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1、Chapter8PolishSpacesandAnalyticSetsTheBorelsubsetsofacompleteseparablemetricspacehaveanumberofinterestingandusefulcharacteristics.Forexample,ifAandBareuncountableBorelsubsetsofcompleteseparablemetricspaces,thenAandBareBorelisomorphicthatis,thereisabijectionf:A→Bsuchthatfandf−1arebothBorel
2、measurable.ArelatedresultsaysthatifAisaBorelsubsetofacompleteseparablemetricspace,ifYisacompleteseparablemetricspace,andiff:A→YisinjectiveandBorelmeasurable,thenf(A)isaBorelsubsetofY.Ifthefunctionfisnotinjective,thenf(A)maynotbeaBorelset,butitwillbeμ-measurableforeveryfiniteBorelmeasureμon
3、Y(thatis,therewillbeBorelsubsetsB1andB2ofYsuchthatB1⊆f(A)⊆B2andμ(B2−B1)=0).Thischapterisdevotedtoprovingsuchresultsandtoshowingthecontextinwhichtheyarise.8.1PolishSpacesAPolishspaceisaseparabletopologicalspacethatcanbemetrizedusingacompletemetric.Thissectioncontainsanumberofelementaryprop
4、ertiesofPolishspaces.InSects.8.3through8.6wewillusetheseproperties,plustheconceptofananalyticset(definedinSect.8.2),toderivesomedeepandusefulresultsaboutmeasurablesetsandfunctions.TherearemanytopologicalspacesthatarePolish,buthavenocompletemetricthatisparticularlynaturalorsimple.Furthermor
5、e,manyconstructionsandfactsofinterestinmeasuretheorydependontheexistenceofacompletemetric,butnotonthechoiceofaparticularmetric.IthasthusbecomerathercommontodealwiththeclassofPolishspaces,ratherthanwiththeclassofcompleteseparablemetricspaces.D.L.Cohn,MeasureTheory:SecondEdition,BirkhauserA
6、dvanced¨239TextsBaslerLehrb¨ucher,DOI10.1007/978-1-4614-6956-88,©SpringerScience+BusinessMedia,LLC20132408PolishSpacesandAnalyticSetsExamples8.1.1.(a)ForeachdthespaceRd,withitsusualtopology,isPolish.(b)Moregenerally,eachseparableBanachspace,withthetopologyinducedbyitsnorm,isPolish.(c)Each
7、compactmetrizablespaceisPolish(seeTheoremD.39andCorollaryD.40).ItamountstothesamethingtosaythateachcompactHausdorffspacethathasacountablebaseisPolish(seeProposition7.1.13).Weneedthefollowingresultsbeforewelookatsomeadditionalexamples.Proposition8.1.2.Eachclosedsubs
