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1、MEASURE,INTEGRATION&PROBABILITYIvanFWildeMathematicsDepartmentKing'sCollegeLondonChapter1¾-algebrasandBorelfunctionsPreliminarydiscussion²SupposethatXisacontinuousrandomvariable.ThenProb(X=s)=0foranys2R.However,theeventfX2RghasprobabilityoneandisthedisjointunionoftheeventsfX=sgfors2R;[fX2Rg=fX=sg
2、:sEacheventontherighthandsidehasprobabilityzero,sotheprobabilitiesoftheeventsontherighthandsidedonotaddup"tothatofthelefthandside.Wewishtounderstandthis.²SupposethatXisarandomvariableonasamplespace•,andsupposethatXtakesvaluesx1;x2;:::.PutAi=f!2•:X(!)=xig.ThenXEX=xiProb(X=xi)iX=xiProb(Ai):i(1;!2A
3、iInparticular,for1lAi(!)=,0;!=2AiE1lAi=1Prob(1lAi=1)=Prob(Ai):PAlsoE(xi1lAi)=xiProb(Ai).ButwecanwriteX=ixi1lAiandwerecoverEXasXXEX=xiProb(Ai)=E(xi1lAi):iiHereXisastep-function"on•.Thisformulaformsthebasisforthegeneral"expectation,i.e.,thatforanarbitraryrandomvariable.12Chapter1²Onemust(sometime
4、s)askwhichsubsetsofasamplespacearedeemedtobeevents.Canonetakeallsubsetsofthesamplespacetobeevents?Theanswerissometimesyesandsometimesno.Forexample,inthecasewhentheprobabilityofaneventwithinaboundedregionofR3isrequiredtobeproportionaltothevolumeassociatedwiththeevent,thenonenaturallyaskswhethereve
5、rysubsetof(aboundedregion)ofR3actuallyhasavolume.ThatthisisnotsoisdemonstratedbytheBanach-Tarskitheorem.(ThissaysthataballofunitradiusinR3canbecutupintoa¯nitenumberofpieceswhichcanthenbereassembledtoformaballofradius2.Themeaningofvolume"forthesepiecesisnotclear.)Wemustbepreciseabouttheconceptof
6、event".Inthemodern"(Kol-mogorov)theoryofprobability,thisisformulatedintermsof¾-algebras.De¯nition1.1.Acollection§ofsubsetsofanon-emptysetXiscalleda¾-algebraif(i)X2§,(ii)ifA2§,thenAc=XnA2§,S1(iii)ifAn2§forn=1;2;:::,thenn=1An2§.Thesetsin§arecalledmeasurablesets,and(X;§)iscalledamea-surablespace".
7、Remarks1.2.1.Since?=Xc,itfollowsthat?2§.2.ForanyA1;A2;:::;An2S§,putAn+1=An+2=¢¢¢=?.Thenwe1seethatA1[¢¢¢[An=k=1Ak2§,by(1)above,and(iii).T1¡S1c¢c3.LetTA1;A2;¢¢¢2§.Thensincen=1An=n=1An,weseethat1n=1An2§.IfwetakeAn+1=An+2=
