0.66867 0 Td [(est)-397(in)24(t)24(\266)469(egrable)-391(par)-391(h)24(yp)-36(oth)24(\265)469(ese)-391(on)-391(a)-391(une)-397(relation)-391(de)-391(domination)]TJ -11.88 -1.1988 Td [(du)-337(t)24(yp)-30(e)-337(souhait)18(\266)475(ee.)]TJ ET 512.64 283.5 m 519.84 283.5 l 519.84 291.92 l 512.64 291.92 l f BT /F5 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 3.8882 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(126)]TJ ET endstream endobj 414 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 413 0 R /ProcSet 2 0 R >> /Contents 412 0 R >> endobj 413 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F13 39 0 R /F15 68 0 R /F18 107 0 R >> endobj 416 0 obj << /Length 13906 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.461 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-16138(IX.)-391(Les)-385(espaces)-391(L)]TJ /F20 1 Tf 8.352 0 0 8.352 0 0 Tm 62.879 93.245 Td [(p)]TJ /F0 1 Tf 20.592 0 0 20.592 0 0 Tm 7.1888 31.746 Td [(Chapitre)-377(IX.)-1129(Les)-381(espaces)-374(L)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 30.603 46.625 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 50.141 Td [(Dans)-337(tout)-337(ce)-337(c)24(hapitre)-331(on)-337(\257xe)-337(un)-337(espace)-337(mesur)30(\266)469(e)-337(\()]TJ /F8 1 Tf 21.404 0 Td [(X)-30(;)]TJ /F6 1 Tf 1.3012 0 Td [(T)]TJ /F8 1 Tf 0.79518 0 Td [(;)-168(m)]TJ /F2 1 Tf 1.3253 0 Td [(\).)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 37.992 Td [(IX.1.)-236(Rapp)-35(els)-376(sur)-376(les)-376(fonctions)-376(con)30(v)30(exes.)]TJ 0 -2.6533 Td [(IX.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 42.364 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83133 0.084335 Td [(\266)]TJ -0.0060234 -0.084335 Td [(efinition.)]TJ /F12 1 Tf 5.2651 0 Td [(Soien)24(t)]TJ /F8 1 Tf 3.0181 0 Td [(I)]TJ /F12 1 Tf 0.84337 0 Td [(un)-325(in)24(terv)54(alle)-319(de)]TJ /F3 1 Tf 7.0843 0 Td [(R)]TJ /F12 1 Tf 1.0482 0 Td [(et)]TJ /F8 1 Tf 1.1566 0 Td [(f)]TJ /F12 1 Tf 0.91566 0 Td [(une)-325(application)-331(de)]TJ /F8 1 Tf 8.3614 0 Td [(I)]TJ /F12 1 Tf 0.84338 0 Td [(dans)]TJ /F3 1 Tf 2.3253 0 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)-445(On)]TJ -37.108 -1.1988 Td [(dit)-331(que)]TJ /F8 1 Tf 3.4096 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(est)]TJ /F1 1 Tf 1.5542 0 Td [(con)30(v)24(exe)]TJ /F12 1 Tf 4.259 0 Td [(si)-337(p)-30(our)-331(tout)]TJ /F8 1 Tf 5.5301 0 Td [(a)]TJ /F12 1 Tf 0.85542 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(b)]TJ /F12 1 Tf 0.75904 0 Td [(dans)]TJ /F8 1 Tf 2.3374 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(et)-331(tout)]TJ /F8 1 Tf 3.3253 0 Td [(u)]TJ /F12 1 Tf 0.90361 0 Td [(et)]TJ /F8 1 Tf 1.1627 0 Td [(v)]TJ /F12 1 Tf 0.8494 0 Td [(p)-30(ositifs)-331(tels)-331(que)]TJ /F8 1 Tf 7.1446 0 Td [(u)]TJ /F2 1 Tf 0.78916 0 Td [(+)]TJ /F8 1 Tf 0.98795 0 Td [(v)]TJ /F2 1 Tf 0.8012 0 Td [(=)-277(1)]TJ /F12 1 Tf -37.608 -1.2048 Td [(on)-331(a)-337(:)]TJ /F2 1 Tf 10.657 -1.1988 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(\))]TJ /F8 1 Tf 2.3855 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(ua)]TJ /F2 1 Tf 1.3253 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(v)-42(b)]TJ /F2 1 Tf 0.95181 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(uf)]TJ /F2 1 Tf 1.1687 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))-228(+)]TJ /F8 1 Tf 1.6145 0 Td [(v)-36(f)]TJ /F2 1 Tf 1.1205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(\))]TJ /F12 1 Tf 0.71687 0 Td [(.)]TJ /F1 1 Tf -26.56 -2.0542 Td [(Remarques.)]TJ /F2 1 Tf 0 -1.7048 Td [(Les)-367(couples)-373(\()]TJ /F8 1 Tf 5.759 0 Td [(u;)-168(v)]TJ /F2 1 Tf 1.5422 0 Td [(\))-367(consid)24(\266)475(er)18(\266)475(es)-373(son)24(t)-373(les)-367(\()]TJ /F8 1 Tf 9.5843 0 Td [(t;)]TJ /F2 1 Tf 0.8012 0 Td [(1)]TJ /F6 1 Tf 0.74699 0 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(t)]TJ /F2 1 Tf 0.36145 0 Td [(\))-373(p)-30(our)]TJ /F8 1 Tf 3.1566 0 Td [(t)]TJ /F2 1 Tf 0.72892 0 Td [(parcouran)24(t)-373([0)]TJ /F8 1 Tf 5.9036 0 Td [(;)]TJ /F2 1 Tf 0.44579 0 Td [(1])-367(;)-385(p)-30(our)]TJ /F8 1 Tf 4.2048 0 Td [(a)]TJ /F2 1 Tf 0.89759 0 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(b)]TJ /F2 1 Tf 0.79518 0 Td [(dans)]TJ /F8 1 Tf -37.157 -1.1988 Td [(I)]TJ /F2 1 Tf 0.90964 0 Td [(les)]TJ /F8 1 Tf 1.512 0 Td [(ua)]TJ /F2 1 Tf 1.3675 0 Td [(+)]TJ /F8 1 Tf 1.0361 0 Td [(v)-42(b)]TJ /F2 1 Tf 1.3494 0 Td [(consid)30(\266)469(er)24(\266)469(es)-391(son)24(t)-403(les)-391(p)-30(oin)24(ts)-397(du)-397(segmen)24(t)-397([)]TJ /F8 1 Tf 17.295 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)]TJ /F8 1 Tf 0.44578 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(].)-626(La)-403(propri)24(\266)475(et)24(\266)469(e)-397(de)-397(con)30(v)24(exit)24(\266)475(e)-403(des)]TJ -24.873 -1.1988 Td [(in)24(terv)60(alles)-337(assure)-337(que)-337(ces)-337(p)-24(oin)24(ts)-337(son)24(t)-337(dans)]TJ /F8 1 Tf 18.717 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(et)-331(donc)-337(que)]TJ /F8 1 Tf 5.4157 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(ua)]TJ /F2 1 Tf 1.3253 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(v)-36(b)]TJ /F2 1 Tf 0.95181 0 Td [(\))-337(est)-337(bien)-337(d)30(\266)469(e\257ni.)]TJ -29.253 -1.7048 Td [(Lorsque)]TJ /F8 1 Tf 3.8795 0 Td [(u)]TJ /F2 1 Tf 1.0301 0 Td [(=)-457(0)-439(\(et)-445(donc)]TJ /F8 1 Tf 6.3373 0 Td [(v)]TJ /F2 1 Tf 0.9759 0 Td [(=)-457(1\))-439(la)-445(condition)-439(\()]TJ /F8 1 Tf 8.6687 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(\))-439(se)-445(r)30(\266)469(eduit)-439(\265)500(a)]TJ /F8 1 Tf 6.1024 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F11 1 Tf 0.8494 0 Td [(6)]TJ /F8 1 Tf 1.2349 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F2 1 Tf 0.43373 0 Td [(\),)-469(elle)-445(est)-445(tou-)]TJ -32.699 -1.1988 Td [(jours)-367(r)30(\266)469(ealis)24(\266)475(ee.)-542(On)-367(p)-30(eut)-367(faire)-373(une)-361(remarque)-367(analogue)-367(lorsque)]TJ /F8 1 Tf 26.855 0 Td [(v)]TJ /F2 1 Tf 0.8494 0 Td [(=)-331(0,)-373(ainsi,)-373(p)-30(our)-361(v)48(\266)475(eri\257er)-367(la)]TJ -27.705 -1.1988 Td [(condition)-337(\()]TJ /F8 1 Tf 4.7771 0 Td [(C)]TJ /F2 1 Tf 0.78916 0 Td [(\),)-337(il)-337(su\261t)-337(de)-331(le)-337(faire)-343(a)30(v)24(ec)]TJ /F8 1 Tf 11.235 0 Td [(u)]TJ /F2 1 Tf 0.90361 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(v)]TJ /F2 1 Tf 0.85542 0 Td [(strictemen)24(t)-337(p)-30(ositifs.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 21.123 Td [(IX.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 25.322 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 7.1988 0 Td [(Soit)]TJ /F8 1 Tf 2.1566 0 Td [(f)]TJ /F12 1 Tf 1.0361 0 Td [(une)-439(fonction)-445(con)24(tin)24(ue)-439(sur)-439(un)-445(in)30(terv)54(alle)]TJ /F8 1 Tf 17.843 0 Td [(I)]TJ /F12 1 Tf 0.95783 0 Td [(et)-439(deux)-439(fois)]TJ -33.892 -1.1988 Td [(d)30(\266)469(eriv)54(able)-445(sur)-445(l'in)24(t)24(\266)469(erieur)-439(de)]TJ /F8 1 Tf 12.283 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(.)-771(Alors,)-469(p)-30(our)-439(que)]TJ /F8 1 Tf 8.5482 0 Td [(f)]TJ /F12 1 Tf 1.0422 0 Td [(soit)-439(con)24(v)24(exe)-439(il)-445(faut)-445(et)-439(il)-445(su\261t)-445(que)-445(sa)]TJ -22.392 -1.2048 Td [(d)30(\266)469(eriv)54(\266)469(ee)-337(seconde)]TJ /F8 1 Tf 7.0663 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 18.595 33.392 Td [(00)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 13.837 22.918 Td [(soit)-337(p)-30(ositiv)30(e.)]TJ /F1 1 Tf -8.506 -2.0542 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 0 -1.6988 Td [(Supp)-24(osons)]TJ /F8 1 Tf 4.9699 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(con)30(v)24(exe.)-566(P)30(our)]TJ /F8 1 Tf 6.6988 0 Td [(a)]TJ /F2 1 Tf 0.90362 0 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(b)]TJ /F2 1 Tf 0.8012 0 Td [(distincts)-379(dans)-373(l'in)24(t)24(\266)475(erieur)-373(de)]TJ /F8 1 Tf 12.434 0 Td [(I)]TJ /F2 1 Tf 0.89157 0 Td [(la)-373(relation)-373(de)-373(con)30(v)24(exit)24(\266)475(e)]TJ -28.873 -1.7229 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.2289 0 Td [(u)]TJ /F2 1 Tf 0.90964 0 Td [(=)]TJ /F8 1 Tf 1.1145 0 Td [(v)]TJ /F2 1 Tf 0.86145 0 Td [(=)-337(1)]TJ /F8 1 Tf 1.6145 0 Td [(=)]TJ /F2 1 Tf 0.5 0 Td [(2)-367(donne)-373(\242)-337(=)]TJ /F8 1 Tf 6.253 0.6747 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-222(+)]TJ /F8 1 Tf 1.6084 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ ET 0.504 w 224.86 211.46 m 283.82 211.46 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.03 16.756 Td [(2)]TJ /F6 1 Tf 3.0843 0.68675 Td [(\241)]TJ /F8 1 Tf 1.0241 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.50602 0.6747 Td [(a)]TJ /F2 1 Tf 0.75301 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(b)]TJ ET 313.63 211.46 m 339.77 211.46 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.084 16.756 Td [(2)]TJ 1.4578 0.68675 Td [(\))]TJ /F11 1 Tf 0.72892 0 Td [(>)]TJ /F2 1 Tf 1.1145 0 Td [(0.)-554(La)-373(form)30(ule)-373(de)-367(T)78(a)24(ylor)-373(mon)24(tre)]TJ -25.054 -2.2169 Td [(alors)-361(que,)-373(si)]TJ /F8 1 Tf 5.6325 0 Td [(b)]TJ /F2 1 Tf 0.78916 0 Td [(tend)-367(v)24(ers)]TJ /F8 1 Tf 4.3976 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(,)-367(le)-367(rapp)-30(ort)]TJ 6.8133 0.68072 Td [(\242)]TJ ET 265.39 184.96 m 306.14 184.96 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.205 14.539 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F6 1 Tf 0.65663 0 Td [(\241)]TJ /F8 1 Tf 1 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 36.009 21.219 Td [(2)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.09 15.226 Td [(tend)-367(v)24(ers)]TJ /F8 1 Tf 4.5241 0.68072 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 44.664 23.28 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.723 15.906 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ ET 365.9 184.96 m 394.85 184.96 l S BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 31.572 14.539 Td [(8)]TJ 1.5783 0.68675 Td [(,)-367(la)-361(p)-30(ositivit)24(\266)475(e)-367(de)]TJ /F8 1 Tf 7.5663 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 59.121 22.383 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.825 15.226 Td [(\()]TJ /F8 1 Tf 0.39156 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))-361(en)]TJ -37.416 -1.7289 Td [(d)30(\266)469(ecoule.)]TJ 0 -2.4096 Td [(Recipro)-30(quemen)30(t,)-506(supp)-30(osons)]TJ /F8 1 Tf 12.783 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 26.776 16.461 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.693 11.087 Td [(d)30(\266)469(e\257nie)-469(et)-469(p)-30(ositiv)30(e)-475(sur)-469(l'in)24(t)24(\266)469(erieur)-469(de)]TJ /F8 1 Tf 16.44 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-849(Prenons)]TJ /F8 1 Tf 5.1205 0 Td [(a)]TJ /F2 1 Tf 0.99398 0 Td [(et)]TJ /F8 1 Tf 1.3012 0 Td [(b)]TJ /F2 1 Tf -38.735 -1.1988 Td [(dans)-463(l'in)24(t)24(\266)469(erieur)-469(de)]TJ /F8 1 Tf 8.6627 0 Td [(I)]TJ /F2 1 Tf 0.98795 0 Td [(et)]TJ /F8 1 Tf 1.2952 0 Td [(u)]TJ /F2 1 Tf 1.0422 0 Td [(et)]TJ /F8 1 Tf 1.3012 0 Td [(v)]TJ /F2 1 Tf 0.98795 0 Td [(p)-30(ositifs)-469(de)-463(somme)-469(1.)-843(P)24(osons)]TJ /F8 1 Tf 13.614 0 Td [(c)]TJ /F2 1 Tf 0.93373 0 Td [(=)]TJ /F8 1 Tf 1.2771 0 Td [(ua)]TJ /F2 1 Tf 1.4157 0 Td [(+)]TJ /F8 1 Tf 1.0904 0 Td [(v)-36(b)]TJ /F2 1 Tf 0.94579 0 Td [(,)-500(il)-469(s'agit)-469(de)]TJ -33.554 -1.1988 Td [(prouv)30(er)]TJ /F8 1 Tf 3.747 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(\))]TJ /F11 1 Tf 0.79518 0 Td [(6)]TJ /F8 1 Tf 1.1867 0 Td [(uf)]TJ /F2 1 Tf 1.1687 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))-277(+)]TJ /F8 1 Tf 1.7169 0 Td [(v)-42(f)]TJ /F2 1 Tf 1.1205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(b)]TJ /F2 1 Tf 0.43374 0 Td [(\).)-680(Comme)]TJ /F8 1 Tf 5.0843 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 34.216 13.021 Td [(00)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.416 8.6894 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-415(est)-415(p)-30(ositiv)30(e,)-433(la)-415(form)30(ule)-415(de)-409(T)78(a)24(ylor)-415(p)-30(ermet)]TJ -20.048 -1.1988 Td [(d')24(\266)475(ecrire)-439(:)]TJ /F8 1 Tf 4.5843 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(\))]TJ /F11 1 Tf 0.83133 0 Td [(>)]TJ /F8 1 Tf 1.2229 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(c)]TJ /F2 1 Tf 0.43374 0 Td [(\))-289(+)-295(\()]TJ /F8 1 Tf 2.1325 0 Td [(a)]TJ /F6 1 Tf 0.81928 0 Td [(\241)]TJ /F8 1 Tf 1.0663 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(\))]TJ /F8 1 Tf 0.38554 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 29.112 11.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.62 7.4906 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(\))-433(et)]TJ /F8 1 Tf 2.0843 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F11 1 Tf 0.83735 0 Td [(>)]TJ /F8 1 Tf 1.2169 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(c)]TJ /F2 1 Tf 0.43374 0 Td [(\))-295(+)-289(\()]TJ /F8 1 Tf 2.1325 0 Td [(b)]TJ /F6 1 Tf 0.71687 0 Td [(\241)]TJ /F8 1 Tf 1.0723 0 Td [(c)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F8 1 Tf 0.39157 0 Td [(f)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.31 11.306 Td [(0)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.042 7.4906 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(\).)-746(En)-433(m)24(ultiplian)18(t)-433(ces)]TJ -29.53 -1.2048 Td [(relations)-457(resp)-30(ectiv)24(emen)30(t)-457(par)]TJ /F8 1 Tf 13.054 0 Td [(u)]TJ /F2 1 Tf 1.0301 0 Td [(et)]TJ /F8 1 Tf 1.2892 0 Td [(v)]TJ /F2 1 Tf 0.9759 0 Td [(et)-457(en)-457(les)-457(a)-54(joutan)24(t)-463(on)-451(obtien)24(t)-457(\(puisque)]TJ /F8 1 Tf 17.705 0 Td [(u)]TJ /F2 1 Tf 0.87952 0 Td [(+)]TJ /F8 1 Tf 1.0783 0 Td [(v)]TJ /F2 1 Tf 1.006 0 Td [(=)-481(1\))]TJ -18.355 -2.3976 Td [(127)]TJ /F5 1 Tf 6.512 0 Td [(U.P.M.C.)-530(Cours)-530(de)-524(P.)-530(MAZET)]TJ ET endstream endobj 418 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 417 0 R /ProcSet 2 0 R >> /Contents 416 0 R >> endobj 417 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F8 22 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F20 415 0 R >> endobj 419 0 obj << /Length 18268 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.461 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-16138(IX.)-391(Les)-385(espaces)-391(L)]TJ /F20 1 Tf 8.352 0 0 8.352 0 0 Tm 62.879 93.245 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 62.22 Td [(uf)]TJ /F2 1 Tf 1.1687 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(a)]TJ /F2 1 Tf 0.5241 0 Td [(\))-216(+)]TJ /F8 1 Tf 1.5964 0 Td [(v)-36(f)]TJ /F2 1 Tf 1.1145 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(\))]TJ /F11 1 Tf 0.66867 0 Td [(>)]TJ /F8 1 Tf 1.0542 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(c)]TJ /F2 1 Tf 0.43373 0 Td [(\),)-331(ce)-331(qui)-331(est)-331(la)-331(relation)-331(souhait)24(\266)469(ee)-331(lorsque)-331(ni)]TJ /F8 1 Tf 19.211 0 Td [(a)]TJ /F2 1 Tf 0.85542 0 Td [(ni)]TJ /F8 1 Tf 1.1627 0 Td [(b)]TJ /F2 1 Tf 0.75904 0 Td [(n'est)-325(une)-331(extr)24(\266)475(emit)18(\266)475(e)]TJ -30.747 -1.1988 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-445(On)-337(obtien)30(t)-337(le)-343(cas)-331(g)24(\266)475(en)24(\266)469(eral)-331(par)-337(passage)-331(\265)500(a)-337(la)-331(limite)-337(gr)0(^)500(ace)-337(\265)500(a)-331(la)-337(con)30(tin)18(uit)24(\266)469(e)-337(de)]TJ /F8 1 Tf 32.735 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(.)]TJ ET 512.64 728.1 m 519.84 728.1 l 519.84 736.52 l 512.64 736.52 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 58.991 Td [(En)-331(particulier)-337(cette)-337(prop)-30(osition)-337(mon)24(tre)-337(la)-337(con)30(v)24(exit)24(\266)469(e)-337(des)-337(fonctions)-331(suiv)48(an)30(tes)-343(:)]TJ /F8 1 Tf 0 -1.6145 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.431 82.702 Td [(x)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 8.8253 57.376 Td [(sur)]TJ /F3 1 Tf 1.6747 0 Td [(R)]TJ /F2 1 Tf 1.0542 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F3 1 Tf 0.94578 0 Td [(R)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23.466 82.702 Td [(\244)]TJ /F4 1 Tf 0 -0.94828 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.392 57.376 Td [(,)]TJ /F8 1 Tf 1.753 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F8 1 Tf 1.2771 0 Td [(x)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.259 82.702 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.639 57.376 Td [(sur)]TJ /F3 1 Tf 1.6747 0 Td [(R)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 35.828 81.892 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.036 57.376 Td [(p)-30(our)-337(1)]TJ /F11 1 Tf 3.1446 0 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(p)-283(<)]TJ /F2 1 Tf 1.8373 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.3374 0 Td [(,)]TJ /F8 1 Tf 1.747 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(7!)]TJ /F2 1 Tf 1.2831 0 Td [(ln)0(\()]TJ 1.3735 0.6747 Td [(1)]TJ ET 0.504 w 471.02 688.75 m 477.86 688.75 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.41 56.689 Td [(x)]TJ /F2 1 Tf 0.68675 0.68674 Td [(\))-337(sur)]TJ /F3 1 Tf 2.3976 0 Td [(R)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 61.845 82.702 Td [(\244)]TJ /F4 1 Tf 0 -0.94828 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 57.376 Td [(.)]TJ -38.886 -1.8253 Td [(Nous)-331(utiliserons)-337(le)-343(dernier)-331(exemple)-343(par)-331(son)-343(corollaire)-331(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 43.912 Td [(IX.1.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 52.641 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(a)]TJ /F12 1 Tf 0.86144 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(b)]TJ /F12 1 Tf 0.76506 0 Td [(dans)]TJ /F3 1 Tf 2.3374 0 Td [(R)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 36.043 75.926 Td [(\244)]TJ /F4 1 Tf 0 -0.94827 Td [(+)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.187 52.641 Td [(et)-331(p)-30(our)]TJ /F8 1 Tf 3.5301 0 Td [(u)]TJ /F12 1 Tf 0.90362 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(v)]TJ /F12 1 Tf 0.85542 0 Td [(p)-30(ositifs)-337(de)-331(somme)-337(1)-331(on)-337(a)]TJ /F8 1 Tf -11.157 -1.1988 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.509 74.211 Td [(u)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.542 51.442 Td [(b)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.871 74.211 Td [(v)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 23.723 51.442 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(ua)]TJ /F2 1 Tf 1.3253 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(v)-36(b)]TJ /F12 1 Tf 0.94578 0 Td [(.)]TJ /F1 1 Tf -22.723 -2.1446 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(La)-469(con)30(v)24(exit)24(\266)475(e)-469(de)-469(ln)0(\()]TJ 8.9578 0.68072 Td [(1)]TJ ET 275.76 592.27 m 282.6 592.27 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.072 48.611 Td [(x)]TJ /F2 1 Tf 0.69277 0.68675 Td [(\))-463(assure)-469(ln)0(\()]TJ 6.7289 0.68072 Td [(1)]TJ ET 347.9 592.27 m 387.07 592.27 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.108 48.611 Td [(ua)]TJ /F2 1 Tf 1.3193 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(v)-42(b)]TJ /F2 1 Tf 1.0723 0.68675 Td [(\))]TJ /F11 1 Tf 0.88554 0 Td [(6)]TJ /F8 1 Tf 1.2771 0 Td [(u)]TJ /F2 1 Tf 0.74096 0 Td [(ln)0(\()]TJ 1.3554 0.68072 Td [(1)]TJ ET 439.2 592.27 m 445.54 592.27 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.747 48.611 Td [(a)]TJ /F2 1 Tf 0.64458 0.68675 Td [(\))-313(+)]TJ /F8 1 Tf 1.7892 0 Td [(v)]TJ /F2 1 Tf 0.68675 0 Td [(ln)-6(\()]TJ 1.3434 0.68072 Td [(1)]TJ ET 492.55 592.27 m 498.6 592.27 l S BT /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.247 48.611 Td [(b)]TJ /F2 1 Tf 0.58434 0.68675 Td [(\))-469(;)-530(en)]TJ -36.5 -1.5602 Td [(prenan)24(t)-307(l'exp)-30(onen)30(tielle)-313(puis)-301(les)-313(in)30(v)24(erses)-313(des)-307(deux)-301(mem)24(bres)-313(on)-307(obtien)30(t)-307(la)-307(relation)-307(annonc)24(\266)469(ee)]TJ ET 512.64 555.01 m 519.84 555.01 l 519.84 563.43 l 512.64 563.43 l f BT /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 35.203 Td [(IX.2.)-236(Les)-381(semi-normes)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.777 42.201 Td [(k)-451(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.948 60.176 Td [(p)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 17.633 35.203 Td [(.)]TJ -13.186 -2.4271 Td [(IX.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 39.292 Td [(1.)]TJ /F15 1 Tf 1.753 0 Td [(D)]TJ 0.83133 0.084339 Td [(\266)]TJ -0.0060234 -0.084339 Td [(efinition.)]TJ /F12 1 Tf 5.2711 0 Td [(Soit)]TJ /F8 1 Tf 2.0542 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(une)-337(fonction)-337(mesurable)-337(sur)]TJ /F8 1 Tf 12.163 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(\265)499(a)-331(v)48(aleurs)-337(dans)]TJ ET 466.85 479.59 m 475.49 479.59 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 39.06 39.292 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)]TJ -34.452 -2.3614 Td [(P)30(our)]TJ /F8 1 Tf 2.4337 0 Td [(p)]TJ /F6 1 Tf 0.78313 0 Td [(2)]TJ /F2 1 Tf 0.66265 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [([)]TJ /F12 1 Tf 0.60843 0 Td [(on)-337(p)-30(ose)]TJ /F6 1 Tf 5.6446 0 Td [(k)]TJ /F8 1 Tf 0.50603 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.724 52.271 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.759 36.93 Td [(=)]TJ /F9 1 Tf 1.0602 1.4096 Td [(\265)]TJ 0.90362 -0.048191 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.595 53.435 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 25.506 36.93 Td [(dm)]TJ /F9 1 Tf 1.3976 1.4096 Td [(\266)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 39.552 54.564 Td [(1)]TJ /F7 1 Tf 0.56897 0 Td [(=p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 28.91 36.93 Td [(,)]TJ -23.578 -2.3313 Td [(\(a)24(v)24(ec)-337(la)-331(con)24(v)24(en)30(tion)]TJ /F2 1 Tf 8.6988 0 Td [(\(+)]TJ /F6 1 Tf 1.1627 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.733 50.099 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.325 34.599 Td [(=)-277(\(+)]TJ /F6 1 Tf 2.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 29.966 50.099 Td [(1)]TJ /F7 1 Tf 0.56897 0 Td [(=p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 22.488 34.599 Td [(=)-277(+)]TJ /F6 1 Tf 1.8313 0 Td [(1)]TJ /F12 1 Tf 1.006 0 Td [(\).)]TJ -19.994 -1.5422 Td [(De)-325(m)24(^)469(eme)-319(on)-325(note)]TJ /F6 1 Tf 8.006 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 21.371 46.728 Td [(1)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 16.054 33.057 Td [(la)-325(b)-30(orne)-325(inf)30(\266)469(erieure)-325(des)]TJ /F8 1 Tf 10.036 0 Td [(M)]TJ /F12 1 Tf 1.4036 0 Td [(de)]TJ ET 344.38 405 m 353.02 405 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 28.813 33.057 Td [(R)]TJ /F12 1 Tf 1.0482 0 Td [(qui)-325(ma)-54(joren)18(t)]TJ /F6 1 Tf 5.9578 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F12 1 Tf 0.60241 0 Td [(presque)-325(partout.)]TJ /F1 1 Tf -31.964 -1.7892 Td [(Remarques.)]TJ 0.81325 -1.5422 Td [(1.)]TJ /F2 1 Tf 1.8614 0 Td [(Il)-487(existe)-487(des)]TJ /F8 1 Tf 5.9699 0 Td [(M)]TJ /F2 1 Tf 1.5663 0 Td [(qui)-481(ma)-60(joren)24(t)]TJ /F6 1 Tf 6.2771 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.76506 0 Td [(presque)-487(partout,)-530(par)-481(exemple)]TJ /F8 1 Tf 13.831 0 Td [(M)]TJ /F2 1 Tf 1.6084 0 Td [(=)-530(+)]TJ /F6 1 Tf 2.0843 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)-903(On)]TJ -36.651 -1.1988 Td [(p)-30(eut)-367(donc)-373(trouv)24(er)-367(une)-373(suite)-367(de)-367(tels)-337(\266)469(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 20.024 0 Td [(M)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.672 40.607 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.235 28.527 Td [(qui)-367(tend)-373(v)24(ers)-367(la)-367(b)-30(orne)-367(inf)24(\266)469(erieure)]TJ /F6 1 Tf 14.584 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.138 40.245 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 28.527 Td [(.)]TJ -38.886 -1.2048 Td [(P)30(our)-493(c)24(haque)-487(indice)]TJ /F8 1 Tf 9.1265 0 Td [(n)]TJ /F2 1 Tf 1.0904 0 Td [(il)-493(y)-493(a)-487(donc)-493(un)-487(ensem)18(ble)-487(n)24(\266)475(egligeable)]TJ /F8 1 Tf 16.886 0 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.491 38.892 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.217 27.322 Td [(en)-493(dehors)-493(duquel)-493(on)-487(a)]TJ /F6 1 Tf -28.886 -1.3434 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(M)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.509 36.961 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.681 25.979 Td [(.)-445(Alors)]TJ /F8 1 Tf 3.3614 0 Td [(A)]TJ /F2 1 Tf 1.0301 0 Td [(=)]TJ /F9 1 Tf 1.0542 0.95181 Td [([)]TJ /F8 1 Tf 1.2771 -0.95181 Td [(A)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.983 36.961 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 19.024 25.979 Td [(est)-331(un)-325(ensem)24(ble)-331(n)30(\266)469(egligeable)-325(et,)-331(en)-325(dehors)-331(de)]TJ /F8 1 Tf 19.596 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(,)-325(on)-331(a,)-325(p)-30(our)]TJ -34.042 -1.3976 Td [(tout)]TJ /F8 1 Tf 2.247 0 Td [(n)]TJ /F2 1 Tf 0.60241 0 Td [(,)]TJ /F6 1 Tf 0.71084 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.69277 0 Td [(6)]TJ /F8 1 Tf 1.1928 0 Td [(M)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.991 34.961 Td [(n)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.928 24.581 Td [(;)-451(en)-415(passan)24(t)-421(\265)500(a)-409(la)-415(limite)-415(on)-415(obtien)24(t)]TJ /F6 1 Tf 15.813 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F11 1 Tf 0.68675 0 Td [(6)]TJ /F6 1 Tf 1.1928 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.155 34.599 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.657 24.581 Td [(en)-415(dehors)-421(de)]TJ /F8 1 Tf 6.0904 0 Td [(A)]TJ /F2 1 Tf -38.416 -1.2048 Td [(donc)-319(presque)-319(partout.)-445(Ainsi)]TJ /F6 1 Tf 12.578 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 27.914 32.883 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 20.627 23.376 Td [(est)-319(lui-m)24(^)469(eme)-313(un)-325(ma)-54(joran)24(t)-319(de)]TJ /F6 1 Tf 12.934 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.59639 0 Td [(presque)-319(partout,)-325(c'est)]TJ -29.729 -1.1988 Td [(\266)469(evidemmen)24(t)-337(le)-337(plus)-331(p)-30(etit.)]TJ /F1 1 Tf 0.84337 -1.5422 Td [(2.)]TJ /F2 1 Tf 1.6386 0 Td [(La)-379(mo)-30(di\257cation)-373(de)]TJ /F8 1 Tf 8.6627 0 Td [(f)]TJ /F2 1 Tf 0.96988 0 Td [(sur)-379(un)-373(ensem)18(ble)-373(n)30(\266)469(egligeable)-373(ne)-373(mo)-30(di\257e)-349(\266)475(evidemmen)24(t)-379(pas)]TJ /F6 1 Tf 25.066 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.086 28.952 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 19.25 Td [(\(p)-30(our)-289(0)]TJ /F8 1 Tf 3.4819 0 Td [(<)-283(p)]TJ /F11 1 Tf 1.8373 0 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(\),)-301(on)-289(p)-24(eut)-295(donc)-289(encore)-289(donner)-289(un)-289(sens)-289(\265)500(a)]TJ /F6 1 Tf 17.524 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.672 26.969 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.313 19.25 Td [(lorsque)]TJ /F8 1 Tf 3.3795 0 Td [(f)]TJ /F2 1 Tf 0.87952 0 Td [(n'est)-289(pas)-295(d)30(\266)469(e\257nie)]TJ -32.241 -1.3916 Td [(partout)-283(mais)-277(seulemen)30(t)-283(presque)-277(partout.)-433(On)-277(p)-30(eut)-253(\266)469(egalemen)30(t)-283(d)30(\266)469(e\257nir)]TJ /F6 1 Tf 29.572 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.233 24.987 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.187 17.858 Td [(lorsque)]TJ /F8 1 Tf 3.3735 0 Td [(f)]TJ /F2 1 Tf 0.87349 0 Td [(est)-283(une)]TJ -36.102 -1.3313 Td [(classe)-337(de)-331(fonctions)-337(mo)-30(dulo)-337(l')24(\266)469(egalit)24(\266)475(e)-337(presque)-337(partout.)]TJ /F1 1 Tf 0.81325 -1.5422 Td [(3.)]TJ /F2 1 Tf 1.6386 0 Td [(Lorsque)]TJ /F6 1 Tf 3.8133 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.888 20.875 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.982 14.985 Td [(est)-379(\257ni,)]TJ /F8 1 Tf 3.6566 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))-379(est)-379(\257ni)-373(presque)-379(partout.)-566(C'est)-355(\266)475(eviden)18(t)-373(lorsque)]TJ /F8 1 Tf 21.554 0 Td [(p)]TJ /F2 1 Tf 0.8494 0 Td [(=)-343(+)]TJ /F6 1 Tf 1.8976 0 Td [(1)]TJ /F2 1 Tf -38.163 -1.3855 Td [(et,)-457(p)-30(our)]TJ /F8 1 Tf 4.0361 0 Td [(p)]TJ /F2 1 Tf 0.93373 0 Td [(\257ni,)-463(cela)-439(signi\257e)-433(que)]TJ /F6 1 Tf 9.6325 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.172 20.056 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 21.982 13.599 Td [(est)-439(in)24(t)24(\266)475(egrable)-433(donc)-439(\257ni)-433(presque)-439(partout.)-746(On)-439(p)-30(eut)]TJ -16.651 -1.1988 Td [(donc,)-349(dans)-349(ce)-349(cas,)-355(se)-349(ramener)-349(au)-349(cas)-349(d'une)-349(fonction)-355(\265)500(a)-343(v)48(aleurs)-349(dans)]TJ /F3 1 Tf 29.627 0 Td [(R)]TJ /F2 1 Tf 1.0663 0 Td [(\(i.e.)-493(\257nies\))-349(par)-349(une)]TJ -30.693 -1.1988 Td [(mo)-30(di\257cation)-337(sur)-337(un)-337(ensem)24(ble)-337(n)30(\266)469(egligeable.)]TJ /F1 1 Tf 0.81325 -2 Td [(4.)]TJ /F2 1 Tf 1.6807 0 Td [(La)-397(relation)]TJ /F6 1 Tf 5.247 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21 12.59 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 15.464 9.2015 Td [(=)-379(0)-391(lorsque)-397(0)]TJ /F8 1 Tf 6.4157 0 Td [(<)-373(p)-385(<)]TJ /F2 1 Tf 3.1928 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.3916 0 Td [(signi\257e)-397(que)]TJ /F9 1 Tf 5.3253 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.681 13.754 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 35.349 9.2015 Td [(dm)]TJ /F2 1 Tf 1.7892 0 Td [(est)-397(n)24(ul)-397(donc)-391(que)]TJ /F8 1 Tf -31.807 -1.7108 Td [(f)]TJ /F2 1 Tf 0.98193 0 Td [(est)-391(n)30(ulle)-391(presque)-385(partout.)-602(De)-391(m)24(^)475(eme,)-397(lorsque)]TJ /F8 1 Tf 20.247 0 Td [(p)]TJ /F2 1 Tf 0.86747 0 Td [(=)-361(+)]TJ /F6 1 Tf 1.9157 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(;)-409(elle)-391(imp)-30(ose)]TJ /F6 1 Tf 5.9337 0 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F11 1 Tf 0.63856 0 Td [(6)]TJ /F2 1 Tf 1.1446 0 Td [(0)-385(presque)]TJ -34.964 -1.2048 Td [(partout,)-337(elle)-337(est)-343(donc)-307(\266)475(equiv)48(alen)30(te)-337(\265)500(a)-331(la)-337(n)24(ullit)30(\266)469(e)-337(de)]TJ /F8 1 Tf 21.187 0 Td [(f)]TJ /F2 1 Tf 0.92771 0 Td [(presque)-337(partout.)]TJ /F5 1 Tf -22.114 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(128)]TJ ET endstream endobj 421 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 420 0 R /ProcSet 2 0 R >> /Contents 419 0 R >> endobj 420 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F20 415 0 R >> endobj 422 0 obj << /Length 21766 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.461 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-16138(IX.)-391(Les)-385(espaces)-391(L)]TJ /F20 1 Tf 8.352 0 0 8.352 0 0 Tm 62.879 93.245 Td [(p)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 6.1446 62.22 Td [(5.)]TJ /F2 1 Tf 1.5663 0 Td [(Il)-337(est)-337(clair)-337(d'apr)24(\265)469(es)-337(cette)-337(d)24(\266)475(e\257nition)-343(que)]TJ /F6 1 Tf 16.982 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.621 88.469 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.042 62.22 Td [(ne)-331(d)24(\266)469(ep)-24(end)-337(que)-343(de)]TJ /F6 1 Tf 8.012 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(.)-451(En)-337(outre,)-343(lorsque)]TJ /F6 1 Tf -30.873 -1.3855 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F6 1 Tf 1.0602 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(j)]TJ /F2 1 Tf 0.27711 0 Td [(,)-337(on)-331(a)]TJ /F6 1 Tf 2.8373 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 19.06 86.487 Td [(p)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 14.012 60.834 Td [(6)]TJ /F6 1 Tf 1.0542 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.724 86.487 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.994 60.834 Td [(.)]TJ /F1 1 Tf -10.849 -1.7952 Td [(6.)]TJ /F2 1 Tf 1.5602 0 Td [(Dans)-337(la)-337(d)30(\266)469(e\257nition)-337(pr)30(\266)469(ec)24(\266)469(eden)30(te)]TJ /F8 1 Tf 13.084 0 Td [(p)]TJ /F2 1 Tf 0.83735 0 Td [(n'est)-337(pas)-337(supp)-30(os)24(\266)475(e)-337(en)24(tier.)]TJ -16.295 -2.3374 Td [(La)-337(notation)]TJ /F6 1 Tf 5.4578 0 Td [(k)-337(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 17.353 80.573 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.253 56.701 Td [(est)-343(justi\257)24(\266)475(ee)-337(par)-337(l')24(\266)475(enonc)24(\266)475(e)-343(suiv)54(an)24(t)-337(:)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 44.59 Td [(IX.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 53.454 Td [(2.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Soit)]TJ /F8 1 Tf 2.0542 0 Td [(f)]TJ /F12 1 Tf 0.92169 0 Td [(une)-337(fonction)-331(mesurable)-331(sur)]TJ /F8 1 Tf 12.151 0 Td [(X)]TJ /F12 1 Tf 0.90964 0 Td [(,)-331(on)-331(supp)-30(ose)-331(l'existence)-331(de)]TJ /F8 1 Tf -27.602 -1.1988 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F2 1 Tf 0.83133 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([)]TJ /F12 1 Tf 0.61446 0 Td [(tel)-337(que)]TJ /F6 1 Tf 3.3072 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.164 74.211 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.199 52.256 Td [(<)]TJ /F2 1 Tf 1.0602 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F12 1 Tf 1 0 Td [(.)-445(Alors,)-337(lorsque)]TJ /F8 1 Tf 7.0723 0 Td [(p)]TJ /F12 1 Tf 0.83735 0 Td [(tend)-343(v)24(ers)]TJ /F2 1 Tf 4.3434 0 Td [(+)]TJ /F6 1 Tf 0.78313 0 Td [(1)]TJ /F12 1 Tf 1 0 Td [(,)]TJ /F6 1 Tf 0.60843 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.483 74.211 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 36.024 52.256 Td [(tend)-337(v)18(ers)]TJ /F6 1 Tf 4.3434 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 60.06 74.211 Td [(1)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 42.765 52.256 Td [(.)]TJ /F1 1 Tf -37.434 -2.1867 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(On)-331(a)-331(p)-30(our)-337(presque)-331(tout)]TJ /F8 1 Tf 10.675 0 Td [(x)]TJ /F2 1 Tf 0.90361 0 Td [(la)-331(ma)-60(joration)]TJ /F6 1 Tf 6.253 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F11 1 Tf 0.55422 0 Td [(6)]TJ /F6 1 Tf 1.0542 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 53.543 71.081 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 38.548 50.069 Td [(d'o)-30(\265)530(u)-337(l'on)-331(tire,)]TJ -33.217 -1.8614 Td [(p)-30(our)]TJ /F8 1 Tf 2.3313 0 Td [(a)-283(<)-283(p)-277(<)]TJ /F2 1 Tf 3.7048 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)]TJ /F6 1 Tf 0.59036 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.233 69.581 Td [(p)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 16.976 48.208 Td [(6)]TJ /F6 1 Tf 1.0542 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 29.388 69.581 Td [(a)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 21.018 48.208 Td [(\242)]TJ /F2 1 Tf 0.44578 0 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 33.56 68.418 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 24.247 48.208 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.259 69.581 Td [(p)]TJ /F10 1 Tf 0.58621 0 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.157 48.208 Td [(,)-313(donc)]TJ /F9 1 Tf 2.9518 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 44.741 69.581 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 31.892 48.208 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(\()]TJ /F6 1 Tf 0.38554 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 52.397 68.418 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.41 48.208 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.095 69.581 Td [(p)]TJ /F10 1 Tf 0.58621 0 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(a)]TJ /F9 1 Tf 11.952 0 0 11.952 0 0 Tm 39.151 49.569 Td [(Z)]TJ /F6 1 Tf 1.006 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 59.112 69.581 Td [(a)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 41.958 48.208 Td [(dm)]TJ /F2 1 Tf 1.7048 0 Td [(et)]TJ /F6 1 Tf -38.331 -1.8976 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.9138 65.693 Td [(p)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6205 46.31 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 15.259 65.693 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 11.458 46.31 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 16.957 66.857 Td [(1)]TJ /F10 1 Tf 0.56897 0 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(a=p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 14.343 46.31 Td [(\()]TJ /F6 1 Tf 0.39157 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.371 65.693 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.765 46.31 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.552 66.857 Td [(a=p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.464 46.31 Td [(.)]TJ -13.133 -2.0422 Td [(Lorsque)]TJ /F8 1 Tf 3.759 0 Td [(p)]TJ /F2 1 Tf 0.81928 0 Td [(tend)-325(v)24(ers)-325(+)]TJ /F6 1 Tf 5.0964 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-319(dans)-319(le)-325(ma)-60(joran)24(t)-319(trouv)54(\266)469(e)-325(le)-325(facteur)-319(\()]TJ /F6 1 Tf 16.175 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.336 62.78 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.211 44.268 Td [(\))]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.509 63.943 Td [(a=p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 36.223 44.268 Td [(tend)-325(v)24(ers)-325(0)-319(si)]TJ /F8 1 Tf 6.1265 0 Td [(f)]TJ /F2 1 Tf 0.91566 0 Td [(est)]TJ -37.934 -1.4398 Td [(n)30(ulle)-325(presque)-319(partout)-325(et)-319(v)24(ers)-319(1)-319(sinon.)-439(P)24(ar)-319(ailleurs)-319(le)-325(facteur)-319(\()]TJ /F6 1 Tf 26.741 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 48.19 60.719 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.47 42.828 Td [(\))]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 49.888 61.883 Td [(1)]TJ /F10 1 Tf 0.56896 0 Td [(\241)]TJ /F7 1 Tf 0.89655 0 Td [(a=p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 37.506 42.828 Td [(tend)-325(v)24(ers)]TJ /F6 1 Tf 4.3133 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 62.138 60.719 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 44.217 42.828 Td [(.)]TJ -38.886 -1.1988 Td [(Il)-331(s'ensuit)-337(que,)-337(p)-30(our)]TJ /F8 1 Tf 9.1024 0 Td [(u)-277(>)]TJ /F6 1 Tf 1.9036 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 25.664 58.995 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.735 41.629 Td [(,)-331(d)24(\265)469(es)-331(que)]TJ /F8 1 Tf 4.1988 0 Td [(p)]TJ /F2 1 Tf 0.83735 0 Td [(est)-343(su\261sammen)24(t)-331(grand)-343(on)-331(a)]TJ /F6 1 Tf 12.669 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 54.431 58.995 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.723 41.629 Td [(<)-283(u)]TJ /F2 1 Tf 1.6325 0 Td [(.)]TJ -35.024 -1.8554 Td [(De)-367(m)24(^)469(eme,)-373(p)-30(our)]TJ /F8 1 Tf 7.1807 0 Td [(v)-367(<)]TJ /F6 1 Tf 1.9639 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 23 56.349 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 16.867 39.774 Td [(,)-379(prenons)]TJ /F8 1 Tf 4.4157 0 Td [(w)]TJ /F2 1 Tf 1.1145 0 Td [(tel)-367(que)]TJ /F8 1 Tf 3.3675 0 Td [(v)-373(<)-331(w)-367(<)]TJ /F6 1 Tf 4.1506 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 45.095 56.349 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.675 39.774 Td [(et)-367(notons)]TJ /F8 1 Tf 4.4639 0 Td [(A)]TJ /F2 1 Tf 1.1145 0 Td [(l'ensem)24(ble)-367(des)]TJ /F8 1 Tf -32.922 -1.1988 Td [(x)]TJ /F2 1 Tf 0.98193 0 Td [(tels)-403(que)]TJ /F6 1 Tf 3.8494 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.68072 0 Td [(>)-403(w)]TJ /F2 1 Tf 1.9217 0 Td [(.)-674(On)-409(a)-409(alors)-415(la)-409(minoration)]TJ /F6 1 Tf 12.458 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.68073 0 Td [(>)]TJ /F8 1 Tf 1.1807 0 Td [(w)]TJ /F2 1 Tf 0.74698 0 Td [(1)301(I)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47 54.987 Td [(A)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.904 38.575 Td [(et)-409(donc,)-433(p)-30(our)]TJ /F8 1 Tf 6.4458 0 Td [(p)-409(<)]TJ /F2 1 Tf 2.0904 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)]TJ /F9 1 Tf -38.886 -0.3494 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.638 53.34 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.4639 36.864 Td [(dm)]TJ /F11 1 Tf 1.8253 0 Td [(>)]TJ /F8 1 Tf 1.2108 0 Td [(w)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.948 53.34 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.705 36.864 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\),)-451(d'o)-30(\265)524(u)-421(l'on)-427(tire)]TJ /F6 1 Tf 7.3916 0 Td [(k)]TJ /F8 1 Tf 0.50603 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 35.371 52.176 Td [(p)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 25.56 36.864 Td [(>)]TJ /F8 1 Tf 1.2048 0 Td [(w)]TJ /F9 1 Tf 0.74096 0.80723 Td [(\241)]TJ /F8 1 Tf 0.46385 -0.80723 Td [(m)]TJ /F2 1 Tf 0.87349 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))]TJ /F9 1 Tf 0.38554 0.80723 Td [(\242)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 44.121 53.616 Td [(1)]TJ /F7 1 Tf 0.56897 0 Td [(=p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.102 36.864 Td [(.)-722(Le)-427(cas)]TJ /F8 1 Tf 4.259 0 Td [(p)]TJ /F2 1 Tf 0.93373 0 Td [(=)]TJ /F8 1 Tf 1.2048 0 Td [(a)]TJ /F2 1 Tf 0.95181 0 Td [(mon)24(tre)-427(que)]TJ /F8 1 Tf -34.12 -1.7108 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(A)]TJ /F2 1 Tf 0.75301 0 Td [(\))-295(est)-295(\257ni.)-439(P)30(ar)-295(ailleurs)]TJ /F8 1 Tf 9.5542 0 Td [(A)]TJ /F2 1 Tf 1.0422 0 Td [(n'est)-295(pas)-295(n)24(\266)475(egligeable)-295(\(sans)-301(quoi)]TJ /F8 1 Tf 13.849 0 Td [(w)]TJ /F2 1 Tf 1.0361 0 Td [(ma)-60(jore)]TJ /F6 1 Tf 3.3253 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F2 1 Tf 0.56627 0 Td [(presque)-295(partout)]TJ -32.271 -1.4819 Td [(et)]TJ /F6 1 Tf 1.1386 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 11.543 47.616 Td [(1)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 9.1446 33.671 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(w)]TJ /F2 1 Tf 0.74699 0 Td [(\),)-313(on)-313(a)-307(donc)]TJ /F8 1 Tf 5.512 0 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F8 1 Tf 0.66867 0 Td [(>)]TJ /F2 1 Tf 1.0542 0 Td [(0)-307(d'o)-30(\265)524(u)]TJ /F9 1 Tf 3.006 0.81325 Td [(\241)]TJ /F8 1 Tf 0.45783 -0.81325 Td [(m)]TJ /F2 1 Tf 0.87952 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(A)]TJ /F2 1 Tf 0.74699 0 Td [(\))]TJ /F9 1 Tf 0.39157 0.81325 Td [(\242)]TJ /F4 1 Tf 8.352 0 0 8.352 0 0 Tm 37.966 49.047 Td [(1)]TJ /F7 1 Tf 0.56896 0 Td [(=p)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 28.078 33.671 Td [(!)]TJ /F2 1 Tf 1.2771 0 Td [(1)-307(lorsque)]TJ /F8 1 Tf 4.2108 0 Td [(p)]TJ /F2 1 Tf 0.80723 0 Td [(tend)-313(v)24(ers)-313(+)]TJ /F6 1 Tf 5.0723 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)-439(Comme)]TJ /F8 1 Tf -35.114 -1.1988 Td [(v)-445(<)-403(w)]TJ /F2 1 Tf 2.8494 0 Td [(,)-433(la)-409(minoration)-415(obten)24(ue)-415(assure)-409(alors)-415(que,)-433(d)24(\265)475(es)-415(que)]TJ /F8 1 Tf 22.464 0 Td [(p)]TJ /F2 1 Tf 0.91566 0 Td [(est)-415(su\261sammen)24(t)-415(grand,)-433(on)-409(a)]TJ /F6 1 Tf -26.229 -1.1988 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 9.9138 44.176 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 7.6205 31.274 Td [(>)-277(v)]TJ /F2 1 Tf 1.5723 0 Td [(.)]TJ -3.8614 -1.8554 Td [(Les)-313(deux)-319(r)24(\266)475(esultats)-319(que)-319(l'on)-325(vien)24(t)-319(d')24(\266)469(etablir)-313(signi\257en)24(t)-319(bien)-319(que)]TJ /F6 1 Tf 26.723 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 48.155 41.53 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 34.38 29.418 Td [(tend)-319(v)24(ers)]TJ /F6 1 Tf 4.3072 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 57.655 41.53 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 41.404 29.418 Td [(lorsque)]TJ /F8 1 Tf -36.072 -1.3313 Td [(p)]TJ /F2 1 Tf 0.83735 0 Td [(tend)-337(v)24(ers)-337(+)]TJ /F6 1 Tf 5.1205 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)]TJ ET 512.64 334.47 m 519.84 334.47 l 519.84 342.9 l 512.64 342.9 l f BT /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 25.087 Td [(Plus)-337(g)30(\266)469(en)24(\266)475(eralemen)24(t)-337(on)-337(a)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 18.324 Td [(IX.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 21.967 Td [(3.)]TJ /F15 1 Tf 1.753 0 Td [(Pr)24(oposition.)]TJ /F12 1 Tf 6.8675 0 Td [(Soit)]TJ /F8 1 Tf 2.0542 0 Td [(f)]TJ /F12 1 Tf 0.93373 0 Td [(une)-337(application)-337(mesurable)-331(de)]TJ /F8 1 Tf 13.127 0 Td [(X)]TJ /F12 1 Tf 1.241 0 Td [(dans)]TJ ET 0.504 w 437.33 272.44 m 445.97 272.44 l S BT /F3 1 Tf 11.952 0 0 11.952 0 0 Tm 36.59 21.967 Td [(R)]TJ /F12 1 Tf 0.72289 0 Td [(.)-445(Alors)]TJ -31.982 -1.6687 Td [({)-331(l'ensem)24(ble)]TJ /F8 1 Tf 5.6446 0 Td [(I)]TJ /F12 1 Tf 0.85542 0 Td [(des)]TJ /F8 1 Tf 1.7289 0 Td [(p)]TJ /F12 1 Tf 0.83132 0 Td [(tels)-337(que)]TJ /F6 1 Tf 3.7048 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.181 28.478 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 20.44 20.298 Td [(est)-337(\257ni)-337(est)-343(un)-337(in)30(terv)54(alle)-337(de)]TJ /F2 1 Tf 11.964 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.61446 0 Td [(;)]TJ -30.687 -1.8494 Td [({)-355(p)-30(our)-361(tout)]TJ /F8 1 Tf 5.4458 0 Td [(a)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F2 1 Tf 0.66867 0 Td [(]0;)-162(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(,)-367(l'application)]TJ /F8 1 Tf 6.3976 0 Td [(p)]TJ /F6 1 Tf 0.81928 0 Td [(7!)-325(k)]TJ /F8 1 Tf 1.8253 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.793 25.823 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 26.482 18.448 Td [(est)-367(con)30(tin)18(ue)-361(\265)500(a)-355(gauc)24(he)-361(\(resp.)-524(\265)500(a)-361(droite\))-361(au)]TJ -21.151 -1.3916 Td [(p)-30(oin)30(t)]TJ /F8 1 Tf 2.6084 0 Td [(a)]TJ /F12 1 Tf 0.86747 0 Td [(sauf)-331(si)]TJ /F8 1 Tf 3.0904 0 Td [(a)]TJ /F12 1 Tf 0.86747 0 Td [(est)-337(l'extr)24(\266)475(emit)18(\266)475(e)-337(inf)24(\266)469(erieure)-337(\(resp.)-445(sup)0(\266)469(erieure\))-337(de)]TJ /F8 1 Tf 20.458 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(a)]TJ /F6 1 Tf 0.80722 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(.)]TJ -32.181 -1.6627 Td [(En)-524(particulier,)-572(si)]TJ /F8 1 Tf 8.3072 0 Td [(I)]TJ /F12 1 Tf 1.0422 0 Td [(n'est)-524(pas)-530(vide,)-566(l'application)]TJ /F8 1 Tf 13.127 0 Td [(p)]TJ /F6 1 Tf 1.0964 0 Td [(7!)-596(k)]TJ /F8 1 Tf 2.0964 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.931 21.452 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 33.03 15.394 Td [(est)-530(con)24(tin)24(ue)-524(\265)500(a)-524(gauc)30(he)-524(en)]TJ -27.699 -1.3313 Td [(l'extr)30(\266)469(emit)24(\266)469(e)-487(sup)0(\266)469(erieure)-487(de)]TJ /F8 1 Tf 11.633 0 Td [(I)]TJ /F12 1 Tf 0.51807 0 Td [(,)-524(elle)-487(est)-487(con)30(tin)18(ue)-481(\265)500(a)-481(droite)-487(en)-481(l'extr)24(\266)469(emit)24(\266)469(e)-487(inf)30(\266)469(erieure)-487(de)]TJ /F8 1 Tf 25.337 0 Td [(I)]TJ /F12 1 Tf 1.006 0 Td [(si)]TJ -38.494 -1.2048 Td [(celle-ci)-331(n'est)-337(pas)-337(n)24(ulle)-331(et)-337(elle)-337(est)-343(con)30(tin)18(ue)-331(en)-331(tous)-337(les)-337(autres)-337(p)-24(oin)24(ts)-337(de)]TJ /F2 1 Tf 30.693 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.28313 0 Td [(.)]TJ -33.976 -1.6627 Td [(Ainsi,)-331(si)-337(une)-331(suite)-331(d')24(\266)469(el)24(\266)469(emen)24(ts)]TJ /F8 1 Tf 13.072 0 Td [(p)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 27.052 15.806 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 19.777 11.195 Td [(de)]TJ /F8 1 Tf 1.3313 0 Td [(I)]TJ /F12 1 Tf 0.8494 0 Td [(a)-331(une)-337(limite)]TJ /F8 1 Tf 5.5542 0 Td [(a)]TJ /F12 1 Tf 0.85542 0 Td [(dans)]TJ /F2 1 Tf 2.3374 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1.006 0 Td [(])]TJ /F12 1 Tf 0.60843 0 Td [(alors)-331(la)-331(suite)-337(des)]TJ /F6 1 Tf 7.6205 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 62.302 15.443 Td [(p)]TJ 0.59483 -0.26724 Td [(n)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 9.7557 Td [(tend)-337(v)24(ers)]TJ /F6 1 Tf 4.3434 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 16.129 13.392 Td [(a)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 11.705 9.7557 Td [(.)]TJ -6.3735 -1.6627 Td [(En)-331(outre)-343(on)-331(est)-337(dans)-337(l'un)-337(des)-337(cas)-337(suiv)54(an)24(ts:)]TJ /F1 1 Tf 0.81325 -1.6687 Td [(1.)]TJ /F12 1 Tf 1.5602 0 Td [(l'application)]TJ /F8 1 Tf 5.7229 0 Td [(p)]TJ /F6 1 Tf 0.78313 0 Td [(7!)-277(k)]TJ /F8 1 Tf 1.7771 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.457 8.6159 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.831 6.4244 Td [(est)-343(croissan)30(te)-337(;)]TJ /F2 1 Tf 6.1084 -2.5361 Td [(129)]TJ /F5 1 Tf 6.4458 0 Td [(U.P.M.C.)-530(Cours)-530(de)-530(P.)-530(MAZET)]TJ ET endstream endobj 424 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 423 0 R /ProcSet 2 0 R >> /Contents 422 0 R >> endobj 423 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F20 415 0 R >> endobj 425 0 obj << /Length 22728 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.461 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-16138(IX.)-391(Les)-385(espaces)-391(L)]TJ /F20 1 Tf 8.352 0 0 8.352 0 0 Tm 62.879 93.245 Td [(p)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 6.1446 62.22 Td [(2.)]TJ /F12 1 Tf 1.4699 0 Td [(l'application)]TJ /F8 1 Tf 5.6807 0 Td [(p)]TJ /F6 1 Tf 0.77711 0 Td [(7!)-283(k)]TJ /F8 1 Tf 1.7831 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.259 88.469 Td [(p)]TJ /F12 1 Tf 11.952 0 0 11.952 0 0 Tm 17.651 62.22 Td [(attein)24(t)-289(un)-295(minim)24(um)-289(en)-289(un)-289(p)-30(oin)24(t)]TJ /F8 1 Tf 14.289 0 Td [(a)]TJ /F12 1 Tf 0.81928 0 Td [(de)]TJ /F2 1 Tf 1.2892 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(,)-301(elle)-289(est)-295(d)24(\266)475(ecrois-)]TJ -31.994 -1.3855 Td [(san)24(te)-337(sur)]TJ /F2 1 Tf 4.2651 0 Td [(]0;)]TJ /F8 1 Tf 1.2229 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(])]TJ /F12 1 Tf 0.60843 0 Td [(et)-337(croissan)24(te)-331(sur)]TJ /F2 1 Tf 7.4458 0 Td [([)]TJ /F8 1 Tf 0.27711 0 Td [(a)]TJ /F2 1 Tf 0.53012 0 Td [(;)-162(+)]TJ /F6 1 Tf 1.2229 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])]TJ /F12 1 Tf 0.27711 0 Td [(.)]TJ /F2 1 Tf -17.38 -2.241 Td [(Commen)-6(\273)439(cons)-337(par)-307(\266)475(etablir)-337(un)-337(lemme.)]TJ /F0 1 Tf 14.328 0 0 14.328 0 0 Tm 4.4472 46.329 Td [(IX.2.)]TJ /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 8.2771 55.539 Td [(4.)]TJ /F15 1 Tf 1.753 0 Td [(Lemme.)]TJ /F12 1 Tf 4.1024 0 Td [(Si)]TJ /F2 1 Tf 1.1687 0 Td [(0)]TJ /F8 1 Tf 0.77711 0 Td [(<)-283(p)]TJ /F11 1 Tf 1.8373 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(r)]TJ /F11 1 Tf 0.75301 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(q)]TJ /F11 1 Tf 0.75904 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F12 1 Tf 1 0 Td [(,)-337(p)-30(our)]TJ /F8 1 Tf 2.9819 0 Td [(f)]TJ /F12 1 Tf 0.92771 0 Td [(mesurable)-337(on)-331(a)-337(:)]TJ /F6 1 Tf -8.7229 -1.506 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.293 76.745 Td [(r)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 21.837 54.033 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(max)0(\()]TJ /F6 1 Tf 2.253 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.267 76.745 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.157 54.033 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.784 76.745 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.59 54.033 Td [(\).)]TJ /F1 1 Tf -24.259 -2.0602 Td [(D)24(\266)548(emonstration.)]TJ /F2 1 Tf 8.8193 0 Td [(On)-331(p)-30(eut)-337(bien)-337(s)-30(^)524(ur)-331(supp)-30(oser)]TJ /F8 1 Tf 11.97 0 Td [(p)-283(<)-277(r)-313(<)-277(q)]TJ /F2 1 Tf 4.4699 0 Td [(et)-331(donc)]TJ /F8 1 Tf 3.5542 0 Td [(p)]TJ /F2 1 Tf 0.83735 0 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(r)]TJ /F2 1 Tf 0.81326 0 Td [(\257nis.)]TJ -31.633 -1.6205 Td [(P)30(osons)]TJ /F8 1 Tf 3.2711 0 Td [(M)]TJ /F2 1 Tf 1.3614 0 Td [(=)-277(max)0(\()]TJ /F6 1 Tf 3.3072 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.276 71.478 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 15.277 50.352 Td [(;)]TJ /F6 1 Tf 0.44578 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.784 71.478 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 17.711 50.352 Td [(\).)-433(Le)-277(r)24(\266)475(esultat)-283(est)-253(\266)469(eviden)24(t)-277(p)-30(our)]TJ /F8 1 Tf 13.313 0 Td [(M)]TJ /F2 1 Tf 1.3614 0 Td [(=)-277(+)]TJ /F6 1 Tf 1.8313 0 Td [(1)]TJ /F2 1 Tf 1.2771 0 Td [(et)-277(p)-30(our)]TJ /F8 1 Tf 3.4157 0 Td [(M)]TJ /F2 1 Tf 1.3554 0 Td [(=)-283(0)-277(\(dans)]TJ -34.934 -1.3855 Td [(ce)-253(cas)]TJ /F8 1 Tf 2.7229 0 Td [(f)]TJ /F2 1 Tf 0.8494 0 Td [(est)-253(n)24(ulle)-253(presque)-246(partout\).)-421(Supp)-30(osons)-253(donc)]TJ /F8 1 Tf 18.892 0 Td [(M)]TJ /F2 1 Tf 1.3313 0 Td [(\257ni)-253(non)-253(n)30(ul)-253(et)-253(p)-30(osons)]TJ /F8 1 Tf 9.3735 0 Td [(g)]TJ /F2 1 Tf 0.78916 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(f)-108(=)54(M)]TJ /F2 1 Tf 2.1205 0 Td [(.)-421(On)]TJ -37.133 -1.7169 Td [(a)-325(alors)]TJ /F6 1 Tf 3.2169 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F6 1 Tf 0.51807 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.405 67.047 Td [(p)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 10.753 47.25 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(1)-331(et)]TJ /F6 1 Tf 1.994 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.914 67.047 Td [(q)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 15.982 47.25 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(1,)-331(c'est-)0(\265)500(a-dire)]TJ /F9 1 Tf 6.2229 1.3614 Td [(Z)]TJ /F6 1 Tf 1.006 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.181 68.211 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 27.313 47.25 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(1)-331(et)]TJ /F9 1 Tf 1.988 1.3614 Td [(Z)]TJ /F6 1 Tf 1.006 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 50.733 68.211 Td [(q)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 36.06 47.25 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(1)-325(si)]TJ /F8 1 Tf 1.8253 0 Td [(q)]TJ /F6 1 Tf 0.75903 0 Td [(6)]TJ /F2 1 Tf [(=)-283(+)]TJ /F6 1 Tf 1.8374 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)]TJ -38.886 -2.3193 Td [(et)-337(il)-331(s'agit)-337(de)-331(prouv)24(er)]TJ /F6 1 Tf 9.7349 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.853 63.719 Td [(r)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 17.337 44.93 Td [(6)]TJ /F8 1 Tf 1.0542 0 Td [(M)]TJ /F2 1 Tf 1.0783 0 Td [(,)-337(c'est-)0(\265)500(a-dire)]TJ /F6 1 Tf 5.7349 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 38.233 63.719 Td [(r)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 27.392 44.93 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(1,)-337(ou)-331(encore)]TJ /F9 1 Tf 5.6145 1.3614 Td [(Z)]TJ /F6 1 Tf 1.006 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39156 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 53.638 64.883 Td [(r)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 38.09 44.93 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(1.)]TJ -35.494 -2.1325 Td [(Lorsque)]TJ /F8 1 Tf 3.8675 0 Td [(q)]TJ /F2 1 Tf 0.91566 0 Td [(=)-433(+)]TJ /F6 1 Tf 1.994 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-451(on)-427(a)-427(presque)-427(partout)]TJ /F6 1 Tf 10.59 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F11 1 Tf 0.71687 0 Td [(6)]TJ /F2 1 Tf 1.2108 0 Td [(1)-427(et)-427(donc)]TJ /F6 1 Tf 4.6747 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 49.879 61.84 Td [(r)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 35.735 42.798 Td [(6)]TJ /F6 1 Tf 1.2108 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 56.336 61.84 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.831 42.798 Td [(,)-451(par)-427(suite)]TJ /F9 1 Tf -34.5 -0.3494 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.517 59.383 Td [(r)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 9.3614 41.087 Td [(dm)]TJ /F11 1 Tf 1.6747 0 Td [(6)]TJ /F9 1 Tf 1.0542 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.198 59.383 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.139 41.087 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F2 1 Tf 1.0542 0 Td [(1.)]TJ -13.542 -2.0783 Td [(Supp)-24(osons)-277(main)24(tenan)18(t)]TJ /F8 1 Tf 10.09 0 Td [(q)]TJ /F2 1 Tf 0.75301 0 Td [(\257ni.)-427(Comme)]TJ /F8 1 Tf 5.6988 0 Td [(r)]TJ /F2 1 Tf 0.75301 0 Td [(est)-283(compris)-271(en)24(tre)]TJ /F8 1 Tf 7.6446 0 Td [(p)]TJ /F2 1 Tf 0.77711 0 Td [(et)]TJ /F8 1 Tf 1.1084 0 Td [(q)]TJ /F2 1 Tf 0.48193 0 Td [(,)-289(il)-271(p)-30(eut)-277(s')24(\266)469(ecrire)]TJ /F8 1 Tf 6.9819 0 Td [(r)]TJ /F2 1 Tf 0.75903 0 Td [(=)]TJ /F8 1 Tf 1.0542 0 Td [(up)]TJ /F2 1 Tf 1.1807 0 Td [(+)]TJ /F8 1 Tf 0.87952 0 Td [(v)-36(q)]TJ /F2 1 Tf -38.163 -1.1988 Td [(a)30(v)24(ec)]TJ /F8 1 Tf 2.3373 0 Td [(u)]TJ /F2 1 Tf 1.0542 0 Td [(et)]TJ /F8 1 Tf 1.3133 0 Td [(v)]TJ /F2 1 Tf 1 0 Td [(p)-30(ositifs)-481(de)-481(somme)-481(1.)-879(La)-487(con)30(v)24(exit)24(\266)475(e)-487(de)]TJ /F8 1 Tf 17.855 0 Td [(t)]TJ /F6 1 Tf 0.87952 0 Td [(7!)]TJ /F8 1 Tf 1.5241 0 Td [(a)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 45.534 54.693 Td [(t)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 32.651 37.81 Td [(p)-30(our)]TJ /F8 1 Tf 2.5121 0 Td [(a)]TJ /F6 1 Tf 1.0542 0 Td [(2)]TJ /F3 1 Tf 1.1867 0 Td [(R)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 54.56 54.693 Td [(\244)]TJ /F4 1 Tf 0 -0.93966 Td [(+)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.271 37.81 Td [(assure)-475(alors)]TJ /F6 1 Tf -33.94 -1.1988 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 11.086 52.978 Td [(r)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 8.5181 36.611 Td [(6)]TJ /F8 1 Tf 1.1084 0 Td [(u)]TJ /F6 1 Tf 0.57229 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 18.052 52.978 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 13.319 36.611 Td [(+)]TJ /F8 1 Tf 1.0241 0 Td [(v)]TJ /F6 1 Tf 0.51807 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.733 52.978 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 18.084 36.611 Td [(lorsque)]TJ /F6 1 Tf 3.4578 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F2 1 Tf 0.64458 0 Td [(est)-367(\257ni)-367(non)-367(n)30(ul.)-542(P)30(ar)-361(ailleurs)-367(cette)-367(ma)-54(joration)]TJ -18.994 -1.2048 Td [(est)-307(\266)469(eviden)24(te)-337(si)]TJ /F6 1 Tf 6.512 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F2 1 Tf 0.60843 0 Td [(est)-337(n)24(ul)-337(ou)-331(in\257ni.)]TJ -9.2651 -2.1325 Td [(En)-337(in)24(t)24(\266)475(egran)24(t)-343(on)-337(obtien)30(t)-343(donc)]TJ /F9 1 Tf 13.066 1.3675 Td [(Z)]TJ /F6 1 Tf 1 -1.3675 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51807 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.224 48.211 Td [(r)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 22.428 33.274 Td [(dm)]TJ /F11 1 Tf 1.6807 0 Td [(6)]TJ /F8 1 Tf 1.0663 0 Td [(u)]TJ /F9 1 Tf 0.74096 1.3675 Td [(Z)]TJ /F6 1 Tf 1 -1.3675 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56626 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.974 48.211 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 29.964 33.274 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F8 1 Tf 1.006 0 Td [(v)]TJ /F9 1 Tf 0.68675 1.3675 Td [(Z)]TJ /F6 1 Tf 1 -1.3675 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(g)]TJ /F2 1 Tf 0.51205 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.509 48.211 Td [(q)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 37.301 33.274 Td [(dm)]TJ /F11 1 Tf 1.6867 0 Td [(6)]TJ /F8 1 Tf 1.0602 0 Td [(u)]TJ /F2 1 Tf 0.8012 0 Td [(+)]TJ /F8 1 Tf 1 0 Td [(v)]TJ /F2 1 Tf 0.80723 0 Td [(=)-283(1,)]TJ -37.325 -1.7108 Td [(d'o)-30(\265)530(u)]TJ /F6 1 Tf 2.2229 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(g)]TJ /F6 1 Tf 0.51205 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.974 44.599 Td [(r)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7349 31.563 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(1,)-331(ce)-337(qui)-337(ac)24(h)24(\265)475(ev)24(e)-337(la)-337(preuv)30(e.)]TJ ET 512.64 376.09 m 519.84 376.09 l 519.84 384.51 l 512.64 384.51 l f BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 29.328 Td [(Preuv)30(e)-385(de)-391(la)-385(prop)-36(osition)-385(IX.2.3.)]TJ /F2 1 Tf 0 -1.6205 Td [(Mon)24(trer)-367(que)]TJ /F8 1 Tf 5.8193 0 Td [(I)]TJ /F2 1 Tf 0.87952 0 Td [(est)-367(un)-367(in)24(terv)54(alle)-361(revien)24(t)-367(\265)499(a)-367(mon)24(trer)-361(qu'il)-367(a)-361(la)-367(propri)24(\266)469(et)24(\266)475(e)-367(de)-367(con)30(v)24(exit)24(\266)475(e,)-379(c'est-)]TJ -6.6988 -1.2048 Td [(\265)499(a-dire,)-385(p)-24(our)-373(0)]TJ /F8 1 Tf 6.4036 0 Td [(<)-337(p)-343(<)-337(r)-373(<)-343(q)]TJ /F11 1 Tf 5.8313 0 Td [(6)]TJ /F2 1 Tf 1.1205 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-379(si)]TJ /F8 1 Tf 1.6988 0 Td [(p)]TJ /F2 1 Tf 0.87349 0 Td [(et)]TJ /F8 1 Tf 1.2048 0 Td [(q)]TJ /F2 1 Tf 0.85542 0 Td [(appartiennen)24(t)-373(\265)500(a)]TJ /F8 1 Tf 7.3313 0 Td [(I)]TJ /F2 1 Tf 0.88554 0 Td [(alors)]TJ /F8 1 Tf 2.4337 0 Td [(r)]TJ /F2 1 Tf 0.8494 0 Td [(appartien)24(t)-373(aussi)-373(\265)500(a)]TJ /F8 1 Tf -31.265 -1.1988 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)-614(Autremen)24(t)-391(dit,)-403(il)-391(s'agit)-397(de)-385(prouv)24(er)-397(que)]TJ /F6 1 Tf 18.627 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.31 35.642 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.873 25.304 Td [(et)]TJ /F6 1 Tf 1.2289 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 42.5 35.642 Td [(q)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 30.482 25.304 Td [(\257nis)-391(en)30(tra)108(^)385(\263nen)24(t)]TJ /F6 1 Tf 7.012 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.94 35.642 Td [(r)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.873 25.304 Td [(\257nis.)-614(Cela)]TJ -34.542 -1.3313 Td [(d)30(\266)469(ecoule)-313(\266)475(evidemmen)24(t)-337(du)-337(lemme)-337(pr)24(\266)475(ec)24(\266)469(eden)30(t.)]TJ 0 -2.241 Td [(Etudions)-331(alors)-337(la)-337(con)30(tin)18(uit)24(\266)469(e)-337(\265)500(a)-331(gauc)24(he)-331(en)]TJ /F8 1 Tf 17.898 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F2 1 Tf 0.83133 0 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(])-337(de)]TJ /F8 1 Tf 1.9458 0 Td [(p)]TJ /F6 1 Tf 0.78313 0 Td [(7!)-277(k)]TJ /F8 1 Tf 1.7771 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 47.897 30.53 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.886 21.732 Td [(.)]TJ -28.554 -1.8012 Td [(Si)]TJ /F8 1 Tf 1.2169 0 Td [(a)]TJ /F2 1 Tf 0.89157 0 Td [(=)-361(+)]TJ /F6 1 Tf 1.9157 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(,)-397(la)-385(prop)-30(osition)]TJ /F1 1 Tf 7.1506 0 Td [(IX.2.2.)]TJ /F2 1 Tf 3.8012 0 Td [(a\261rme)-385(cette)-385(con)24(tin)24(uit)18(\266)475(e)-391(p)-30(ourvu)-385(que)]TJ /F6 1 Tf 16.012 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.69 27.943 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 39.717 19.93 Td [(ne)-379(soit)-385(pas)]TJ -34.386 -1.3916 Td [(iden)24(tiquemen)24(t)-307(+)]TJ /F6 1 Tf 7.2771 0 Td [(1)]TJ /F2 1 Tf 1.3072 0 Td [(sur)-307(]0;)-168(+)]TJ /F6 1 Tf 3.6446 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [([.)-439(Si)-307(c'est)-307(le)-307(cas,)-313(la)-307(limite)-307(\265)499(a)-301(gauc)24(he)-301(en)]TJ /F8 1 Tf 16.614 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)-277(+)]TJ /F6 1 Tf 1.8313 0 Td [(1)]TJ /F2 1 Tf 1.3072 0 Td [(de)]TJ /F8 1 Tf 1.3072 0 Td [(p)]TJ /F6 1 Tf 0.77711 0 Td [(7!)-277(k)]TJ /F8 1 Tf 1.7771 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 63.086 25.961 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 17.153 Td [(est)-409(+)]TJ /F6 1 Tf 2.4096 0 Td [(1)]TJ /F2 1 Tf 1.4036 0 Td [(et)-409(il)-403(y)-409(a)-403(con)24(tin)24(uit)18(\266)475(e)-409(\265)500(a)-409(gauc)30(he)-403(sauf)-409(si)]TJ /F6 1 Tf 16.301 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F10 1 Tf 8.352 0 0 8.352 0 0 Tm 38.698 23.978 Td [(1)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 28.247 17.153 Td [(est)-409(\257ni.)-662(Dans)-409(ce)-409(cas)]TJ /F8 1 Tf 9.6145 0 Td [(I)]TJ /F2 1 Tf 0.92169 0 Td [(est)-415(r)30(\266)469(eduit)-403(au)]TJ -33.452 -1.1988 Td [(singleton)]TJ /F6 1 Tf 4.2289 0 Td [(f)]TJ /F2 1 Tf 0.5 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1g)]TJ /F2 1 Tf 1.5 0 Td [(,)]TJ /F8 1 Tf 0.61446 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)-277(+)]TJ /F6 1 Tf 1.8313 0 Td [(1)]TJ /F2 1 Tf 1.3373 0 Td [(est)-337(l'extr)24(\266)475(emit)18(\266)475(e)-337(inf)24(\266)475(erieure)-343(de)]TJ /F8 1 Tf 12.349 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(et)]TJ /F8 1 Tf 1.1626 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F8 1 Tf 0.94579 0 Td [(I)]TJ /F2 1 Tf 0.51807 0 Td [(.)]TJ -28.235 -2.4398 Td [(Si)]TJ /F8 1 Tf 1.1687 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(est)-337(\257ni,)-337(il)-337(revien)24(t)-337(au)-331(m)24(^)469(eme)-331(d')24(\266)469(etudier)-337(la)-331(con)24(tin)24(uit)18(\266)475(e)-343(\265)499(a)-331(gauc)30(he)-337(de)]TJ /F8 1 Tf 27.548 0 Td [(p)]TJ /F6 1 Tf 0.78313 0 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.991 19.926 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.753 13.515 Td [(dm)]TJ /F2 1 Tf 1.4036 0 Td [(.)]TJ -35.825 -2.1325 Td [(P)30(our)-343(cela)-343(notons)]TJ /F8 1 Tf 7.6807 0 Td [(A)]TJ /F2 1 Tf 1.0904 0 Td [(\(resp.)]TJ /F8 1 Tf 2.9157 0 Td [(B)]TJ /F2 1 Tf 0.80723 0 Td [(\))-343(l'ensem)24(ble)-337(des)]TJ /F8 1 Tf 7.2831 0 Td [(x)]TJ /F2 1 Tf 0.90964 0 Td [(de)]TJ /F8 1 Tf 1.3434 0 Td [(X)]TJ /F2 1 Tf 1.247 0 Td [(tels)-337(que)]TJ /F6 1 Tf 3.7108 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F11 1 Tf 0.56626 0 Td [(6)]TJ /F2 1 Tf 1.0602 0 Td [(1)-343(\(resp.)]TJ /F6 1 Tf 3.2952 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.60241 0 Td [(\()]TJ /F8 1 Tf 0.38554 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F8 1 Tf 0.56024 0 Td [(>)]TJ /F2 1 Tf 1.0663 0 Td [(1\).)]TJ -37.994 -1.7108 Td [(On)-337(a)-331(alors)]TJ /F9 1 Tf 4.8976 1.3614 Td [(Z)]TJ /F6 1 Tf 1 -1.3614 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 17.724 14.426 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 13.012 9.6714 Td [(dm)]TJ /F2 1 Tf 1.6747 0 Td [(=)]TJ /F9 1 Tf 1.0602 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 23.328 12.53 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 16.952 9.6714 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 25.914 14.426 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 18.735 9.6714 Td [(dm)]TJ /F2 1 Tf 1.6205 0 Td [(+)]TJ /F9 1 Tf 1 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 31.362 12.53 Td [(B)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.602 9.6714 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.991 14.426 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 24.386 9.6714 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)]TJ -20.452 -2.1807 Td [(Remarquons)-343(alors)-355(que)-349(l'application)]TJ /F8 1 Tf 15.807 0 Td [(p)]TJ /F6 1 Tf 0.8012 0 Td [(7!)-301(j)]TJ /F8 1 Tf 1.5783 0 Td [(f)]TJ /F2 1 Tf 0.59638 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.57229 0 Td [(\))]TJ /F6 1 Tf 0.38554 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 36.845 11.306 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.554 7.4906 Td [(est)-349(d)24(\266)475(ecroissan)24(te)-349(p)-30(our)]TJ /F8 1 Tf 9.5602 0 Td [(x)]TJ /F6 1 Tf 0.87349 0 Td [(2)]TJ /F8 1 Tf 0.96386 0 Td [(A)]TJ /F2 1 Tf 1.0964 0 Td [(et)-349(croissan)24(te)]TJ -33.717 -1.2048 Td [(p)-30(our)]TJ /F8 1 Tf 2.3614 0 Td [(x)]TJ /F6 1 Tf 0.8494 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(B)]TJ /F2 1 Tf 0.81325 0 Td [(.)]TJ /F5 1 Tf -4.9699 -2.3976 Td [(U.P.M.C.)-524(Cours)-536(de)-524(P.)-530(MAZET)]TJ /F2 1 Tf 19 0 Td [(130)]TJ ET endstream endobj 427 0 obj << /Type /Page /Parent 4 0 R /Resources << /Font 426 0 R /ProcSet 2 0 R >> /Contents 425 0 R >> endobj 426 0 obj << /F0 5 0 R /F1 12 0 R /F2 13 0 R /F3 14 0 R /F4 15 0 R /F5 16 0 R /F6 20 0 R /F7 21 0 R /F8 22 0 R /F9 29 0 R /F10 33 0 R /F11 34 0 R /F12 35 0 R /F15 68 0 R /F20 415 0 R >> endobj 428 0 obj << /Length 21399 >> stream BT /F1 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 64.461 Td [(In)30(t)24(\266)548(egration)-385(2004-2005-2006)-16138(IX.)-391(Les)-385(espaces)-391(L)]TJ /F20 1 Tf 8.352 0 0 8.352 0 0 Tm 62.879 93.245 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 5.3313 61.858 Td [(La)-337(propri)24(\266)469(et)24(\266)475(e)-337(de)-337(Bepp)-30(o)-337(Levi)-337(assure)-331(donc)-337(la)-331(con)24(tin)24(uit)18(\266)475(e)-337(\265)500(a)-337(gauc)30(he)-337(de)]TJ /F8 1 Tf 29.072 0 Td [(p)]TJ /F6 1 Tf 0.78313 0 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 52.974 87.219 Td [(B)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 37.711 61.858 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 55.612 89.116 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 39.494 61.858 Td [(dm)]TJ /F2 1 Tf 1.3976 0 Td [(.)]TJ -35.56 -2.8193 Td [(De)-319(m)30(^)469(eme)-313(le)-319(th)18(\266)475(eor)24(\265)469(eme)-313(de)-313(con)24(v)24(ergence)-319(domin)30(\266)469(ee)-319(assure)-313(la)-313(con)24(tin)24(uit)18(\266)475(e)-319(en)]TJ /F8 1 Tf 30.554 0 Td [(a)]TJ /F2 1 Tf 0.84338 0 Td [(de)]TJ /F8 1 Tf 1.3193 0 Td [(p)]TJ /F6 1 Tf 0.77711 0 Td [(7!)]TJ /F9 1 Tf 1.2831 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 58.19 83.185 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 41.313 59.039 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 60.767 85.081 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 43.096 59.039 Td [(dm)]TJ /F2 1 Tf -37.765 -2.3735 Td [(p)-30(ourvu)-397(qu'il)-391(existe)]TJ /F8 1 Tf 8.6988 0 Td [(b)-379(<)-379(a)]TJ /F2 1 Tf 2.8916 0 Td [(v)54(\266)469(eri\257an)24(t)]TJ /F9 1 Tf 3.9819 1.3615 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 30.707 79.788 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 22.108 56.665 Td [(j)]TJ /F8 1 Tf 0.28313 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 33.293 81.685 Td [(b)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 23.831 56.665 Td [(dm)-385(<)]TJ /F2 1 Tf 2.9398 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1.3976 0 Td [(\(en)-397(e\256et)-397(on)-397(a)-397(alors,)-415(p)-30(our)]TJ /F8 1 Tf 11.572 0 Td [(x)]TJ /F6 1 Tf 0.95181 0 Td [(2)]TJ /F8 1 Tf 1.0482 0 Td [(A)]TJ /F2 1 Tf 1.1446 0 Td [(et)]TJ /F8 1 Tf -38.331 -1.9096 Td [(p)]TJ /F11 1 Tf 0.92169 0 Td [(6)]TJ /F8 1 Tf 1.1928 0 Td [(b)]TJ /F2 1 Tf 0.42771 0 Td [(,)-439(la)-421(relation)-415(de)-415(domination)]TJ /F6 1 Tf 12.452 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 32.664 78.952 Td [(p)]TJ /F11 1 Tf 11.952 0 0 11.952 0 0 Tm 23.705 54.756 Td [(6)]TJ /F6 1 Tf 1.1988 0 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F2 1 Tf 0.59639 0 Td [(\()]TJ /F8 1 Tf 0.39157 0 Td [(x)]TJ /F2 1 Tf 0.56627 0 Td [(\))]TJ /F6 1 Tf 0.39157 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 39.216 78.952 Td [(b)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 27.807 54.756 Td [(\).)-698(Cela)-421(a)-421(lieu)-415(en)-415(particulier)-421(d)24(\265)475(es)-421(qu'il)]TJ -22.476 -1.1988 Td [(existe)]TJ /F8 1 Tf 2.8133 0 Td [(b)-277(<)-277(a)]TJ /F2 1 Tf 2.6205 0 Td [(tel)-337(que)]TJ /F6 1 Tf 3.3072 0 Td [(k)]TJ /F8 1 Tf 0.50602 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 22.431 76.064 Td [(b)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 16.301 53.557 Td [(<)]TJ /F2 1 Tf 1.0602 0 Td [(+)]TJ /F6 1 Tf 0.77711 0 Td [(1)]TJ /F2 1 Tf 1 0 Td [(.)]TJ -13.807 -1.6446 Td [(En)-361(passan)24(t)-361(\265)500(a)-355(la)-361(somme)-361(de)-355(ces)-361(applications)-361(on)-361(en)-355(d)24(\266)475(eduit)-361(la)-361(con)30(tin)18(uit)24(\266)469(e)-361(\265)500(a)-361(gauc)30(he)-361(souhait)24(\266)469(ee)]TJ 0 -1.1988 Td [(sauf)-313(si)]TJ /F6 1 Tf 3.0482 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.60241 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 14.284 71.995 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 10.705 50.714 Td [(est)-313(iden)24(tiquemen)18(t)-313(+)]TJ /F6 1 Tf 8.8253 0 Td [(1)]TJ /F2 1 Tf 1.3133 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3434 0 Td [(p)-283(<)-277(a)]TJ /F2 1 Tf 2.3675 0 Td [(.)-433(Si)-313(tel)-319(est)-313(le)-319(cas,)-319(la)-313(limite)-313(\265)500(a)-307(gauc)24(he)-313(en)]TJ /F8 1 Tf 17.102 0 Td [(a)]TJ /F2 1 Tf 0.83735 0 Td [(de)]TJ /F8 1 Tf -38.163 -1.3916 Td [(p)]TJ /F6 1 Tf 0.77711 0 Td [(7!)-283(k)]TJ /F8 1 Tf 1.7831 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 12.862 70.012 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 9.7048 49.322 Td [(est)-313(+)]TJ /F6 1 Tf 2.3133 0 Td [(1)]TJ /F2 1 Tf 1.3072 0 Td [(et)-313(il)-307(y)-307(a)-307(con)24(tin)24(uit)24(\266)469(e)-313(\265)500(a)-307(gauc)30(he)-307(en)]TJ /F8 1 Tf 13.681 0 Td [(a)]TJ /F2 1 Tf 0.83735 0 Td [(sauf)-307(si)]TJ /F6 1 Tf 3.0422 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 46.483 70.012 Td [(a)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 33.223 49.322 Td [(est)-313(\257ni.)-439(Dans)-313(ce)-307(cas)]TJ /F8 1 Tf 9 0 Td [(a)]TJ /F6 1 Tf 0.80723 0 Td [(2)]TJ /F8 1 Tf 0.94578 0 Td [(I)]TJ /F2 1 Tf -38.645 -1.3855 Td [(et)]TJ /F8 1 Tf 1.1687 0 Td [(a)]TJ /F2 1 Tf 0.86145 0 Td [(est)-337(l'extr)24(\266)475(emit)18(\266)475(e)-337(inf)24(\266)469(erieure)-337(de)]TJ /F8 1 Tf 12.349 0 Td [(I)]TJ /F2 1 Tf 0.85542 0 Td [(\(puisque)]TJ /F6 1 Tf 4.0361 0 Td [(k)]TJ /F8 1 Tf 0.5 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 37.491 68.03 Td [(b)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 26.825 47.936 Td [(=)-283(+)]TJ /F6 1 Tf 1.8373 0 Td [(1)]TJ /F2 1 Tf 1.3313 0 Td [(p)-30(our)]TJ /F8 1 Tf 2.3675 0 Td [(b)-277(<)-283(a)]TJ /F2 1 Tf 2.2892 0 Td [(\).)]TJ -29.319 -1.6446 Td [(Nous)-265(a)24(v)24(ons)-265(donc)-240(\266)469(etabli)-271(la)-265(prop)-30(osition)-271(concernan)24(t)-265(la)-265(con)24(tin)24(uit)18(\266)475(e)-271(\265)499(a)-265(gauc)24(he.)-421(P)30(our)-271(ce)-265(qui)-265(est)-271(de)]TJ 0 -1.1988 Td [(la)-289(con)24(tin)24(uit)18(\266)475(e)-295(\265)500(a)-295(droite,)-295(le)-301(cas)]TJ /F8 1 Tf 12.265 0 Td [(a)]TJ /F2 1 Tf 0.80723 0 Td [(=)-283(+)]TJ /F6 1 Tf 1.8373 0 Td [(1)]TJ /F2 1 Tf 1.2892 0 Td [(est)-295(sans)-295(ob)-60(jet)-289(et,)-301(dans)-295(le)-295(cas)]TJ /F8 1 Tf 12.554 0 Td [(a)]TJ /F2 1 Tf 0.81927 0 Td [(\257ni,)-301(on)-295(p)-30(eut)-295(raisonner)]TJ -29.572 -1.2048 Td [(de)-361(fa)-6(\273)439(con)-361(analogue)-367(en)-361(utilisan)24(t)-367(la)-361(propri)24(\266)469(et)24(\266)475(e)-367(de)-361(Bepp)-36(o)-361(Levi)-367(p)-30(our)-361(l')24(\266)469(etude)-367(de)-361(la)-361(con)24(tin)24(uit)18(\266)475(e)-367(\265)500(a)]TJ 0 -1.6506 Td [(droite)-391(de)-391(l'application)]TJ /F8 1 Tf 10.12 0 Td [(p)]TJ /F6 1 Tf 0.87349 0 Td [(7!)]TJ /F9 1 Tf 1.3675 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 26.121 59.133 Td [(A)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 18.904 42.238 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 28.698 61.03 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 20.687 42.238 Td [(dm)]TJ /F2 1 Tf 1.7892 0 Td [(et)-391(le)-391(th)18(\266)475(eor)24(\265)469(eme)-391(de)-385(con)24(v)24(ergence)-391(domin)24(\266)475(ee)-397(p)-24(our)-391(celle)]TJ -17.145 -2.3735 Td [(de)-331(l'application)]TJ /F8 1 Tf 7.0602 0 Td [(p)]TJ /F6 1 Tf 0.77711 0 Td [(7!)]TJ /F9 1 Tf 1.2771 1.3614 Td [(Z)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 21.474 55.737 Td [(B)]TJ /F6 1 Tf 11.952 0 0 11.952 0 0 Tm 15.693 39.864 Td [(j)]TJ /F8 1 Tf 0.27711 0 Td [(f)]TJ /F6 1 Tf 0.59639 0 Td [(j)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 24.103 57.633 Td [(p)]TJ /F8 1 Tf 11.952 0 0 11.952 0 0 Tm 17.476 39.864 Td [(dm)]TJ /F2 1 Tf 1.7349 0 Td [(.)]TJ -13.88 -2.8554 Td [(Etudions)-331(alors)-337(la)-337(monotonie)-337(de)-331(l'application)]TJ /F8 1 Tf 19.705 0 Td [(p)]TJ /F6 1 Tf 0.77711 0 Td [(7!)-283(k)]TJ /F8 1 Tf 1.7831 0 Td [(f)]TJ /F6 1 Tf 0.59638 0 Td [(k)]TJ /F7 1 Tf 8.352 0 0 8.352 0 0 Tm 41.06 52.392 Td [(p)]TJ /F2 1 Tf 11.952 0 0 11.952 0 0 Tm 29.102 37.009 Td [(.)]TJ -23.771 -1.7711 Td [(Supp)-24(osons)-481(tout)-481(d'ab)-30(ord)-481(que)-481(cette)-481(application)-481(atteigne)-481(un)-481(minim)24(um)-481(en)-475(un)-481(p)-30(oin)24(t)]TJ /F8 1 Tf 37.157 0 Td [(a)]TJ /F2 1 Tf 1.006 0 Td [(de)]TJ -38.163 -1.2048 Td [(]0;)-168(+)]TJ /F6 1 Tf 2 0 Td [(1)]TJ /F2 1 Tf 1 0 Td 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/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
43.663 50.28 Td
[(et)]TJ
-38.331 -1.1988 Td
[(l'in)24(\266)475(egalit)18(\266)475(e)-337(triangulaire)]TJ
/F6 1 Tf
9.9518 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F2 1 Tf
0.81928 0 Td
[(+)]TJ
/F8 1 Tf
1 0 Td
[(g)]TJ
/F6 1 Tf
0.51807 0 Td
[(k)]TJ
/F10 1 Tf
8.352 0 0 8.352 0 0 Tm
26.647 69.659 Td
[(1)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
19.693 49.081 Td
[(6)]TJ
/F6 1 Tf
1.0602 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F6 1 Tf
0.59639 0 Td
[(k)]TJ
/F10 1 Tf
8.352 0 0 8.352 0 0 Tm
31.983 69.659 Td
[(1)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
23.367 49.081 Td
[(+)]TJ
/F6 1 Tf
1 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(k)]TJ
/F10 1 Tf
8.352 0 0 8.352 0 0 Tm
37.034 69.659 Td
[(1)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
26.681 49.081 Td
[(.)]TJ
-21.349 -1.7048 Td
[(Le)-337(cas)]TJ
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3.0723 0 Td
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/F2 1 Tf
1.8373 0 Td
[(+)]TJ
/F6 1 Tf
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[(1)]TJ
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1.3313 0 Td
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14.328 0 0 14.328 0 0 Tm
4.4472 36.877 Td
[(IX.2.)]TJ
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11.952 0 0 11.952 0 0 Tm
8.2771 44.207 Td
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/F6 1 Tf
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[(2)]TJ
/F2 1 Tf
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[([1;)-168(+)]TJ
/F6 1 Tf
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[(1)]TJ
/F2 1 Tf
1 0 Td
[([)]TJ
/F12 1 Tf
0.60843 0 Td
[(et)]TJ
/F8 1 Tf
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[(f)]TJ
/F12 1 Tf
0.92771 0 Td
[(et)]TJ
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[(g)]TJ
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[(f)]TJ
/F2 1 Tf
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[(+)]TJ
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[(k)]TJ
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8.352 0 0 8.352 0 0 Tm
43.19 60.254 Td
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11.952 0 0 11.952 0 0 Tm
30.867 42.503 Td
[(6)]TJ
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[(k)]TJ
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[(f)]TJ
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8.352 0 0 8.352 0 0 Tm
47.974 60.254 Td
[(p)]TJ
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11.952 0 0 11.952 0 0 Tm
34.157 42.503 Td
[(+)]TJ
/F6 1 Tf
1 0 Td
[(k)]TJ
/F8 1 Tf
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/F6 1 Tf
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[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
52.483 60.254 Td
[(p)]TJ
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11.952 0 0 11.952 0 0 Tm
5.3313 40.671 Td
[(p)-30(ourvu)-331(que)]TJ
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[(f)]TJ
/F2 1 Tf
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8.352 0 0 8.352 0 0 Tm
30.819 54.702 Td
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11.952 0 0 11.952 0 0 Tm
22.367 38.623 Td
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[(=)]TJ
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8.352 0 0 8.352 0 0 Tm
39.448 54.702 Td
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11.952 0 0 11.952 0 0 Tm
27.982 38.623 Td
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13.06 0 Td
[(A)]TJ
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/F2 1 Tf
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[(f)]TJ
/F2 1 Tf
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[(+)]TJ
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[(k)]TJ
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[(f)]TJ
/F2 1 Tf
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[(+)]TJ
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[(k)]TJ
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8.352 0 0 8.352 0 0 Tm
26.405 51.073 Td
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/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
19.151 36.093 Td
[(=)]TJ
/F6 1 Tf
1.0663 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
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/F6 1 Tf
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/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
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[(p)]TJ
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11.952 0 0 11.952 0 0 Tm
22.145 36.093 Td
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[(A)]TJ
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/F2 1 Tf
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14.199 0 Td
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/F6 1 Tf
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/F8 1 Tf
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8.352 0 0 8.352 0 0 Tm
32.56 47.254 Td
[(p)]TJ
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11.952 0 0 11.952 0 0 Tm
23.711 32.605 Td
[(sur)]TJ
ET
305.28 399.6 m
313.92 399.6 l
S
BT
/F3 1 Tf
11.952 0 0 11.952 0 0 Tm
25.542 32.605 Td
[(R)]TJ
/F4 1 Tf
8.352 0 0 8.352 0 0 Tm
37.586 46.443 Td
[(+)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
27.422 32.605 Td
[(\(puisque)]TJ
/F8 1 Tf
4.1928 0 Td
[(p)]TJ
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1.0482 0 Td
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/F2 1 Tf
1.3193 0 Td
[(1\))-493(a)30(v)24(ec)]TJ
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3.7349 0 Td
[(u)]TJ
/F2 1 Tf
1.1084 0 Td
[(=)]TJ
/F8 1 Tf
2.4578 0.6747 Td
[(A)]TJ
ET
481.25 392.68 m
514.51 392.68 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
40.265 31.918 Td
[(A)]TJ
/F2 1 Tf
0.9759 0 Td
[(+)]TJ
/F8 1 Tf
1 0 Td
[(B)]TJ
/F2 1 Tf
1.4217 0.68675 Td
[(et)]TJ
/F8 1 Tf
-38.331 -2.2952 Td
[(v)]TJ
/F2 1 Tf
0.79518 0 Td
[(=)]TJ
/F8 1 Tf
2.1627 0.6747 Td
[(B)]TJ
ET
87.336 365.25 m
120.6 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
7.3072 29.623 Td
[(A)]TJ
/F2 1 Tf
0.96988 0 Td
[(+)]TJ
/F8 1 Tf
1 0 Td
[(B)]TJ
/F2 1 Tf
0.92771 0.68675 Td
[(,)-277(on)-253(obtien)24(t)-259(:)]TJ
/F9 1 Tf
5.8916 1.1084 Td
[(\263)]TJ
/F8 1 Tf
1.7289 -0.43373 Td
[(A)]TJ
ET
200.95 365.25 m
234.22 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
16.813 29.623 Td
[(A)]TJ
/F2 1 Tf
0.96988 0 Td
[(+)]TJ
/F8 1 Tf
1 0 Td
[(B)]TJ
/F6 1 Tf
1.2169 1.3614 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(f)]TJ
/F6 1 Tf
0.59638 0 Td
[(j)]TJ
ET
239.04 365.25 m
252.86 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
20.205 29.623 Td
[(A)]TJ
/F2 1 Tf
1.1386 0.68675 Td
[(+)]TJ
/F8 1 Tf
1.9518 0.6747 Td
[(B)]TJ
ET
266.62 365.25 m
299.88 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
22.307 29.623 Td
[(A)]TJ
/F2 1 Tf
0.9759 0 Td
[(+)]TJ
/F8 1 Tf
1 0 Td
[(B)]TJ
/F6 1 Tf
1.2108 1.3614 Td
[(j)]TJ
/F8 1 Tf
0.28313 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(j)]TJ
ET
304.7 365.25 m
317.52 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
25.627 29.623 Td
[(B)]TJ
/F9 1 Tf
1.0602 1.7952 Td
[(\264)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
39.043 44.668 Td
[(p)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
28.024 30.31 Td
[(6)]TJ
/F8 1 Tf
2.1928 0.6747 Td
[(A)]TJ
ET
348.98 365.25 m
382.25 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
29.199 29.623 Td
[(A)]TJ
/F2 1 Tf
0.9759 0 Td
[(+)]TJ
/F8 1 Tf
1 0 Td
[(B)]TJ
/F9 1 Tf
0.92771 1.7952 Td
[(\263)]TJ
/F6 1 Tf
0.71687 -0.43373 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(f)]TJ
/F6 1 Tf
0.59639 0 Td
[(j)]TJ
ET
392.26 365.25 m
406.08 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
33.024 29.623 Td
[(A)]TJ
/F9 1 Tf
1.0663 1.7952 Td
[(\264)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
49.647 44.668 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
35.223 30.31 Td
[(+)]TJ
/F8 1 Tf
1.9518 0.6747 Td
[(B)]TJ
ET
432.5 365.25 m
465.77 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
36.187 29.623 Td
[(A)]TJ
/F2 1 Tf
0.96988 0 Td
[(+)]TJ
/F8 1 Tf
1.006 0 Td
[(B)]TJ
/F9 1 Tf
0.92771 1.7952 Td
[(\263)]TJ
/F6 1 Tf
0.71687 -0.43373 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(j)]TJ
ET
475.78 365.25 m
488.59 365.25 l
S
BT
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
39.94 29.623 Td
[(B)]TJ
/F9 1 Tf
1.0542 1.7952 Td
[(\264)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
59.517 44.668 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
42.054 30.31 Td
[(,)-271(d'o)-36(\265)530(u)]TJ
-36.723 -2.2349 Td
[(l'on)-313(d)30(\266)469(eduit)-313(imm)24(\266)469(ediatemen)30(t)-313(:)]TJ
/F6 1 Tf
13.096 0 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(f)]TJ
/F2 1 Tf
0.77109 0 Td
[(+)]TJ
/F8 1 Tf
0.95783 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(j)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
30.371 40.771 Td
[(p)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
21.964 28.075 Td
[(6)]TJ
/F2 1 Tf
1.0542 0 Td
[(\()]TJ
/F6 1 Tf
0.39157 0 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(f)]TJ
/F6 1 Tf
0.59639 0 Td
[(j)]TJ
/F2 1 Tf
0.45783 0 Td
[(+)]TJ
/F6 1 Tf
0.95181 0 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(j)]TJ
/F2 1 Tf
0.27711 0 Td
[(\))]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
38.853 40.771 Td
[(p)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
27.892 28.075 Td
[(6)]TJ
/F2 1 Tf
1.0542 0 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(A)]TJ
/F2 1 Tf
0.92771 0 Td
[(+)]TJ
/F8 1 Tf
0.95181 0 Td
[(B)]TJ
/F2 1 Tf
0.80723 0 Td
[(\))]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
46.388 40.771 Td
[(p)]TJ
/F10 1 Tf
0.58621 0 Td
[(\241)]TJ
/F4 1 Tf
0.89655 0 Td
[(1)]TJ
/F9 1 Tf
11.952 0 0 11.952 0 0 Tm
33.904 29.189 Td
[(\263)]TJ
/F8 1 Tf
0.59639 -1.1145 Td
[(A)]TJ
/F4 1 Tf
8.352 0 0 8.352 0 0 Tm
50.44 40.771 Td
[(1)]TJ
/F10 1 Tf
0.57759 0 Td
[(\241)]TJ
/F7 1 Tf
0.88793 0 Td
[(p)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
36.735 28.075 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(f)]TJ
/F6 1 Tf
0.60241 0 Td
[(j)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
54.224 40.771 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
38.53 28.075 Td
[(+)]TJ
/F8 1 Tf
0.95181 0 Td
[(B)]TJ
/F4 1 Tf
8.352 0 0 8.352 0 0 Tm
57.655 40.771 Td
[(1)]TJ
/F10 1 Tf
0.57758 0 Td
[(\241)]TJ
/F7 1 Tf
0.88793 0 Td
[(p)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
41.777 28.075 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(j)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
61.319 40.771 Td
[(p)]TJ
/F9 1 Tf
11.952 0 0 11.952 0 0 Tm
43.307 29.189 Td
[(\264)]TJ
/F2 1 Tf
0.90964 -1.1145 Td
[(.)]TJ
-38.886 -2.1566 Td
[(En)-331(in)24(t)24(\266)469(egran)30(t)-337(il)-337(vien)24(t)-337(:)]TJ
0 -1.9398 Td
[(\()]TJ
/F6 1 Tf
0.38554 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F2 1 Tf
0.72892 0 Td
[(+)]TJ
/F8 1 Tf
0.90361 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
12.681 33.737 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
9.2711 23.979 Td
[(\))]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
13.828 34.9 Td
[(p)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
10.404 23.979 Td
[(6)]TJ
/F2 1 Tf
1.0542 0 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(A)]TJ
/F2 1 Tf
0.87349 0 Td
[(+)]TJ
/F8 1 Tf
0.90964 0 Td
[(B)]TJ
/F2 1 Tf
0.80723 0 Td
[(\))]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
21.224 34.9 Td
[(p)]TJ
/F10 1 Tf
0.58621 0 Td
[(\241)]TJ
/F4 1 Tf
0.89655 0 Td
[(1)]TJ
/F9 1 Tf
11.952 0 0 11.952 0 0 Tm
16.313 25.087 Td
[(\263)]TJ
/F8 1 Tf
0.60241 -1.1084 Td
[(A)]TJ
/F4 1 Tf
8.352 0 0 8.352 0 0 Tm
25.276 34.9 Td
[(1)]TJ
/F10 1 Tf
0.56896 0 Td
[(\241)]TJ
/F7 1 Tf
0.89655 0 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
19.151 23.979 Td
[(\()]TJ
/F6 1 Tf
0.38554 0 Td
[(k)]TJ
/F8 1 Tf
0.50603 0 Td
[(f)]TJ
/F6 1 Tf
0.59638 0 Td
[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
30.25 33.737 Td
[(p)]TJ
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[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
34.526 59.737 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
24.59 41.334 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(X)-24(;)-168(m)]TJ
/F2 1 Tf
2.1747 0 Td
[(\))]TJ
/F12 1 Tf
0.6506 0 Td
[(et)]TJ
/F8 1 Tf
1.0964 0 Td
[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
42.336 59.737 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
30.042 41.334 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(X)-24(;)-168(m)]TJ
/F2 1 Tf
2.1747 0 Td
[(\))]TJ
/F12 1 Tf
0.6506 0 Td
[(son)24(t)-265(des)-265(espaces)-265(v)24(ectoriels)]TJ
-27.928 -1.2048 Td
[(et)-337(l'application)]TJ
/F6 1 Tf
6.8916 0 Td
[(k)-331(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
19.397 56.857 Td
[(p)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
14.301 40.129 Td
[(est)-337(une)-337(semi-norme)-337(sur)-337(le)-343(premier)-337(et)-331(une)-337(norme)-337(sur)-337(le)-343(second.)]TJ
/F2 1 Tf
-8.9699 -2.2048 Td
[(Rapp)-30(elons)-337(que,)-331(sur)-337(un)-331(espace)-331(v)24(ectoriel,)-337(une)-331(semi-norme)-337(est)-331(une)-331(application)]TJ
/F8 1 Tf
33.361 0 Td
[(\272)]TJ
/F2 1 Tf
0.88554 0 Td
[(qui)-331(v)48(\266)475(eri\257e)-331(:)]TJ
/F8 1 Tf
-34.247 -1.2048 Td
[(\272)]TJ
/F2 1 Tf
0.55422 0 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(x)]TJ
/F2 1 Tf
0.57229 0 Td
[(\))]TJ
/F11 1 Tf
0.66867 0 Td
[(>)]TJ
/F2 1 Tf
1.0542 0 Td
[(0)-289(;)]TJ
/F8 1 Tf
1.3675 0 Td
[(\272)]TJ
/F2 1 Tf
0.56024 0 Td
[(\()]TJ
/F8 1 Tf
0.38554 0 Td
[(\270x)]TJ
/F2 1 Tf
1.1566 0 Td
[(\))-283(=)]TJ
/F6 1 Tf
1.7229 0 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(\270)]TJ
/F6 1 Tf
0.58434 0 Td
[(j)]TJ
/F8 1 Tf
0.44578 0 Td
[(\272)]TJ
/F2 1 Tf
0.56024 0 Td
[(\()]TJ
/F8 1 Tf
0.38554 0 Td
[(x)]TJ
/F2 1 Tf
0.57229 0 Td
[(\))-295(et)]TJ
/F8 1 Tf
1.8012 0 Td
[(\272)]TJ
/F2 1 Tf
0.55422 0 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(x)]TJ
/F2 1 Tf
0.70482 0 Td
[(+)]TJ
/F8 1 Tf
0.90964 0 Td
[(y)]TJ
/F2 1 Tf
0.5241 0 Td
[(\))]TJ
/F11 1 Tf
0.66868 0 Td
[(6)]TJ
/F8 1 Tf
1.0542 0 Td
[(\272)]TJ
/F2 1 Tf
0.56024 0 Td
[(\()]TJ
/F8 1 Tf
0.38554 0 Td
[(x)]TJ
/F2 1 Tf
0.57229 0 Td
[(\))-138(+)]TJ
/F8 1 Tf
1.4337 0 Td
[(\272)]TJ
/F2 1 Tf
0.55422 0 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(y)]TJ
/F2 1 Tf
0.5241 0 Td
[(\).)-433(Une)-295(norme)-289(est)-295(alors)-289(une)-289(semi-norme)]TJ
-22.289 -1.1988 Td
[(qui)-331(v)48(\266)475(eri\257e)-337(en)-331(outre)-337(:)-445(\()]TJ
/F8 1 Tf
9.7048 0 Td
[(\272)]TJ
/F2 1 Tf
0.55422 0 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(x)]TJ
/F2 1 Tf
0.57229 0 Td
[(\))-283(=)-277(0\))]TJ
/F6 1 Tf
2.8916 0 Td
[(\))]TJ
/F2 1 Tf
1.2771 0 Td
[(\()]TJ
/F8 1 Tf
0.38554 0 Td
[(x)]TJ
/F2 1 Tf
0.8494 0 Td
[(=)-283(0\).)]TJ
/F1 1 Tf
-16.627 -1.8193 Td
[(Remarques.)]TJ
0.81325 -1.5602 Td
[(1.)]TJ
/F2 1 Tf
1.4337 0 Td
[(La)-271(d)24(\266)475(e\257nition)-277(des)-271(espaces)]TJ
/F6 1 Tf
10.886 0 Td
[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
27.414 46.59 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
19.886 32.141 Td
[(et)]TJ
/F8 1 Tf
1.1084 0 Td
[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
31.017 46.59 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
22.404 32.141 Td
[(p)-30(eut)-246(^)475(etre)-271(utilis)24(\266)469(ee)-271(p)-30(our)-271(0)]TJ
/F8 1 Tf
10.59 0 Td
[(<)-283(p)-277(<)]TJ
/F2 1 Tf
2.8976 0 Td
[(1,)-283(elle)-271(d)24(\266)469(e\257nit)-265(encore)]TJ
-30.56 -1.1988 Td
[(des)-307(espaces)-307(v)24(ectoriels)-313(mais)-307(le)-307(fait)-313(que)]TJ
/F6 1 Tf
16.506 0 Td
[(k)-307(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
33.121 43.702 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
23.861 30.942 Td
[(ne)-307(soit)-307(plus)-301(une)-307(semi-norme)-313(sur)-307(ces)-307(espaces)-313(fait)]TJ
-18.53 -1.3313 Td
[(qu'on)-331(ne)-337(les)-337(consid)30(\265)469(ere)-337(qu'exceptionnellemen)30(t.)]TJ
/F1 1 Tf
0.81325 -1.5602 Td
[(2.)]TJ
/F2 1 Tf
1.5301 0 Td
[(La)-319(d)24(\266)469(e\257nition)-319(des)]TJ
/F6 1 Tf
7.5843 0 Td
[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
22.828 40.737 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
16.735 28.051 Td
[(mon)24(tre)-319(imm)24(\266)469(ediatemen)30(t)-325(que,)-325(si)]TJ
/F8 1 Tf
13.59 0 Td
[(f)]TJ
/F6 1 Tf
0.87952 0 Td
[(2)-283(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
46.991 40.737 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
33.62 28.051 Td
[(alors)]TJ
/F6 1 Tf
2.3795 0 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(f)]TJ
/F6 1 Tf
0.60241 0 Td
[(j)-277(2)-283(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
55.905 40.737 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
39.53 28.051 Td
[(.)-439(Il)-319(s'ensuit)]TJ
-34.199 -1.1988 Td
[(que)-337(les)-337(espaces)]TJ
/F6 1 Tf
6.8253 0 Td
[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
18.379 39.012 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
13.639 26.852 Td
[(son)18(t)-337(des)-337(espaces)-337(v)24(ectoriels)-337(r)30(\266)469(eticul)24(\266)469(es.)]TJ
/F1 1 Tf
-7.494 -1.5602 Td
[(3.)]TJ
/F2 1 Tf
1.5602 0 Td
[(P)30(our)]TJ
/F8 1 Tf
2.4337 0 Td
[(f)]TJ
/F2 1 Tf
0.93373 0 Td
[(et)]TJ
/F8 1 Tf
1.1627 0 Td
[(g)]TJ
/F2 1 Tf
0.8494 0 Td
[(mesurables)-337(a)24(v)24(ec)]TJ
/F8 1 Tf
7.3253 0 Td
[(f)]TJ
/F6 1 Tf
0.87349 0 Td
[(2)-283(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
32.793 36.78 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
23.711 25.292 Td
[(on)-337(a)-307(\266)469(evidemmen)30(t)]TJ
/F6 1 Tf
7.8373 0 Td
[(j)]TJ
/F8 1 Tf
0.28313 0 Td
[(g)]TJ
/F6 1 Tf
0.51205 0 Td
[(j)]TJ
/F11 1 Tf
0.55422 0 Td
[(6)]TJ
/F6 1 Tf
1.0542 0 Td
[(j)]TJ
/F8 1 Tf
0.27711 0 Td
[(f)]TJ
/F6 1 Tf
0.60241 0 Td
[(j)-277(\))]TJ
/F8 1 Tf
1.8313 0 Td
[(g)]TJ
/F6 1 Tf
0.78915 0 Td
[(2)-283(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
55.94 36.78 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
39.554 25.292 Td
[(.)]TJ
-34.223 -2.0783 Td
[(T)78(erminons)-289(cette)-295(section)-289(en)-289(donnan)24(t)-289(des)-295(analogues)-289(du)-289(th)18(\266)475(eor)24(\265)469(eme)-289(de)-289(F)78(atou)-295(et)-289(de)-289(la)-289(propri)24(\266)469(et)24(\266)475(e)]TJ
0 -1.1988 Td
[(de)-331(Bepp)-36(o)-331(Levi)-337(p)-30(our)-337(les)-331(normes)]TJ
/F6 1 Tf
13.94 0 Td
[(k)-331(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
29.483 30.926 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
21.018 22.015 Td
[(.)]TJ
/F0 1 Tf
14.328 0 0 14.328 0 0 Tm
4.4472 15.691 Td
[(IX.2.)]TJ
/F1 1 Tf
11.952 0 0 11.952 0 0 Tm
8.2771 18.81 Td
[(8.)]TJ
/F15 1 Tf
1.753 0 Td
[(Pr)24(oposition.)]TJ
/F12 1 Tf
6.8614 0 Td
[(P)30(our)-319(toute)-313(suite)-319(de)-313(fonctions)]TJ
/F8 1 Tf
12.94 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
43.388 26.702 Td
[(n)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
31.181 18.81 Td
[(mesurables)-319(et)-313(p)-30(ositiv)24(es)-313(et)-319(tout)]TJ
/F8 1 Tf
-25.849 -1.1988 Td
[(p)]TJ
/F6 1 Tf
0.77711 0 Td
[(2)]TJ
/F2 1 Tf
0.66867 0 Td
[(]0;)-168(+)]TJ
/F6 1 Tf
2 0 Td
[(1)]TJ
/F2 1 Tf
1 0 Td
[(])]TJ
/F12 1 Tf
0.61446 0 Td
[(on)-337(a)]TJ
/F6 1 Tf
2.2229 0 Td
[(k)]TJ
/F2 1 Tf
0.66867 0 Td
[(lim)-168(inf)]TJ
/F8 1 Tf
2.9398 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
23.914 24.987 Td
[(n)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
17.253 17.611 Td
[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
25.414 24.625 Td
[(p)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
18.446 17.611 Td
[(6)]TJ
/F2 1 Tf
1.0542 0 Td
[(lim)-174(inf)]TJ
/F6 1 Tf
2.9458 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
33.534 24.987 Td
[(n)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
23.976 17.611 Td
[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
35.026 24.625 Td
[(p)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
24.892 17.611 Td
[(.)]TJ
-19.56 -1.747 Td
[(Si)-337(la)-331(suite)-337(des)]TJ
/F8 1 Tf
6.4036 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
17.491 22.487 Td
[(n)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
13.102 15.864 Td
[(est)-337(croissan)24(te)-337(et)]TJ
/F8 1 Tf
7.3253 0 Td
[(f)]TJ
/F12 1 Tf
0.93374 0 Td
[(est)-337(sa)-337(limite)-337(simple,)-331(on)-337(a)]TJ
/F6 1 Tf
11.241 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F6 1 Tf
0.60241 0 Td
[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
48.948 22.133 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
34.892 15.864 Td
[(=)-283(lim)]TJ
/F6 1 Tf
2.6145 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
55.086 22.487 Td
[(n)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
39.042 15.864 Td
[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
56.586 22.133 Td
[(p)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
39.952 15.864 Td
[(.)]TJ
/F1 1 Tf
-34.62 -2.1928 Td
[(D)24(\266)548(emonstration.)]TJ
/F2 1 Tf
8.8193 0 Td
[(Lorsque)]TJ
/F8 1 Tf
3.9277 0 Td
[(p)-536(<)]TJ
/F2 1 Tf
2.3494 0 Td
[(+)]TJ
/F6 1 Tf
0.78313 0 Td
[(1)]TJ
/F2 1 Tf
1 0 Td
[(,)-524(les)-493(applications)]TJ
/F8 1 Tf
8.1265 0 Td
[(x)]TJ
/F6 1 Tf
1.1084 0 Td
[(7!)]TJ
/F8 1 Tf
1.5361 0 Td
[(x)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
48.017 20.159 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
34.506 13.671 Td
[(et)]TJ
/F8 1 Tf
1.3193 0 Td
[(x)]TJ
/F6 1 Tf
1.1084 0 Td
[(7!)]TJ
/F8 1 Tf
1.5361 0 Td
[(x)]TJ
/F4 1 Tf
8.352 0 0 8.352 0 0 Tm
55.871 20.159 Td
[(1)]TJ
/F7 1 Tf
0.56897 0 Td
[(=p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
40.801 13.671 Td
[(son)24(t)-493(des)]TJ
-35.47 -1.1988 Td
[(bijections)-319(croissan)24(tes)-325(de)]TJ
ET
192.38 158.97 m
201.02 158.97 l
S
BT
/F3 1 Tf
11.952 0 0 11.952 0 0 Tm
16.096 12.473 Td
[(R)]TJ
/F4 1 Tf
8.352 0 0 8.352 0 0 Tm
24.069 17.633 Td
[(+)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
17.807 12.473 Td
[(sur)-325(lui-m)24(^)475(eme,)-325(on)-325(en)-319(d)24(\266)475(eduit)-319(que)-325(l'on)-325(p)-30(eut)-295(\266)469(ec)24(hanger)-325(l')30(\266)469(el)24(\266)469(ev)54(ation)]TJ
-12.476 -1.2048 Td
[(\265)499(a)-337(la)-337(puissance)]TJ
/F8 1 Tf
6.4157 0 Td
[(p)]TJ
/F2 1 Tf
0.83735 0 Td
[(ou)-343(1)]TJ
/F8 1 Tf
1.8976 0 Td
[(=p)]TJ
/F2 1 Tf
1.3434 0 Td
[(a)30(v)24(ec)-343(le)-343(passage)-337(\265)499(a)-337(la)-337(limite)-343(inf)30(\266)469(erieure.)-457(Le)-343(r)30(\266)469(esultat)-343(d)30(\266)469(ecoule)-343(alors)-343(du)]TJ
-10.494 -1.1988 Td
[(th)24(\266)469(eor)24(\265)469(eme)-331(de)-331(F)78(atou)-337(et)-337(de)-331(la)-337(propri)24(\266)469(et)24(\266)475(e)-337(de)-337(Bepp)-30(o)-337(Levi)-337(appliqu)24(\266)475(es)-337(\265)500(a)-331(la)-337(suite)-337(des)-337(\()]TJ
/F8 1 Tf
33.97 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
56.94 14.193 Td
[(n)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
40.337 10.069 Td
[(\))]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
58.276 15.004 Td
[(p)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
41.187 10.069 Td
[(.)]TJ
-35.855 -1.5602 Td
[(Lorsque)]TJ
/F8 1 Tf
3.6687 0 Td
[(p)]TJ
/F2 1 Tf
0.78313 0 Td
[(=)-277(+)]TJ
/F6 1 Tf
1.8313 0 Td
[(1)]TJ
/F2 1 Tf
1 0 Td
[(,)-253(p)-30(osons)]TJ
/F8 1 Tf
3.6867 0 Td
[(g)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
24.009 11.961 Td
[(n)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
17.602 8.5087 Td
[(=)-451(inf)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
26.698 11.245 Td
[(k)]TJ
/F17 1 Tf
0.62931 0 Td
[(>)]TJ
/F7 1 Tf
0.89655 0 Td
[(n)]TJ
/F8 1 Tf
11.952 0 0 11.952 0 0 Tm
20.386 8.5087 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
29.871 11.961 Td
[(k)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
21.596 8.5087 Td
[(et)]TJ
/F8 1 Tf
1.0602 0 Td
[(M)]TJ
/F2 1 Tf
1.3614 0 Td
[(=)-277(lim)-174(inf)]TJ
/F6 1 Tf
3.994 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
41.509 11.961 Td
[(n)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
29.548 8.5087 Td
[(k)]TJ
/F10 1 Tf
8.352 0 0 8.352 0 0 Tm
43 11.607 Td
[(1)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
30.843 8.5087 Td
[(.)-415(P)30(our)]TJ
/F8 1 Tf
3.0181 0 Td
[(n)]TJ
/F11 1 Tf
0.87952 0 Td
[(6)]TJ
/F8 1 Tf
1.0602 0 Td
[(k)]TJ
/F2 1 Tf
0.77711 0 Td
[(on)-234(a)-228(donc)]TJ
/F8 1 Tf
4.3012 0 Td
[(g)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
59.181 11.961 Td
[(n)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
42.181 8.5087 Td
[(6)]TJ
/F8 1 Tf
1.0542 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
62.569 11.961 Td
[(k)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
44.217 8.5087 Td
[(,)]TJ
-38.886 -1.7349 Td
[(d'o)-30(\265)530(u)]TJ
/F6 1 Tf
2.2349 0 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(g)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
12.224 9.478 Td
[(n)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
9.0843 6.7738 Td
[(k)]TJ
/F10 1 Tf
8.352 0 0 8.352 0 0 Tm
13.716 9.1159 Td
[(1)]TJ
/F11 1 Tf
11.952 0 0 11.952 0 0 Tm
10.681 6.7738 Td
[(6)]TJ
/F6 1 Tf
1.0723 0 Td
[(k)]TJ
/F8 1 Tf
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[(\))-379(toute)-379(s)24(\266)469(erie)-379(absolumen)24(t)-379(con)24(v)24(er-)]TJ
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[(gen)30(te)-307(est)-307(con)24(v)24(ergen)24(te.)-439(Plus)-307(pr)24(\266)475(ecis)24(\266)475(emen)24(t,)-313(si)-307(l'on)-307(a)-301(des)-277(\266)469(el)24(\266)469(emen)24(ts)]TJ
/F8 1 Tf
27.367 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
47.491 86.823 Td
[(n)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
34.036 60.822 Td
[(de)]TJ
/F6 1 Tf
1.3012 0 Td
[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
51.56 87.625 Td
[(p)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
36.795 60.822 Td
[(tels)-301(que)]TJ
/F9 1 Tf
3.6386 0.94579 Td
[(X)]TJ
/F6 1 Tf
1.6145 -0.94579 Td
[(k)]TJ
/F8 1 Tf
0.5 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
61.586 86.823 Td
[(n)]TJ
/F6 1 Tf
11.952 0 0 11.952 0 0 Tm
43.584 60.822 Td
[(k)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
63.086 86.461 Td
[(p)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
5.3313 59.473 Td
[(est)-337(\257ni)-337(alors)-337(:)]TJ
0 -1.6747 Td
[({)-680(p)-30(our)-343(presque)-343(tout)]TJ
/F8 1 Tf
9.3855 0 Td
[(x)]TJ
/F12 1 Tf
0.57229 0 Td
[(,)-349(la)-337(s)24(\266)469(erie)-349(de)-337(terme)-349(g)30(\266)469(en)30(\266)469(eral)]TJ
/F8 1 Tf
11.687 0 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
39.302 82.504 Td
[(n)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
28.006 57.798 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(x)]TJ
/F2 1 Tf
0.57229 0 Td
[(\))]TJ
/F12 1 Tf
0.72892 0 Td
[(est)-349(absolumen)24(t)-343(con)24(v)24(ergen)24(te)-343(\(donc)]TJ
-24.367 -1.1988 Td
[(con)30(v)24(ergen)18(te\))-337(;)]TJ
0 -1.8735 Td
[({)-668(la)-331(fonction)]TJ
/F8 1 Tf
6.1386 0 Td
[(f)]TJ
/F12 1 Tf
0.93374 0 Td
[(d)24(\266)475(e\257nie)-337(presque)-337(partout)-337(par)]TJ
/F8 1 Tf
12.211 0 Td
[(f)]TJ
/F2 1 Tf
0.60241 0 Td
[(\()]TJ
/F8 1 Tf
0.38554 0 Td
[(x)]TJ
/F2 1 Tf
0.57229 0 Td
[(\))-283(=)]TJ
/F9 1 Tf
1.7229 0.95181 Td
[(X)]TJ
/F8 1 Tf
1.6145 -0.95181 Td
[(f)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
42.931 78.099 Td
[(n)]TJ
/F2 1 Tf
11.952 0 0 11.952 0 0 Tm
30.542 54.726 Td
[(\()]TJ
/F8 1 Tf
0.39157 0 Td
[(x)]TJ
/F2 1 Tf
0.57229 0 Td
[(\))]TJ
/F12 1 Tf
0.72289 0 Td
[(appartien)24(t)-337(\265)500(a)]TJ
/F6 1 Tf
5.6988 0 Td
[(L)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
55.267 78.909 Td
[(p)]TJ
/F12 1 Tf
11.952 0 0 11.952 0 0 Tm
39.416 54.726 Td
[(;)]TJ
-34.084 -2.1747 Td
[({)-1000(la)-500(suite)-506(des)-506(sommes)-500(partielles)]TJ
/F9 1 Tf
15.596 0.95181 Td
[(X)]TJ
/F7 1 Tf
8.352 0 0 8.352 0 0 Tm
29.983 73.512 Td
[(i