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首页 GTM 180-Srivastava S.M___A Course on Borel Sets …

GTM 180-Srivastava S.M___A Course on Borel Sets (Springer 1998).pdf

GTM 180-Srivastava S.M___A Cour…

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简介:本文档为《GTM 180-Srivastava S.M___A Course on Borel Sets (Springer 1998)pdf》,可适用于人文社科领域

ACourseonBorelSetsSMSrivastavaSpringerAcknowledgmentsIamgratefultomanypeoplewhohavesuggestedimprovementsintheoriginalmanuscriptforthisbookInparticularIwouldliketothankSCBagchi,RBarua,SGangopadhyay(ne´eBhattacharya),JKGhosh,MGNadkarni,andBVRaoMydeepestfeelingsofgratitudeandappreciationarereservedforHSarbadhikariwhoverypatientlyreadseveralversionsofthisbookandhelpedinallpossiblewaystobringthebooktoitspresentformItisapleasuretorecordmyappreciationforAMaitrawhoshowedthebeautyandpowerofBorelsetstoagenerationofIndianmathematiciansincludingmeIalsothankhimforhissuggestionsduringtheplanningstageofthebookIthankPBandyopadhyaywhohelpedmeimmenselytosortoutalltheLATEXproblemsThanksarealsoduetoRKarforpreparingtheLATEXfilesfortheillustrationsinthebookIamindebtedtoSBRao,DirectoroftheIndianStatisticalInstituteforextendingexcellentmoralandmaterialsupportAllmycolleaguesintheStat–MathUnitalsolentamuchneededandinvaluablemoralsupportduringthelonganddifficultperiodthatthebookwaswrittenIthankthemallItakethisopportunitytoexpressmysincerefeelingsofgratitudetomychildren,RosyandRavi,fortheirgreatunderstandingofthetaskItookontomyselfWhattheymissedduringtheperiodthebookwaswrittenwillbeknowntoonlythethreeofusFinally,Ipayhomagetomylatewife,KiranwhoreallyunderstoodwhatmathematicsmeanttomeSMSrivastavaContentsAcknowledgmentsviiIntroductionxiAboutThisBookxvCardinalandOrdinalNumbersCountableSetsOrderofInfinityTheAxiomofChoiceMoreonEquinumerosityArithmeticofCardinalNumbersWellOrderedSetsTransfiniteInductionOrdinalNumbersAlephsTreesInductiononTreesTheSouslinOperationIdempotenceoftheSouslinOperationTopologicalPreliminariesMetricSpacesPolishSpacesCompactMetricSpacesMoreExamplesxContentsTheBaireCategoryTheoremTransferTheoremsStandardBorelSpacesMeasurableSetsandFunctionsBorelGeneratedTopologiesTheBorelIsomorphismTheoremMeasuresCategoryBorelPointclassesAnalyticandCoanalyticSetsProjectiveSetsΣandΠCompleteSetsRegularityPropertiesTheFirstSeparationTheoremOnetoOneBorelFunctionsTheGeneralizedFirstSeparationTheoremBorelSetswithCompactSectionsPolishGroupsReductionTheoremsChoquetCapacitabilityTheoremTheSecondSeparationTheoremCountabletoOneBorelFunctionsSelectionandUniformizationTheoremsPreliminariesKuratowskiandRyllNardzewski’sTheoremDubins–SavageSelectionTheoremsPartitionsintoClosedSetsVonNeumann’sTheoremASelectionTheoremforGroupActionsBorelSetswithSmallSectionsBorelSetswithLargeSectionsPartitionsintoGδSetsReflectionPhenomenonComplementationinBorelStructuresBorelSetswithσCompactSectionsTopologicalVaughtConjectureUniformizingCoanalyticSetsReferencesGlossaryIndexIntroductionTherootsofBorelsetsgobacktotheworkofBaireHewastryingtocometogripswiththeabstractnotionofafunctionintroducedbyDirichletandRiemannAccordingtothem,afunctionwastobeanarbitrarycorrespondencebetweenobjectswithoutgivinganymethodorprocedurebywhichthecorrespondencecouldbeestablishedSinceallthespecificfunctionsthatonestudiedweredeterminedbysimpleanalyticexpressions,BairedelineatedthosefunctionsthatcanbeconstructedstartingfromcontinuousfunctionsanditeratingtheoperationofpointwiselimitonasequenceoffunctionsThesefunctionsarenowknownasBairefunctionsLebesgueandBorelcontinuedthisworkIn,BorelsetsweredefinedforthefirsttimeInhispaper,LebesguemadeasystematicstudyofBairefunctionsandintroducedmanytoolsandtechniquesthatareusedeventodayAmongotherresults,heshowedthatBorelfunctionscoincidewithBairefunctionsThestudyofBorelsetsgotanimpetusfromanerrorinLebesgue’spaper,whichwasspottedbySouslinLebesguewastryingtoprovethefollowing:Supposef:R−→RisaBairefunctionsuchthatforeveryx,theequationf(x,y)=hasauniquesolutionThenyasafunctionofxdefinedbytheaboveequationisBaireThewrongstepintheproofwashiddeninalemmastatingthatasetofrealnumbersthatistheprojectionofaBorelsetintheplaneisBorel(Lebesgueleftthisasatrivialfact!)SouslincalledtheprojectionofaBorelsetanalyticbecausesuchasetcanbeconstructedusinganalyticaloperationsofunionandintersectiononintervalsHeshowedthattherearexiiIntroductionanalyticsetsthatarenotBorelImmediatelyafterthis,SouslinandLusinmadeadeepstudyofanalyticsetsandestablishedmostofthebasicresultsaboutthemTheirresultsshowedthatanalyticsetsareoffundamentalimportancetothetheoryofBorelsetsandgiveititspowerForinstance,SouslinprovedthatBorelsetsarepreciselythoseanalyticsetswhosecomplementsarealsoanalyticLusinshowedthattheimageofaBorelsetunderaonetooneBorelmapisBorelItfollowsthatLebesgue’sthoeremthoughnottheproofwasindeedtrueAroundthesametimeAlexandrovwasworkingonthecontinuumhypothesisofCantor:EveryuncountablesetofrealnumbersisinonetoonecorrespondencewiththereallineAlexandrovshowedthateveryuncountableBorelsetofrealsisinonetoonecorrespondencewiththereallineInotherwords,aBorelsetcannotbeacounterexampletothecontinuumhypothesisUnfortunately,SouslindiedinTheworkonthisnewfoundtopicwascontinuedbyLusinandhisstudentsinMoscowandbySierpin´skiandhiscollaboratorsinWarsawThenextimportantstepwastheintroductionofprojectivesetsbyLusin,,andSierpin´skiin:AsetiscalledprojectiveifitcanbeconstructedstartingwithBorelsetsanditeratingtheoperationsofprojectionandcomplementationSinceBorelsetsaswellasprojectivesetsaresetsthatcanbedescribedusingsimplesetslikeintervalsandsimplesetoperations,theirtheorycametobeknownasdescriptivesettheoryItwasclearfromthebeginningthatthetheoryofprojectivesetswasriddledwithproblemsthatdidnotseemtoadmitsimplesolutionsAsitturnedout,logiciansdidshowlaterthatmostoftheregularitypropertiesofprojectivesets,eg,whethertheysatisfythecontinuumhypothesisornotorwhethertheyareLebesguemeasurableandhavethepropertyofBaireornot,areindependentoftheaxiomsofclassicalsettheoryJustasAlexandrovwastryingtodeterminethestatusofthecontinuumhypothesiswithinBorelsets,Lusinconsideredthestatusoftheaxiomofchoicewithin“Borelfamilies”HeraisedaveryfundamentalanddifficultquestiononBorelsetsthatenricheditstheorysignificantlyLetBbeasubsetoftheplaneAsubsetCofBuniformizesBifitisthegraphofafunctionsuchthatitsprojectiononthelineisthesameasthatofB(SeeFigure)Lusinasked,WhendoesaBorelsetBintheplaneadmitaBoreluniformizationByLusin’stheoremstatedearlier,ifBadmitsaBoreluniformization,itsprojectiontothelinemustbeBorelInBlackwellshowedthatthisconditionisnotsufficientSeveralauthorsconsideredthisproblemandgavesufficientconditionsunderwhichLusin’squestionhasapositiveanswerForinstance,aBorelsetadmitsaBoreluniformizationifthesectionsofBarecountable(Lusin)orcompact(Novikov)orσcompact(ArseninandKunugui)ornonmeager(KechrisandSarbadhikari)EventodaytheseresultsarerankedamongtheIntroductionxiiiYXCBFigureUniformizationfinestresultsonBorelsetsFortheuniformizationofBorelsetsingeneral,themostimportantresultprovedbeforethewarisduetoVonNeumann:ForeveryBorelsubsetBofthesquare,×,,thereisasetNandaBorelfunctionf:,N−→,whosegraphiscontainedinBAsexpected,thisresulthasfoundimportantapplicationsinseveralbranchesofmathematicsSofarwehavemainlybeengivinganaccountofthetheorydevelopedbeforethewarie,uptoThenforsometimetherewasalull,notonlyinthetheoryofBorelsets,butinthewholeofdescriptivesettheoryThiswasmainlybecausemostofthemathematiciansworkinginthisareaatthattimeweretryingtoextendthetheorytohigherprojectiveclasses,which,asweknownow,isnotpossiblewithinZermelo–FraenkelsettheoryFortunately,aroundthesametimesignificantdevelopmentsweretakingplaceinlogicthatbroughtaboutagreatrevivalofdescriptivesettheorythatbenefitedthetheoryofBorelsetstooThefundamentalworkofGo¨delontheincompletenessofformalsystemsultimatelygaverisetoarichandpowerfultheoryofrecursivefunctionsAddisonestablishedastrongconnectionbetweendescriptivesettheoryandrecursivefunctiontheoryThisledtothedevelopmentofamoregeneraltheorycalledeffectivedescriptivesettheory(ThetheoryasdevelopedbyLusinandothershasbecomeknownasclassicaldescriptivesettheory)FromthebeginningitwasapparentthattheeffectivetheoryismorepowerfulthantheclassicaltheoryHowever,thefirstconcreteevidenceofthiscameinthelateseventieswhenLouveauprovedabeautifultheoremonBorelsetsinproductspacesSincethenseveralclassicalresultshavebeenprovedusingeffectivemethodsforwhichnoclassicalproofisknownyetsee,eg,Forcing,apowerfulsettheoretictechnique(inventedbyCohentoshowtheindependenceofthecontinuumhypothesisandtheaxiomofchoicefromotheraxiomsofsettheory),andothersettheoretictoolssuchasdeterminacyandconstructibility,havebeenveryeffectivelyusedtomakethetheoryofBorelsetsaverypowerfultheory(SeeBartoszyn´skiandJudah,Jech,Kechris,andMoschovakis)xivIntroductionMuchoftheinterestinBorelsetsalsostemsfromtheapplicationsthatitstheoryhasfoundinareassuchasprobabilitytheory,mathematicalstatistics,functionalanalysis,dynamicprogramming,harmonicanalysis,representationtheoryofgroups,andC∗algebrasForinstance,BlackwellshowedtheimportanceofthesesetsinavoidingcertaininherentpathologiesinKolmogorov’sfoundationsofprobabilitytheoryinBlackwell’smodelofdynamicprogrammingtheexistenceofoptimalstrategieshasbeenshowntoberelatedtotheexistenceofmeasurableselections(Maitra)Mackeymadeuseofthesesetsinproblemsregardinggrouprepresentations,andinparticularindefiningtopologiesonmeasurablegroupsChoquet,usedthesesetsinpotentialtheoryandsoonThetheoryofBorelsetshasfoundusesindiverseappliedareassuchasoptimization,controltheory,mathematicaleconomics,andmathematicalstatistics,,,,,Theseapplications,inturn,haveenrichedthetheoryofBorelsetsitselfconsiderablyForexample,mostofthemeasurableselectiontheoremsaroseinvariousapplications,andnowthereisarichsupplyofthemSomeofthese,suchasthecrosssectiontheoremsforBorelpartitionsofPolishspacesduetoMackey,Effros,andSrivastavaarebasicresultsonBorelsetsThus,todaythetheoryofBorelsetsstandsonitsownasapowerful,deep,andbeautifultheoryThisbookisanintroductiontothistheoryAboutThisBookThisbookcanbeusedinvariouswaysItcanbeusedasasteppingstonetodescriptivesettheoryFromthispointofview,ouraudiencecanbeundergraduateorbeginninggraduatestudentswhoarestillexploringareasofmathematicsfortheirresearchInthisbooktheywillgetareasonablythoroughintroductiontoBorelsetsandmeasurableselectionsTheywillalsofindthekindofquestionsthatadescriptivesettheoristasksThoughwesticktoBorelsetsonly,wepresentquiteafewimportanttechniques,suchasuniversalsets,prewellordering,andscales,usedindescriptivesettheoryWehopethatstudentswillfindthemathematicspresentedinthisbooksolidandexcitingSecondly,thisbookisaddressedtomathematiciansrequiringBorelsets,measurableselections,etc,intheirworkTherefore,wehavetriedourbesttomakeitaconvenientreferencebookSomeapplicationsarealsogivenjusttoshowthewaythattheresultspresentedhereareusedFinally,wedesirethatthebookbeaccessibletoallmathematiciansHencethebookhasbeenmadeselfcontainedandhasbeenwritteninaneasygoingstyleWehaverefrainedfromdisplayingvariousadvancedtechniquessuchasgames,recursivefunctions,andforcingWeuseonlynaivesettheory,generaltopology,someanalysis,andsomealgebra,whicharecommonlyknownThebookisdividedintofivechaptersInthefirstchapterwegivethesettheoreticpreliminariesInthefirstpartofthischapterwepresentcardinalarithmetic,methodsoftransfiniteinduction,andordinalnumbersThenweintroducetreesandtheSouslinoperationTopologicalpreliminariesarepresentedinChapterWelaterdevelopthetheoryofBorelsetsinthexviAboutThisBookgeneralcontextofPolishspacesHencewegiveafairlycompleteaccountofPolishspacesinthischapterInthelastsectionofthischapterweproveseveraltheoremsthathelpintransferringmanyproblemsfromgeneralPolishspacestothespaceofsequencesNNortheCantorspaceNWeintroduceBorelsetsinChapterHerewedevelopthetheoryofBorelsetsasmuchaspossiblewithoutusinganalyticsetsInthelastsectionofthischapterweintroducetheusualhierarchyofBorelsetsForthefirsttime,readerswillseesomeofthestandardmethodsofdescriptivesettheory,suchasuniversalsets,reduction,andseparationprinciplesChapteriscentraltothisbook,andtheresultsprovedherebringouttheinherentpowerofBorelsetsInthischapterweintroduceanalyticandcoanalyticsetsandprovemostoftheirbasicpropertiesThattheseconceptsareoffundamentalimportancetoBorelsetsisamplydemonstratedinthischapterInChapterwepresentmostofthemajormeasurableselectionanduniformizationtheoremsTheseresultsareparticularlyimportantforapplicationsWeclosethischapterwithadiscussiononVaught’sconjectureanoutstandingopenproblemindescriptivesettheory,andwithaproofofKondoˆ’suniformizationofcoanalyticsetsTheexercisesgiveninthisbookareanintegralpartofthetheory,andreadersareadvisednottoskipthemManyexercisesarelatertreatedasprovedtheoremsSincethisbookisintendedtobeintroductoryonly,manyresultsonBorelsetsthatwewouldhavemuchlikedtoincludehavebeenomittedForinstance,Martin’sdeterminacyofBorelgames,Silver’stheoremoncountingthenumberofequivalenceclassesofaBorelequivalencerelation,andLouveau’stheoremonBorelsetsintheproducthavenotbeenincludedSimilarly,otherresultsrequiringsuchsettheoretictechniquesasconstructibility,largecardinals,andforcingarenotgivenhereInourinsistenceonstickingtoBorelsets,wehavemadeonlyapassingmentionofhigherprojectiveclassesWearesurethatthiswillleavemanydescriptivesettheoristsdissatisfiedWehavenotbeenabletogivemanyapplications,todojusticetowhichwewouldhavehadtoentermanyareasofmathematics,sometimesevendelvingdeepintothetheoriesClearly,thiswouldhaveincreasedthesizeofthebookenormouslyandmadeitunwieldyWehopethatuserswillfindthepassingremarksandreferencesgivenhelpfulenoughtoseehowresultsprovedhereareusedintheirrespectivedisciplinesCardinalandOrdinalNumbersInthischapterwepresentsomebasicsettheoreticalnotionsThefirstfivesectionsaredevotedtocardinalnumbersWeuseZorn’slemmatodevelopcardinalarithmeticOrdinalnumbersandthemethodsoftransfiniteinductiononwellorderedsetsarepresentedinthenextfoursectionsFinally,weintroducetreesandtheSouslinoperationTreesarealsousedinseveralotherbranchesofmathematicssuchasinfinitarycombinatorics,logic,computerscience,andtopologyTheSouslinoperationisofspecialimportancetodescriptivesettheory,andperhapsitwillbenewtosomereadersCountableSetsTwosetsAandBarecalledequinumerousorofthesamecardinality,writtenA≡B,ifthereexistsaonetoonemapffromAontoBSuchanfiscalledabijectionForsetsA,B,andCwecaneasilycheckthefollowingA≡A,A≡B=⇒B≡A,and(A≡BB≡C)=⇒A≡CTheseareproducedherefrommyarticlewiththepermissionoftheIndianAcademyofSciencesCardinalandOrdinalNumbersAsetAiscalledfiniteifthereisabijectionfrom{,,,n−}(nanaturalnumber)ontoA(Forn=wetaketheset{,,,n−}tobetheemptyset∅)IfAisnotfinite,wecallitinfiniteThesetAiscalledcountableifitisfiniteorifthere

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