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1、Chapter10ProbabilityThischapterisdevotedtoanintroductiontoprobabilitytheory.Itcontainssomeofthefundamentalresultsofprobabilitytheorythestronglawoflargenumbers,thecentrallimittheorem,themartingaleconvergencetheorem,theconstructionofBrownianmotionprocesses,andKolmogo
2、rovsconsistencytheorem.Onepurposeofthischapteristogivethereaderachancetoworkthroughsomeapplicationsofmeasuretheoryandtherebytogetsomepracticewiththetechniquespresentedearlier.Another,perhapsmoresigni?cant,goalistogivethereaderabroaderpictureofhowσ-algebras,measures
3、,measurablefunctions,andintegralsarise.10.1BasicsInprobabilitytheoryonedescribesandanalyzesrandomsituations,oftencalledexperiments.Letuslookathowsuchsituationscanbemodeledusingmeasuretheory.Webeginwithsometerminology.Aprobabilityspaceisameasurespace(Ω,A,P)suchthatP
4、(Ω)=1.TheelementsofΩarecalledtheelementaryoutcomesorthesamplepointsofourexperiment,andthemembersofAarecalledevents.IfA∈A,thenP(A)istheprobabilityoftheeventA.Example10.1.1.Weillustratetheseconceptswithaverysimpleexample.Supposewetossafaircoin(oneforwhichaheadhasprob
5、ability1/2)twice.Therearefourpossibleoutcomes:wegettwoheads,wegetaheadandthenatail,wegetatailandthenahead,orwegettwotails.SowecanletoursetΩofelementaryoutcomesbe{HH,HT,TH,TT}.ItisnaturalinthiscasetoletAcontainallthesubsetsofΩ.Forexample,{HT,TH}isoneofthesubsetsofΩ;
6、itcorrespondstothereal-worldeventinwhichwegetaheadonexactlyoneofthetosses.Finally,inthissituationeachelementaryoutcomehasprobability1/4ofoccurring,andsowelettheprobabilityofaneventAbe1/4timesthenumberofelementsofA.D.L.Cohn,MeasureTheory:SecondEdition,BirkhauserAdv
7、anced¨307TextsBaslerLehrb¨ucher,DOI10.1007/978-1-4614-6956-810,?SpringerScience+BusinessMedia,LLC201330810ProbabilityAreal-valuedrandomvariableonaprobabilityspace(Ω,A,P)isanA-measurablefunctionfromΩtoR.Suchavariablerepresentsanumericalobservationormeasurementwhosev
8、aluedependsontheoutcomeoftherandomexperimentrepresentedby(Ω,A,P).Moregenerally,arandomvariablewithvaluesinameasurablespace(S,B)isameasurablefunct
