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1、FOUNDATIONSOFTHETHEORYOFPROBABILITYBYA.N.KOLMOGOROVSecondEnllli.laEditionTRANSLATIONEDITEDBYNATHANMORRISONWITHANADDEDBIBLIOCRAPHYBYA.T.BHARUCHA·REIDUNIVERSITYOFORECONCHELSEAPUBLISHINGCOMPANYNEWYORKCOPYRIGHT1950BYCHELSEAPUBLISHINGCOMPANYCOPYRIGHT©1956,CHELSEAPUBLISHINGCOMP
2、ANYLIBRARYOFCONGRESSCATALOGUECARDNUMBER56-11512PRINTEDINTHEUNITEDSTATESOFAMERICAEDITOR'SNOTEInthepreparationofthisEnglishtranslationofProfessorKolmogorov'sfundamentalwork,theoriginalGermanmonographGrundbeuriffederWahrscheinlichkeitrechnunuwhichappearedintheErgebnisseDerMa
3、thematikin1933,andalsoaRussiantranslationbyG.M.Bavlipublishedin1936havebeenused.Itisapleasuretoacknowledgetheinvaluableassistanceoftwofriendsandformercolleagues,Mrs.IdaRhodesandMr.D.V.Varley,andalsoofmyniece,GizellaGross.ThanksarealsoduetoMr.RoyKueblerwhomadeavailableforc
4、omparisonpurposeshisindependentEnglishtranslationoftheoriginalGermanmonograph.NathanMorriaonPREFACEThepurposeofthismonographistogiveanaxiomaticfoundationforthetheoryofprobability.Theauthorsethimselfthetaskofputtingintheirnaturalplace,amongthegeneralnotionsofmodernmathemat
5、ics,thebasicconceptsofprobabilitytheory-conceptswhichuntilrecentlywereconsideredtobequitepeculiar.ThistaskwouldhavebeenaratherhopelessonebeforetheintroductionofLebesgue'stheoriesofmeasureandintegration.However,afterLebesgue'spublicationofhisinvestigations,theanalogiesbetw
6、eenmeasureofasetandprobabilityofanevent,andbetweenintegralofafunctionandmathematicalexpectationofarandomvariable,becameapparent.Theseanalogiesallowedoffurtherextensions;thus,forexample,variouspropertiesofindependentrandomvariableswereseentobeincompleteanalogywiththecorres
7、pondingpropertiesoforthogonalfunctions.Butifprobabilitytheorywastobebasedontheaboveanalogies,itstillwasnecessarytomakethetheoriesofmeasureandintegrationindependentofthegeometricelementswhichwereintheforegroundwithLebesgue.ThishasbeendonebyFrechet.Whileaconceptionofprobabi
8、litytheorybasedontheabovegeneralviewpointshasbeencurrentforsometimeamongcertainmathematicians,th
