Conditionally Evenly Convex Sets and Evenly QuasiConve - PDF document

quasiconvexityofafunction,i.e.thepropertythattheconditionallowerlevelsetsareevenlyconvex,isaweakerassumptionthanquasiconvexityandlower(orupper)semicontinuity.InSection3weapplythenotionofconditionallyevenlyconvexsettothethedualrepresentationofevenlyquasiconvexmaps,i.e.conditionalmaps:E!L0( ;G;P)withthepropertythattheconditionallowerlevelsetsareevenlyconvex.WeproveinTheorem17thatanevenlyquasiconvexregularmap:E!L0(G)canberepresentedas(X)=sup2L(E;L0(G))R((X););(1)whereR(Y;):=inf2Ef()j()Yg;Y2L0(G);EisatopologicalL0-moduleandL(E;L0(G))isthemoduleofcontinuousL0-linearfunctionalsoverE.Theproofofthisresultisbasedonaversionofthehyperplaneseparationtheoremandnotonsomeapproximationorscalarizationarguments,asithappenedinthevectorspacesetting(see[FM11]).BycarefullyanalyzingtheproofonemayappreciatemanysimilaritieswiththeoriginaldemonstrationinthestaticsettingbyPenotandVolle[PV90].Onekeydierencewith[PV90],inadditiontotheconditionalsetting,isthecontinuityassumptionneededtoobtaintherepresentation(1).Wework,asin[CV09],withevenlyquasiconvexfunctions,anassumptionweakerthanquasiconvexityandlower(orupper)semicontinuity.Asexplainedin[FM11]therepresentationofthetype(1)isacornerstoneinordertoreacharobustrepresentationofQuasi-convexRiskMeasuresorAcceptabilityIndexes.2OnConditionallyEvenlyConvexsetsTheprobabilityspace( ;G;P)isxedthroughoutthispaper.Wheneverwewilldiscusscon-ditionalpropertieswewillalwaysmakereference,evenwithoutexplicitlymentioningitinthenotations-toconditioningwithrespecttothesigmaalgebraG.WedenotewithL0=:L0( ;G;P)thespaceofGmeasurablerandomvariablesthatarePa.s.nite,whereasbyL0thespaceofextendedrandomvariableswhichmaytakevaluesinR[f1g.Weremindthatallequalities/inequalitiesamongrandomvariablesaremeanttoholdP-a.s..AstheexpectedvalueEP[
]ismostlycomputedw.r.t.thereferenceprobabilityP,wewilloftenomitPinthenotation.ForanyA2Gtheelement1A2L0istherandomvariablea.s.equalto1onAand0elsewhere.Ingeneralsince( ;G;P)arexedwewillalwaysomitthem.WedeneL0+=fX2L0jX0gandL0++=fX2L0jX�0g.Theessential(Palmostsurely)supremumesssup(X)ofanarbitraryfamilyofrandomvariablesX2L0( ;F;P)willbesimplydenotedbysup(X),andsimilarlyfortheessentialinmum(see[FS04]SectionA.5forreference).Denition1(Dualpair)Adualpair(E;E0;h
;
i)consistsof:1.(E;+)(resp.(E0;+))isanystructuresuchthattheformalsumx1A+y1ACbelongstoE(resp.x01A+y01AC2E0)foranyx;y2E(resp.x0;y02E0)andA2GwithP(A)�0andthereexistsannullelement02E(resp.02E0)suchthatx+0=xforallx2E(resp.x0+0=x0forallx02E0).2.Amaph
;
i:EE0!L0suchthathx1A+y1AC;x0i=hx;x0i1A+hy;x0i1AChx;x01A+y01ACi=hx;x0i1A+hx;y0i1ACh0;x0i=0andhx;0i=02 Notation4FixasetCE.AstheclassA(C):=fA2GjC1A=E1Agisclosedwithrespecttocountableunion,wedenotewithACtheG-measurablemaximalelementoftheclassA(C)andwithDCthe(P-a.s.unique)complementofAC(seealsotheRemark30).HenceC1AC=E1AC:Wenowgivetheformaldenitionofconditionallyevenlyconvexsetintermsofintersectionsofhyperplanesinthesamespiritof[Fe52].Denition5AsetCEisconditionallyevenlyconvexifthereexistLE0(ingeneralnon-uniqueandemptyifC=E)suchthatC=\x02Lfx2Ejhx;x0iYx0onDCgforsomeYx02L0:(3)Remark6NoticethatforanyarbitraryD2G,LE0thesetC=\x02Lfx2Ejhx;x0iYx0onDgforsomeYx02L0isevenlyconvex,eventhoughingeneralDCD.Remark7WeobservethatsinceEsatises(CSet)thenautomaticallyanyconditionallyevenlyconvexsetsatises(CSet).AsaconsequencetheremightexistasetCwhichfailstobeconditionallyevenlyconvex,sincedoesnotsatisfy(CSet),butCccisconditionallyevenlyconvex.ConsiderforinstanceE=L1G(F);E0=L1G(F),endowedwiththepairinghx;x0i=E[xx0jG].Fixx02L1(F);Y2L0(G)andthesetC=fx2L1(F)jE[xx0jG]Yg:ClearlyCisnotconditionallyevenlyconvexsinceC$Ccc;ontheotherhandCcc=fx2L1G(F)jE[xx0jG]Ygwhichisbydenitionevenlyconvex.Remark8RecallthatasetCEisL0-convexifx+(1�)y2Cforanyx;y2Cand2L0with01.Supposethatalltheelementsx02E0satisfy:hx+(1�)y;x0ihx;x0i+(1�)hy;x0i,forallx;y2E,2L0:01:IfEisL0�convextheneveryconditionallyevenlyconvexsetisalsoL0�convex.Inordertoseparateonepointx2EfromasetCEinaconditionalwayweneedthefollowingdenition:Denition9Forx2EandasubsetCofE;wesaythatxisoutsideCif1Afxg\1AC=?foreveryA2GwithADCandP(A)&#x-278;0.Thisisofcourseamuchstrongerrequirementthanx=2C.Denition10ForCEwedenethepolarandbipolarsetsasfollowsC:=fx02E0jhx;x0i1onDCforallx2Cg;C:=fx2Ejhx;x0i1onDCforallx02Cg=\x02Cfx2Ejhx;x0i1onDCg:4 Let:E!L0be(REG).TheremightexistasetA2Gonwhichthemapisinnite,inthesensethat()1A=+11Aforevery2E.ForthisreasonweintroduceM:=fA2Gj()1A=+11A82Eg:ApplyingLemma36inAppendixwithF:=f()j2EgandY0=+1wecandeducetheexistenceoftwomaximalsetsT2Gand2GforwhichP(T\)=0,P(T[)=1and()=+1onforevery2E;()+1onTforsome2E:(5)Denition15Amap:E!L0(G)is(QCO)conditionallyquasiconvexifUY=f2Ej()1TYgareL0-convex(accordingtoRemark8)foreveryY2L0(G).(EQC)conditionallyevenlyquasiconvexifUY=f2Ej()1TYgareconditionallyevenlyconvexforeveryY2L0(G).Remark16For:E!L0(G)thequasiconvexityofisequivalenttothecondition(x1+(1�)x2)(x1)_(x2);(6)foreveryx1;x22E,2L0(G)and01.Inthiscasethesetsf2Ej()1DYgareL0(G)-convexforeveryY2L0(G)andD2G(Thisfollowsimmediatelyfrom(6)).MoreoverunderthefurtherstructuralpropertyofRemark8wehavethat(EQC)implies(QCO).WewillseeintheL0-modulesframeworkthatifthemapiseitherlowersemicontinuousoruppersemicontinuousthenthereverseimplicationholdstrue(seeProposition23,Corollary26andProposition27).WenowstatethemainresultofthisSection.Theorem17Let(E;E0;h
;
i)beadualpairingintroducedinDenition1.If:E!L0(G)is(REG)and(EQC)then(x)=supx02E0R(hx;x0i;x0);(7)whereforY2L0(G)andx0,R(Y;x0):=inf2Ef()jh;x0iYg:(8)4ConditionalEvenlyconvexityinL0-modulesThissectionisinspiredbythecontributiongiventothetheoryofL0-modulesbyFilipovicetal.[FKV09]ononehandandontheothertotheextendedresearchprovidedbyGuofrom1992untiltoday(seethereferencesin[Gu10]).ThefollowingProposition23showsthatthedenitionofaconditionallyevenlyconvexsetistheappropriategeneralization,inthecontextoftopologicalL0module,ofthenotionofanevenlyconvexsubsetofatopologicalvectorspace,asinbothsettingconvex(resp.L0-convex)setsthatareeitherclosedoropenareevenly(resp.conditionallyevenly)convex.Thisisakeyresultthatallowstoshowthattheassumption(EQC)istheweakestthatallowstoreachadualrepresentationofthemap.WewillconsiderL0,withtheusualoperationsamongrandomvariables,asapartiallyorderedringandwewillalwaysassumeinthesequelthat0isatopologyonL0suchthat(L0;0)isatopologicalring.Wedonotrequirethat0isalineartopologyonL0(sothat(L0;0)maynotbeatopologicalvectorspace)northat0islocallyconvex.6 Lemma22.1.LetEbeatopologicalL0-module.IfCiE,i=1;2;areopenandnonemptyandA2G,thenthesetC11A+C21ACisopen.2.Let(E;Z;)beL0-moduleassociatedtoZ.ThenforanynetfgE,2E,2EandA2G!=)(1A+1AC)!(1A+1AC):Proof.1.Toshowthisclaimletx:=x11A+x21ACwithxi2CiandletU0beaneighborhoodof0satisfyingxi+U0Ci.ThenthesetU:=(x1+U0)1A+(x2+U0)1AC=x+U01A+U01ACiscontainedinC11A+C21ACanditisaneighborhoodofx,sinceU01A+U01ACcontainsU0andisthereforeaneighborhoodof0.2.Observethataseminormsatisesk1A(�)k=1Ak�kk�kandtherefore,bycondition3.inDenition21theclaimfollows.Inparticular,!=)(1A)!(1A). Proposition23Let(E;Z;)beL0-moduleassociatedtoZandsupposethatCEsatises(CSet).1.SupposethatthestrictlypositiveconeL0++is0-openandthatthereexistx002Eandx02Esuchthatx00(x0)�0:UnderAssumptionS-Open,ifCisopenandL0-convexthenCisconditionallyevenlyconvex.2.UnderAssumptionS-Closed,ifCisclosedandL0-convexthenitisconditionallyevenlyconvex.Proof.1.LetCEbeopen,L0-convex,C6=?andletAC2GbethemaximalsetgivenintheNotation4,beingDCitscomplement.SupposethatxisoutsideC,i.e.x2Esatisesfxg1A\C1A=?foreveryA2G,ADC,P(A)�0.DenetheL0-convexsetE:=f2Ejx00()�x00(x)g=(x00)�1(x00(x)+L0++)andnoticethatfxg1A\E1A=;foreveryA2G.AsL0++is0-open,EisopeninE:Asx00(x0)�0;then(x+x0)2EandEisnon-empty.ThenthesetC0=C1DC+E1ACisL0-convex,open(byLemma22)andsatisesfxg1A\C01A=;foreveryA2Gs.t.P(A)�0.AssumptionS-Openguaranteestheexistenceofx2Es.t.x(x)�x()82C0;whichimpliesx(x)�x()onDC,82C.Hence,byTheorem11,Cisconditionallyevenlyconvex.2.LetCEbeclosed,L0-convex,C6=?andsupposethatx2Esatisesfxg1A\C1A=?foreveryA2G,ADC,P(A)�0.LetC0=C1DC+fx+"g1ACwhere"2L0++.ClearlyC0isL0-convex.InordertoprovethatC0isclosedconsideranynet!,fgC0.Then=Z1DC+fx+"g1AC,withZ2C,and(x+")1AC=1AC:Takeany2C.AsCisL0-convex,1DC+1AC=Z1DC+1AC2Cand,byLemma22,1DC+1AC!1DC+1AC:=Z2C,asCisclosed.Therefore,=Z1DC+fx+"g1AC2C0.SinceC0isclosed,L0-convexandfxg1A\C01A=;foreveryA2G,assumptionS-Closedguaranteestheexistenceofx2Es.t.x(x)�x()82C0;whichimpliesx(x)�x()onDC,82C.Hence,byTheorem11,Cisconditionallyevenlyconvex. Proposition24Let(E;Z;)andEberespectivelyasindenitions19and21,andlet0beatopologyonL0suchthatthepositiveconeL0+isclosed.ThenanyconditionallyevenlyconvexL0-conecontainingtheoriginisclosed.Proof.From(20)andthebipolarTheorem12weknowthatC=C=\x02Cfx2Ejhx;x0i0onDCg:8 andx002Esuchthatx00(x0)�0.Thisandthenextitem2allowtheapplicationofProposition23.AfamilyZofL0-seminormsonEinducesatopologyonEinthefollowingway.ForanyniteSZand"2L0++deneUS;":=fx2EjkxkS"gU:=fUS;"jSZniteand"2L0++g:Ugivesaconvexneighborhoodbaseof0anditinducesatopologyonEdenotedbyc.Wehavethefollowingproperties:1.(E;Z;c)isa(L0;j
j)-moduleassociatedtoZ,whichisalsoalocallyconvextopologicalL0-module(seeProposition2.7[Gu10]),2.(E;Z;c)satisesS-OpenandS-Closed(seeTheorems2.6and2.8[FKV09]),3.Anytopological(L0;j
j)module(E;)islocallyconvexifandonlyifisinducedbyafamilyofL0-seminorms,i.e.c,(seeTheorem2.4[FKV09]).Aprobabilistictopology;[Gu10]ThesecondtopologyontheL0-moduleEisatopologyofamoreprobabilisticnatureandoriginatedinthetheoryofprobabilisticmetricspaces(see[SS83]).HereL0isendowedwiththetopology;ofconvergenceinprobabilityandsothepositiveconeL0+is0-closed.Accordingto[Gu10],forevery;2RandanitesubfamilySZofL0-seminormsweletVS;;:=fx2EjP(kxkS)&#x-278;1�gV:=fUS;;jSZnite,&#x-278;0;01g:Vgivesaneighborhoodbaseof0anditinducesalineartopologyonE,alsodenotedby;(indeedifE=L0thenthisisexactlythetopologyofconvergenceinprobability).Thistopologymaynotbelocallyconvex,buthasthefollowingproperties:1.(E;Z;;)becomesa(L0;;)-moduleassociatedtoZ(seeProposition2.6[Gu10]),2.(E;Z;;)satisesS-Closed(seeTheorems3.6and3.9[Gu10]).ThereforeProposition23canbeapplied.5OnConditionallyEvenlyQuasi-ConvexmapsonL0-moduleAsanimmediateconsequenceofProposition23wehavethatlower(resp.upper)semicontinuityandquasiconvexityimplyevenlyquasiconvexityof.FromTheorem17wethendeducetherepresentationforlower(resp.upper)semicontinuousquasiconvexmaps.(LSC)Amap:E!L0(G)islowersemicontinuousifforeveryY2L0thelowerlevelsetsUY=f2Ej()1TYgare-closed.Corollary26Let(E;Z;)andE0=Eberespectivelyasindenitions19and21,satisfyingS-Closed.If:E!L0(G)is(REG),(QCO)and(LSC)then(7)holdstrue.Intheuppersemicontinuouscasewecansaymore(theproofispostponedtoSection6).(USC)Amap:E!L0(G)isuppersemicontinuousifforeveryY2L0thelowerlevelsetsUY=f2Ej()1TYgare-open.10 Proof.1.Lemma31showsthatP(HC;x)�0.Since1ACE=1ACC,ifx=2Cwenecessarilyhave:P(HC;x\AC)=0andthereforeHC;xDC.2.IfxisoutsidejCthenxisoutsidejDCCandx=2C.ThethesisfollowsfromHC;xDCandthefactthatHC;xisthelargestsetD2GforwhichxisoutsidejDC.3.isaconsequenceofLemma35(seeAppendix)item1. ProofofTheorem11.(1))(2).LetLE0,Yx02L0andletC=:\x02Lf2Ejh;x0iYx0onDCg;whichclearlysatisesCcc=C.Bydenition,ifthereexistsx2Es.t.xisoutsideCthen1Afxg\1AC=?8ADC;A2G,P(A)&#x-278;0,andthereforebythedenitionofCthereexistsx02Ls.t.hx;x0iYx0onDC:Hence:hx;x0iYx0&#x-278;h;x0ionDCforall2C:(2))(1)WeareassumingthatCis(CSet),andthereexistsx2Es.t.x=2C(otherwiseC=E).From(28)weknowthat=fy2EjyisoutsideCgisnonempty.Byassumption,forally2thereexists0y2E0suchthath;0yihy;0yionDC;82C.DeneBy:=f2Ejh;0yihy;0yionDCg:Byclearlydependsalsoontheselectionofthe0y2E0associatedtoyandonC,butthisnotationwillnotcauseanyambiguity.Wehave:CByforally2,andCTy2By:Wenowclaimthatx=2Cimpliesx=2Ty2By,thusshowingC=\y2By=\0y2Lf2Ejh;0yiY0yonDCg;(17)whereL:=0y2E0jy2 ,Y0y:=hy;0yi2L0,andthethesisisproved.Supposethatx=2C,then,byLemma31,xisoutsidejHC,wherewesetforsimplicityH=HC;x.Takeanyy26=?anddeney0:=x1H+y1 nH2.TakeBy0=f2Ejh;0y0ihy0;0y0ionDCgwhere0y02E0istheelementassociatedtoy0.Ifx2By0thenwewouldhave:hx;0y0ihy0;0y0i=hx;0y0ionHDC,byLemma32item1,whichisacontradiction,sinceP(H)&#x]TJ/;༔ ; .96;& T; -4;.2;Q -;.9;U T; [0;0.Hencex=2By0Ty2By: Proposition33UnderthesameassumptionsofTheorem11,thefollowingareequivalent:1.Cisconditionallyevenlyconvex2.foreveryx2E,x=2C,thereexistsx02E0suchthath;x0ihx;x0ionHC;x82C;whereHC;xisdenedinLemma31.Proof.(1)=)(2):WeknowthatCsatises(CSet).Asx=2C,from(28)andLemma31weknowthatthereexistsy2Es.t.yisoutsideCandthatH=:HC;xsatisesP(H)&#x]TJ/;༔ ; .96;& T; 10;&#x.516;&#x 0 T; [0;0.Dene~x=x1H+y1 nH.Then~xisoutsideCandbyTheorem11item2thereexistsx02E0h;x0ih~x;x0ionDC;82C:Thisimpliesthethesissinceh~x;x0i=hx;x0i1H+hy;x0i1 nHandHDC.(2)=)(1):Weshowthatitem2ofTheorem11holdstrue.ThisistrivialsinceifxisoutsideCthenx=2CandHC;x=DC: ProofofTheorem12.Item(1)isstraightforward;thefactthatCisconditionallyevenlyconvexfollowsfromthedenition;theproofofCCisalsoobvious.WenowsupposethatCis12 ProofofTheorem17.Let:E!L0(G).TheremightexistasetA2Gonwhichthemapisconstant,inthesensethat()1A=()1Aforevery;2E.ForthisreasonweintroduceA:=fB2Gj()1B=()1B8;2Eg:ApplyingLemma36inAppendixwithF:=f()�()j;2Eg(weconsidertheconvention+1�1=0)andY0=0wecandeducetheexistenceoftwomaximalsetsA2GandA`2GforwhichP(A\A`)=0,P(A[A`)=1and()=()onAforevery;2E;(1)(2)onA`forsome1;22E:(21)Recallthat2Gisthemaximalsetonwhich()1=+11forevery2EandTitscomplement.NoticethatA.Fixx2EandG=f(x)+1g.Forevery"2L0++(G)wesetY"=:01+(x)1An+((x)�")1G\A`+"1GC\A`(22)andforevery"2L0(G)++wesetC"=f2Ej()1T"Y"g:(23)Step1:onthesetA,(x)=R(hx;x0i;x0)foranyx02E0andtherepresentation(x)1A=maxx02E0R(hx;x0i;x0)1A(24)triviallyholdstrueonA.Step2:bythedenitionofY"wededucethatifC"=;forevery"2L0++then(x)()onthesetA`forevery2Eand(x)1A`=R(hx;x0i;x0)1A`foranyx0.Therepresentation(x)1A`=maxx02E0R(hx;x0i;x0)1A`(25)triviallyholdstrueonA`.Thethesisfollowspastingtogetherequations(24)and(25)Step3:wenowsupposethatthereexists"2L0++suchthatC"6=;.ThedenitionofY"impliesthatC"1A=E1AandAisthemaximalelementi.e.A=AC"(givenbyDenition4).MoreoverthissetisconditionallyevenlyconvexandxisoutsideC".Thedenitionofevenlyconvexsetguaranteesthatthereexistsx0"2E0suchthathx;x0"i&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;h;x0"ionDC"=A`;82C":(26)Claim:f2Ejhx;x0"i1A`h;x0"i1A`gf2Ej()&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;((x)�")1G+"1GConA`g.(27)Inordertoprovetheclaimtake2Esuchthathx;x0"i1A`h;x0"i1A`.BycontrawesupposethatthereexistsaFA`,F2GandP(F)&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0suchthat()1F((x)�")1G\F+"1GC\F.Take2C"anddene =1FC+1F2C"sothatweconcludethathx;x0"i�h ;x0"ionA`.Sinceh ;x0"i=h;x0"ionFwereachacontradiction.Oncetheclaimisprovedweendtheargumentobservingthat(x)1A`supx02E0R(hx;x0i;x0)1A`=R(hx;x0"i;x0")1A`(28)=inf2Ef()1A`jhx;x0"i1Ah;x0"i1A`ginf2Ef()1A`j()�((x)�")1G+"1GConA`g((x)�")1G\A`+"1GC\A`;(29)14 Proof.ThetwoclassesAandA`areclosedwithrespecttocountableunion.Indeed,forthefamilyA`,supposethatBi2A`;yi2Dccs.t.yiisoutsidejBiCcc:DeneeB1:=B1,eBi:=BinBi�1,B:=1Si=1eBi=1Si=1Bi.ThenyiisoutsidejeBiCcc,eBiaredisjointelementsofA`andy:=P11yi1eBi2Dcc.Sinceyi1eBi=y1eBi,yisoutsidejeBiCccforalliandsoyisoutsidejBCcc.ThusB2A`.SimilarlyfortheclassA.TheRemark30guaranteestheexistenceofthetwomaximalsetsAM2AandA`M2A`,sothat:B2AimpliesBAM;B`2A`impliesB`A`M.Obviously,P(AM\A`M)=0;asAM2AandA`M2A`.WeclaimthatP(AM[A`M)=1:(31)Toshow(31)letD:= nAM[A`M 2G.BycontradictionsupposethatP(D)�0.FromD(AM)CandthemaximalityofAMwegetD=2A.Thisimpliesthatthereexistsy2Dccsuchthat1Dfyg\1DCcc=?(32)andobviouslyy=2Ccc,asP(D)�0.BytheLemma31thereexistsasetHCcc;y:=H2GsatisfyingP(H)�0,(14)and(15)withCreplacedbyCcc:Condition(15)impliesthatH2A`andthenHA`M.From(14)wededucethatthereexists2Cccs.t.1Ay=1AforallA nH:Then(32)impliesthatDisnotcontainedin nH;sothat:P(D\H)�0:ThisisacontradictionsinceD\HD(A`M)CandD\HHA`M.Item1isatrivialconsequenceofthedenitions. Lemma36WiththesymbolDdenoteanyoneofthebinaryrelations;;=;�,andwithCitsnegation.ConsideraclassFL0(G)ofrandomvariables,Y02L0(G)andtheclassesofsetsA:=fA2Gj8Y2FYDY0onAg;A`:=fA`2Gj9Y2Fs.t.YCY0onA`g:SupposethatforanysequenceofdisjointsetsA`i2A`andtheassociatedr.v.Yi2FwehaveP11Yi1A`i2F.ThenthereexisttwomaximalsetsAM2AandA`M2A`suchthatP(AM\A`M)=0,P(AM[A`M)=1andYDY0onAM,8Y2F YCY0onA`M,forsome Y2F:Proof.NoticethatAandA`areclosedwithrespecttocountableunion.ThisclaimisobviousforA.ForA`,supposethatA`i2A`andthatYi2FsatisesP(fYiCY0g\A`i)=P(A`i):DeningB1:=A`i,Bi:=A`inBi�1,A`1:=1Si=1A`i=1Si=1BiweseethatBiaredisjointelementsofA`andthatY:=P11Yi1Bi2FsatisesP(fYCY0g\A`1)=P(A`1)andsoA`12A`.TheRemark30guaranteestheexistenceoftwosetsAM2AandA`M2A`suchthat:(a)P(A\(AM)C)=0forallA2A,(b)P(A`\(A`M)C)=0forallA`2A`.Obviously,P(AM\A`M)=0;asAM2AandA`M2A`.ToshowthatP(AM[A`M)=1,letD:= nAM[A`M 2G.BycontradictionsupposethatP(D)�0.AsD(AM)C,fromcondition(a)wegetD=2A.Therefore,9 Y2Fs.t.P( YDY0 \D)P(D);i.e.P( YCY0 \D)�0.IfwesetB:= YCY0 \DthenitsatisesP( YCY0 \B)=P(B)�0and,bydenitionofA`,BbelongstoA`.Ontheotherhand,asBD(A`M)C,P(B)=P(B\(A`M)C);andfromcondition(b)P(B\(A`M)C)=0,whichcontradictsP(B)�0. 16