AmodicumofmeasuretheorySECTION1de - PDF document

Chapter2 AmodicumofmeasuretheorySECTION1deÞnesmeasuresandsigma-Þelds.SECTION2deÞnesmeasurablefunctions.SECTION3deÞnestheintegralwithrespecttoameasureasalinearfunctionalonaconeofmeasurablefunctions.ThedeÞnitionsidestepsthedetailsoftheconstructionofintegralsfrommeasures.SECTION*4constructsintegralsofnonnegativemeasurablefunctionswithrespecttoacountablyadditivemeasure.SECTION5establishestheDominatedConvergencetheorem,theSwissArmyknifeofmeasuretheoreticprobability.SECTION6collectstogetheranumberofsimplefactsrelatedtosetsofmeasurezero.SECTION*7presentsafewfactsaboutspacesoffunctionswithintegrablethpowers,withemphasisonthecasep=2,whichdeÞnesaHilbertspace.SECTION8deÞnesuniformintegrability,aconditionslightlyweakerthandomination.Convergenceinischaracterizedasconvergenceinprobabilityplusuniformintegrability.SECTION9deÞnestheimagemeasure,whichincludestheconceptofthedistributionofarandomvariableasaspecialcase.SECTION10explainshowgeneratingclassarguments,forclassesofsets,makemeasuretheoryeasy.SECTION*11extendsgeneratingclassargumentstoclassesoffunctions.   AspromisedinChapter1,webeginwithmeasuresassetfunctions,thenworkquicklytowardstheinterpretationofintegralsaslinearfunctionals.Oncewearepastthepurelyset-theoreticpreliminaries,IwillstartusingthedeFinettinotation)inearnest,writingthesamesymbolforasetanditsindicatorfunction.Ourstartingpointisameasurespace:atriple,withaset,aclassofsubsetsof,andafunctionthatattachesanonnegativenumber(possiblytoeachsetin.Theclassandthesetfunctionarerequiredtohavepropertiesthatfacilitatecalculationsinvolvinglimitsalongsequences. Chapter2:AmodicumofmeasuretheoryCallaclassasigma-fieldofsubsetsof(i)theemptysetandthewholespacebothbelongto(ii)ifbelongstothensodoesitscomplement(iii)ifisacountablecollectionofsetsinthenboththeunionandtheintersectionarealsoinSomeoftherequirementsareredundantasstated.Forexample,oncewehavethen(ii)implies.Whenwecometoestablishpropertiesaboutsigma-ÞeldsitwillbeconvenienttohavethelistofdeÞningpropertiespareddowntoaminimum,toreducetheamountofmechanicalchecking.Thetheoremswillbeassparingaspossibleintheamounttheworktheyrequireforestablishingthesigma-Þeldproperties,butfornowredundancydoesnothurt.Thecollectionneednotcontaineverysubsetof,afactforceduponusingeneralifwewanttohavethepropertiesofacountablyadditivemeasure.Afunctiondefinedonthesigma-fieldiscalleda(countablyadditive,nonnegative)measureif:foreach(iii)ifisacountablecollectionofpairwisedisjointsetsinAmeasureforwhich1iscalledaprobabilitymeasure,andtheiscalledaprobabilityspace.Forthisspecialcaseitistraditionaltouseasymbollikeforthemeasure,asymbollikefortheset,andasymbollikeforthesigma-Þeld.Atriple willalwaysdenoteaprobabilityspace.UsuallythequaliÞcationsÒcountablyadditive,nonnegativeÓareomitted,onthegroundsthatthesepropertiesarethemostcommonlyassumedÑthemostcommoncasesdeservetheshortestnames.OnlywhenthereissomedoubtaboutwhetherthemeasuresareassumedtohaveallthepropertiesofDeÞnitionshouldthequaliÞersbeattached.Forexample,onespeaksofÒÞnitelyadditivemeasuresÓwhenananalogofproperty(iii)isassumedonlyforÞnitedisjointcollections,orÒsignedmeasuresÓwhenthevalueofisnotnecessarilynonnegative.WhenÞnitelyadditiveorsignedmeasuresareunderdiscussionitmakessensetomentionexplicitlywhenaparticularmeasureisnonnegativeorcountablyadditive,but,ingeneral,youshouldgowiththeshortername.Wheredomeasurescomefrom?ThemostbasicconstructionsstartfromsetdeÞnedonsmallcollectionsofsubsets,suchasthecollectionofallsubintervalsoftherealline.OnechecksthathaspropertiesconsistentwiththerequirementsofDeÞnition.OneseekstoextendthedomainofdeÞnitionwhilepreservingthecountableadditivitypropertiesofthesetfunction.AsyousawinChapter1,TheoremsguaranteeingexistenceofsuchextensionsweretheculminationofalongsequenceofreÞnementsintheconceptofintegration(Hawkins1979).Theyrepresentoneofthegreatachievementsofmodernmathematics,eventhoughthosetheoremsnowoccupyonlyahandfulofpagesinmostmeasuretheorytexts. 2.1Measuresandsigma-Finiteadditivityhasseveralappealinginterpretations(suchasthefair-pricesofSection)thathavegivenitreadyacceptanceasanaxiomforamodelofreal-worlduncertainty.Countableadditivityissometimesregardedwithsuspicion,orjustiedasamatterofmathematicalconvenience.(However,seeProblemProblemforanequivalentformofcountableadditivity,whichhassomeclaimtointuitiveappeal.)Itisdifculttodevelopasimpleprobabilitytheorywithoutcountableadditivity,whichgivesonethelicence(foronlyasmallfee)tointegrateseriesterm-by-term,differentiateunderintegrals,andinterchangeotherlimitingoperations.Theclassicalconstructionsaresignicantformyexpositionmostlybecausetheyensureexistenceofthemeasuresneededtoexpressthebasicresultsofprobabilitytheory.IwillrelegatethedetailstotheProblemsandtoAppendixA.Ifyoucraveamoresystematictreatmentyoumightconsultoneofthemanyexcellenttextsonmeasuretheory,suchasRoyden(1968).Theconstructionsdonotindeedcannot,ingeneralleadtocountablyadditivemeasuresontheclassofallsubsetsofagiven.Typically,theyextendasetfunctiondenedonaclassofsetstoameasuredenedonthegeneratedby,ortoonlyslightlylargersigma-elds.Bydesmallestsigma-eldoncontainingallsetsfromforeverysigma-Therepresentationgivenbythesecondlineensuresexistenceofasmallestsigma-eldcontaining.Themethodofdenitionisanalogoustomanydenitionsofcontainingaxedclassinmathematicsthinkofgeneratedsubgroupsorlinearsubspacesspannedbyacollectionofvectors,forexample.Forthenitiontoworkoneneedstocheckthatsigma-eldshavetwoproperties:(i)Ifisanonemptycollectionofsigma-eldson,thecollectionofallthesubsetsofthatbelongtoevery,isalsoasigma-(ii)Foreachthereexistsatleastonesigma-containingallthesetsinYoushouldcheckproperty(i)asanexercise.Property(ii)istrivial,becausethecollectionofallsubsetsofisasigma-Proofsofexistenceofnonmeasurablesetstypicallydependonsomedeepset-theoreticprinciple,suchastheAxiomofChoice.Mathematicianswhocanlivewithdifferentrulesforsettheorycanhavebiggersigma-elds.SeeDudley(1989,Section3.4)orOxtoby(1971,Section5)fordetails.consistsofvepoints,and.Supposeoftwosets,.Findthesigma-eldgeneratedbyForthissimpleexamplewecanproceedbymechanicalapplicationofthepropertiesthatasigma-mustpossess.Inadditiontotheobvious,itmustcontaineachofthesets Chapter2:AmodicumofmeasuretheoryFurtherexperimentationcreatesnonewmembersof;thesigma-eldconsistsofthesetsThesetsaretheofthesigma-eld;everymemberofisaunionofsomecollection(possiblyempty)of.Theonlymeasurablesubsetsofaretheemptysetanditself.Therearenomeasurableprotonsorneutronshidinginsidetheseatoms.Anunsystematicconstructionmightworkfornitesets,butitcannotgenerateallmembersofasigma-eldingeneral.Indeed,wecannotevenhopetolistallthemembersofaninnitesigma-eld.Insteadwemustndalessexplicitwaytocharacterizeitssets.Bydenition,theBorelsigma-eldontherealline,denotedbyisthesigma-eldgeneratedbytheopensubsets.Wecouldalsodenoteitbystandsfortheclassofallopensubsetsof.Thereareseveralothergeneratingclassesfor.Forexample,asyouwillsoonsee,theclassofallintervals],with,isageneratingclass.Itmightappearahopelesstasktoprovethatifwecannotexplicitlylistthemembersofbothsigma-elds,butactuallytheproofisquiteroutine.Youshouldtrytounderstandthestyleofargumentbecauseitisoftenusedinprobabilitytheory.Theequalityofsigma-eldsisestablishedbytwoinclusions,,bothofwhichfollowfrommoreeasilyestablishedresults.Firstwemustprovethat,showingthatisoneofthesigma-thatenterintotheintersectionde,andhence.TheotherinclusionfollowssimilarlyifweshowthatEachinterval]inhasarepresentation,acountableintersectionofopensets.Thesigma-containsallopensets,anditisstableundercountableintersections.Itthereforecontainseach].Thatis,Theargumentforisonlyslightlyharder.Itdependsonthefactthatanopensubsetofthereallinecanbewrittenasacountableunionofopenintervals.Suchanintervalhasarepresentation,and].Thatis,everyopensetcanbebuiltupfromsetsusingoperationsthatareguaranteednottotakeusoutsidethesigma-Myexplanationhasbeenmoderatelydetailed.Inapublishedpaperthereasoningwouldprobablybeabbreviatedtosomethinglikeageneratingclassargumentshowsthatwiththeroutinedetailslefttothereader.Thegeneratingclassargumentoftenreducestoanassertionlike:isasigma-eldand,thereforeAclassofsubsetsofasetiscalledaifitcontainstheemptysetandisstableundercomplements,niteunions,andniteintersections.Fora,writefortheclassofallpossibleintersectionsofcountablesubclasses,andfortheclassofallpossibleunionsofcountablesubclassesof 2.1Measuresandsigma-Ofcourseifisasigma-eldthen,but,ingeneral,theinclusionswillbeproper.Forexample,ifconsistsofallniteunionsofhalfopenintervals],withpossibly  ,thenthesetofrationalsdoesnotbelongtoandthecomplementofthesamesetdoesnotbelongtobeanitemeasureon.Eventhoughmightbemuchlargerthaneither,ageneratingclassargumentwillshowthatallsetsininnerapproximatedby,inthesensethat,foreachouterapproximatedby,inthesensethat,foreachIncidentally,Ichosetheletterstoremindmyselfofopenandclosedsets,whichhavesimilarapproximationpropertiesforBorelmeasuresonmetricspacesseeProblemProblem.Ithelpstoworkonbothapproximationpropertiesatthesametime.Denotebytheclassofallsetsinthatcanbebothinnnerandouterapproximated.Abelongstoifandonlyif,toeach0thereexistllcallthesets-sandwichforTrivially,becauseeachmemberofbelongstoboth.Theapproximationresultwillfollowifweshowthatisasigma-eld,forthenwewillhaveSymmetryofthedenitionensuresthatisstableundercomplements:ifisan-sandwichfor,thenisan-sandwichforToshowthatisstableundercountableunions,consideracountablecollectionofsetsfrom.Weneedtoslicethebreadthinnerasgetslarger:foreach.Theunionissandwichedbetweenthesets;andthesetsarecloseinCanyouprovethisinequality?Doyouseewhyandwhycountableadditivityimpliesthatthemeasureofacountableunionof(notnecessarilydisjoint)setsissmallerthanthesumoftheirmeasures?Ifnot,justwaituntilSection,afterwhichyoucanarguethatasaninequalitybetweenindicatorfunctions,andbyMonotoneConvergence.Wehavean-sandwich,butthebreadmightnotbeoftherighttype.Itiscertainlytruethat(acountableunionofcountableunionsisacountableunion),butthesetneednotbelongto.However,thesetsbelongto,andcountableadditivityimpliesthatDoyouseewhy?Ifnot,waitforMonotoneConvergenceagain.Ifwechoosealargeenoughwehavea2 Chapter2:AmodicumofmeasuretheoryThemeasureforwhichiscalledLebesguemeasureAnothersortofgeneratingclassargument(seeSection)canbeusedtoshowthatthevaluesareuniquelydeterminedbythevaluesgiventointervals;therecanexistatmostonemeasureonwiththestatedproperty.Itishardertoshowthatatleastonesuchmeasureexists.Despiteanyintuitionsyoumighthaveaboutlength,theconstructionofLebesguemeasureisnottrivialAppendixA.Indeed,HenriLebesguebecamefamousforprovingexistenceofthemeasureandshowinghowmuchcouldbedonewiththenewintegrationtheory.ThenameLebesguemeasureisalsogiventoanextensionoftoameasureonasigma-eld,sometimescalledtheLebesguesigma-eld,whichisslightlylarger.IwillhavemoretosayabouttheextensioninSectionBorelsigma-eldsaredenedinsimilarfashionforanytopologicalspaceThatis,denotesthesigma-eldgeneratedbytheopensubsetsofSetsinasigma-aresaidtobe-measurable.Inprobabilitytheorytheyarealsocalledevents.Goodfunctionswillalsobegiventhetitlemeasurable.Trynottogetconfusedwhenyoureallyneedtoknowwhetheranobjectisasetorafunction.   beasetequippedwithasigma-,andbeasetequippedwitha,andbeafunction(alsocalledamap)from.Wesaythatiftheinverseimagebelongstoforeach.Sometimestheinverseimageisdenotedby.Dontbefooledbythenotationintotreatingasafunctionfrom:itsnot,isone-to-one(andonto,ifyouwanttohavedomain).Sometimesan-measurablemapisreferredtoinabbreviatedformasjust-measurable,or-measurable,orjustmeasurable,ifthereisnoambiguityabouttheunspeci T - 1 B T B (  ,  ) ( ,  ) Forexample,ifequalstheBorelsigma-,itiscommontodropthecationandrefertothemapasbeing-measurable,orasbeingBorelmeasurableifisunderstoodandthereisanydoubtaboutwhicheldtousefortherealline.Inthisbook,youmayassumethatanysigma-isitsBorelsigma-eld,unlessexplicitlyspeciedotherwise.Itcangetconfusingifyoumisinterpretwheretheunspeciedsigma-eldslive.Myadvicewouldbethatyouimagineapictureshowingthetwospacesinvolved,withanymissingeldlabelslledin. 2.2MeasurablefunctionsSometimesthefunctionscomerst,andthesigma-eldsarechosenspecitomakethosefunctionsmeasurable.beaclassoffunctionsonaset.Supposethetypicalintoaspaceequippedwithasigma-field.Thenthesigma-fieldgeneratedbyisdefinedas.Itisthesmallestforwhicheachforsomeclassofsubsetsofthenamap-measurableifandonlyifforevery.Youshouldprovethisassertionbycheckingthatisasigma-eld,andthenarguingfromthedenitionofageneratingclass.Inparticular,toestablish-measurabilityofamapintothereallineitisenoughtochecktheinverseimagesofintervalsoftheform,withrangingover.(Infact,wecouldrestricttoacountabledensesubsetofsuchasthesetofrationals:Howwouldyoubuildanintervalfromintervalswithrational?)Thatis,areal-valuedfunctionisBorel-measurableifforeachreal.Therearemanysimilarassertionsobtainedbyusingothergeneratingclassesfor.Someauthorsuseparticulargeneratingclassesforthedenitionofmeasurability,andthenderivefactsaboutinverseimagesofBorelsetsastheorems.Itwillbeconvenienttoconsidernotjustreal-valuedfunctionsonasetbutalsofunctionsfromintotheextendedrealline ].TheBorel isgeneratedbytheclassofopensets,or,moreexplicitly,byallsetsintogetherwiththetwosingletons.Itisaneasyexercisetoshowthat isgeneratedbytheclassofallsetsoftheform],forandbytheclassofallsetsoftheform[,for.WecouldevenrestricttoanycountabledensesubsetofLetasetbeequippedwithasigma-.Letbeasequenceof-measurablefunctionsfrom.DenefunctionsbytakingpointwisesupremaandinNoticethatmighttakethevalue,andmighttakethevalue,atsomepointsof.Wemayconsiderbothasmapsfrom .(Infact,thewholeargumentisunchangedifthefunctionsthemselvesarealsoallowedtotakenitevalues.)Thefunction -measurablebecauseforeachrealforeachxed,thesupremumoftherealnumbersisstrictlygreaterthanifandonlyifforatleastone.Exampleshowswhywehaveonlytocheckinverseimagesforsuchintervals.ThesamegeneratingclassisnotasconvenientforprovingmeasurabilityofItisnottruethataninmumofasequenceofrealnumbersisstrictlygreaterthanifandonlyifallofthenumbersarestrictlygreaterthan:thinkofthesequence,whoseinmumiszero.Insteadyoushouldargueviatheforeachreal Chapter2:AmodicumofmeasuretheoryFromExampleandtherepresentationslimsupandliminf,itfollowsthatthelimsuporliminfofasequenceofmeasurable(real-orextendedreal-valued)functionsisalsomeasurable.Inparticular,ifthelimitexistsitismeasurable.Measurabilityisalsopreservedbytheusualalgebraicoperationsdifferences,products,andsoonprovidedwetakecaretoavoidillegalpointwisecalculationssuchasor00.Thereareseveralwaystoestablishthesestabilityproperties.Oneofthemoredirectmethodsdependsonthefactthatacountabledensesubset,asillustratedbythefollowingargumentforsums.-measurablefunctions,withpointwisesum.(IexcludeinnitevaluesbecauseIdontwanttogetcaughtupwithinconclusivediscussionsofhowwemightproceedatpoints  ,or  .)Howcanweproveisalsoa-measurablefunction?Itistruethatanditistruethatthesetismeasurableforeach,butsigma-eldsarenotrequiredtohaveanyparticularstabilitypropertiesforuncountableunions.Insteadweshouldarguethatateachforwhichthereexistsarationalnumbersuchthat.Converselyifthereisanlyingstrictlybetween.Thusdenotesthecountablesetofrationalnumbers.Acountableunionofintersectionsofpairsofmeasurablesetsismeasurable.ThesumisameasurableAsalittleexerciseyoumighttrytoextendtheargumentfromthelastExampletothecasewhereareallowedtotakethevalue(butnotthevalueIfyouwantpracticeatplayingwithrationals,trytoprovemeasurabilityofproducts(becarefulwithinequalitiesifdividingbynegativenumbers)ortryProblemProblem,whichshowswhyadirectattackonthelimsuprequirescarefulhandlingofinequalitiesinthelimit.Therealsignicanceofmeasurabilitybecomesapparentwhenoneworksthroughtheconstructionofintegralswithrespecttomeasures,asinSection.Forthemomentitisimportantonlythatyouunderstandthatthefamilyofallmeasurablefunctionsisstableundermostofthefamiliaroperationsofanalysis.Theclass,ororjustforshort,consistsofall -measurablefunctionsfrom .Theclass,orforshort,consistsofthenonnegativefunctionsinIfyoudesiredexquisiteprecisionyoucouldwrite  ,toeliminateallambiguityaboutdomain,range,andsigma-Thecollectionisacone(stableundersumsandmultiplicationoffunctionsbypositiveconstants).Itisalsostableunderproducts,pointwiselimitsofsequences, 2.2Measurablefunctionsandsupremaorinmaofcountablecollectionsoffunctions.Itisnotavectorspace,becauseitisnotstableundersubtraction;butitdoeshavethepropertythatifbelongtotakesonlyrealvalues,thenthepositivepart,debytakingthepointwisemaximumofwith0,alsobelongsto.YoucouldadapttheargumentfromExampletoestablishthelastfact.Itprovesconvenienttoworkwithratherthanwiththewholeof,therebyeliminatingmanyproblemswith.Asyouwillsoonlearn,integralshavesomeconvenientpropertieswhenrestrictedtononnegativefunctions.Forourpurposes,oneofthemostimportantfactsaboutwillbethepossibilityofapproximationbysimplefunctionsthatisbymeasurablefunctionsoftheform,fornitecollectionsofrealnumbersandevents.Ifthearedisjoint,,forsome,andiszerootherwise.Ifthearenotdisjoint,thenonzerovaluestakenbyaresumsofvarioussubsetsofthe.Dontforget:thesymbolgetsinterpretedasanindicatorfunctionwhenwestartdoingalgebra.IwillwritefortheconeofallsimplefunctionsinForeachthesequence,definedbyhasthepropertyateveryThedenitionofinvolvesalgebra,soyoumustinterpretastheindicatorfunctionofthesetofallpointsforwhichProof.Ateach,countthenumberofnonzeroindicatorvalues.If,allsummandscontributea1,giving.If,forsomeinteger,thenexactlyofthesummandscontributea1,giving.(Checkthatthelastassertionmakessensewhenequals0.)Thatis,for0,thefunctionroundsdowntoanintegermultipleof2,fromwhichtheconvergenceandmonotoneincreasingpropertiesfollow. f0 Ifyoudonotndthemonotonicityassertionconvincing,youcouldargue,moreformally,that 2n14ni 12 f2i 2n11 2n144ni 1 f2i 2n1 f2i 1 whichreectstheeffectofdoublingthemaximumvalueandhalvingthestepsizewhengoingfromthenthtothe(n+1)stapproximation. Chapter2:AmodicumofmeasuretheoryAsanexerciseyoumightprovethattheproductoffunctionsinbelongsto,byexpressingtheproductasapointwiselimitofproductsofsimplefunctions.Noticehowtheconvention0 0isneededtoensurethecorrectlimitbehavioratpointswhereoneofthefactorsiszero.  Justasrepresentsasortoflimitingsumofvaluesweightedbysmalllengthsofintervalssignisalong,forsum,andtheisasortoflimitingincrementsocanthegeneralintegralbedenedasalimitofweightedsumsbutwithweightsprovidedbythemeasure.Theformalnitioninvolveslimitingoperationsthatdependontheassumedmeasurabilityofthefunction.Youcanskipthedetailsoftheconstruction(Section)bytakingthefollowingresultasanaxiomaticpropertyoftheintegral.Foreachmeasurethereisauniquelydeterminedfunctional,amapmap],havingthefollowingproperties:foreach,wherethefirstzerostandsforthezerofunction;(iii)fornonnegativerealnumbersandfunctions(iv)ifareineverywherethen(v)ifisasequenceinIwillreferto(iii)as,eventhoughisnotavectorspace.ItwillimplyalinearitypropertywhenisextendedtoavectorsubspaceofProperty(iv)isredundantbecauseitfollowsfrom(ii)andnonnegativity.Property(ii)isalsoredundant:putin(i);or,interpreting0as0,put0and0in(iii).Weneedtomakesurethebadcase,foralldoesnotslipthroughifwestartstrippingawayredundantrequirements.Noticethatthelimitfunctionin(v)automaticallybelongsto.ThelimitassertionitselfiscalledtheMonotoneConvergenceproperty.Itcorrespondsdirectlytocountableadditivityofthemeasure.Indeed,ifisacountablecollectionofdisjointsetsfromthenthefunctionspointwisetotheindicatorfunctionof,sothatMonotoneConvergenceandlinearityimplyYoushouldpondertheroleplayedbyinTheorem.Forexample,whatdoesmeanif0and?Theinterpretationdependsontheconventionthat0 Ingeneralyoushouldbesuspiciousofanyconventioninvolving.Paycarefulattentiontocaseswhereitoperates.Forexample,howwouldtheassertionsbeaffectedifweadoptedanewconvention,whereby0 6?WouldtheTheoremstillhold?Whereexactlywoulditfail?Ifeeluneasyifitisnotclearhowaconventionisdisposingofawkwardcases.Myadvice:bevery,very 2.3Integralscarefulwithanycalculationsinvolvinginnity.Subtleerrorsareeasytomisswhenconcealedwithinaconvention.ThereisacompaniontoTheoremthatshowswhyitislargelyamatteroftastewhetheronestartsfrommeasuresorintegralsasthemoreprimitivemeasuretheoreticconcept.beamapfromfrom]thatsatisfiesproperties(ii)through(v)ofTheorem.Thenthesetfunctiondefinedonthesigma-fieldby(i)isa(countablyadditive,nonnegative)measure,withthefunctionalthatitprovidesthelinkbetweenthemeasureandthefunctionalForagiven,letbethesequencedenedbytheLemma.ThenrstequalitybyMonotoneConvergence,thesecondbylinearity.Thevalueisuniquelydeterminedby,asasetfunctionon.Itisevenpossibletousetheequality,orsomethingverysimilar,asthebasisforadirectconstructionoftheintegral,fromwhichproperties(i)through(v)arethenderived,asyouwillseefromSectionInsummary:Thereisaone-to-onecorrespondence A A I   +
def f ned h defined
here betweenmeasuresonthesigma-andincreasinglinearfunctionalsonwiththeMonotoneConvergenceproperty.ToeachmeasurethereisauniquelydeterminedforwhichforeveryThefunctionalisusuallycalledanwithrespect,andisvariouslydenotedby.WiththedeFinettinotation,whereweidentifyasetwithitsindicatorfunction,theisjustanextensionoffromasmallerdomain(indicatorsofsetsin)toalargerdomain(allofAccordingly,weshouldhavenoqualmsaboutdenotingitbythesamesymbol.Iwillwritefortheintegral.Withthisnotation,assertion(i)ofTheoremforall.Youprobablycanttellthattheontheleft-handsideisanindicatorfunctionandtheisanintegral,butyoudontneedtobeabletotellthatispreciselywhat(i)asserts.Inelementaryalgebrawerelyonparentheses,orprecedence,tomakeourmeaningclear.Forexample,bothhavethesamemeaning,becausemultiplicationhashigherprecedencethanaddition.Withtraditionalnotation,andtheactlikeparentheses,enclosingtheintegrandandseparatingitfromfollowingterms.Withlinearfunctionalnotation,wesometimesneedexplicitparenthesestomakethemeaningunambiguous.Asawayofeliminatingsomeparentheses,Ioftenworkwiththeconventionthatintegrationhaslowerprecedencethanexponentiation,multiplication,anddivision,buthigherprecedencethanadditionorsubtraction.ThusIintendyoutoread6as6.Iwouldwriteifthe6werepartoftheintegrand. Chapter2:AmodicumofmeasuretheorySomeofthetraditionalnotationsalsoremoveambiguitywhenfunctionsofseveralvariablesappearintheintegrand.Forexample,invariableisheldxedwhiletheoperatesontherstargumentofthefunction.Whenasimilarambiguitymightarisewithlinearfunctionalnotation,Iwillappendasuperscript,asin,tomakeclearwhichvariableisinvolvedintheintegration.isanitemeasure(thatis,)andisafunction.ThenifandonlyifTheassertionisjustapointwiseinequalityindisguise.Byconsideringseparatelyvaluesforwhich1,for,youcanverifythepointwiseinequalitybetweenfunctions,Infact,thesumontheleft-handsidede,thelargestintegerandtheright-handsidedenotesthesmallestinteger.Fromtheleftmostinequality,MonotoneConvergenceAsimilarargumentgivesacompanionupperbound.Thusthepointwiseinequalityintegratesoutto,fromwhichtheassertedequivalencefollows.ExtensionoftheintegraltoalargerclassoffunctionsEveryfunctioncanbedecomposedintoadifferenceoftwofunctionsin,where.ToextendtoalinearfunctionalonweshoulddeThisdenitionworksifatleastoneofnite;otherwisewegetthedreaded.Ifboth(orequivalently,measurableand)thefunctionissaidtobe-integrable.Thelinearityproperty(iii)ofTheoremcarriesoverpartiallytoproblemsareexcluded,althoughitbecomestedioustohandlealltheawkwardcasesinvolving.Theconstantsneednolongerbenonnegative.Alsoifbothareintegrableandif,withobviousextensionstocertaincasesinvolvingThesetofallreal-valued,-integrablefunctionsinisdenotedby,orThesetisavectorspace(stableunderpointwiseadditionandmultipli-cationbyrealnumbers).Theintegralnesanincreasinglinearfunctionalon,inthesensethatpointwise.TheMonotoneConvergencepropertyimpliesotherpowerfullimitresultsforfunctionsin,asdescribedin.Byrestricting,weeliminateproblemswith 2.3IntegralsForeach,itsisdenedas.Strictlyspeaking,isonlyaseminorm,because0neednotimplythatisthezeroasyouwillseeinSection,itimpliesonlythat0.Itiscommonpracticetoignorethesmalldistinctionandrefertoasanormbeaconvex,real-valuedfunctionon.Thefunctionmeasurable(becauseisanintervalforeachreal),andforeachthereisaconstantsuchthatforall(AppendixC).beaprobabilitymeasure,andbeanintegrablerandomvariable..Fromtheinequality deducethat.Thusweshouldhavenoworriesintakingexpectations(thatis,integratingwithrespectto)todeducethat,aresultknownasJensen.Onewaytorememberthedirectionoftheinequalityistonotethatvar,whichcorrespondstothecaseIntegralswithrespecttoLebesguemeasureLebesguemeasurecorrespondstolength:length:ab] b aforeachinterval.Iwilloccasionallyreverttothetraditionalwaysofwritingsuchintegrals,tworryaboutconfusingtheLebesgueintegralwiththeRiemannintegraloverniteintervals.WhenevertheRiemanniswelldened,soistheLebesgue,andthetwosortsofintegralhavethesamevalue.TheLebesgueisamoregeneralconcept.Indeed,factsabouttheRiemannareoftenestablishedbyanappealtotheoremsabouttheLebesgue.Youdonothavetoabandonwhatyoualreadyknowaboutintegrationoverniteintervals.TheimproperRiemannintegral,,alsoagreeswiththeLebesgueintegralprovided.If,asinthecaseofthefunction,theimproperRiemannintegralmightexistasanitelimit,whiletheLebesgueintegraldoesnotexist.!"#       Toconstructtheintegralasafunctionalon,startingfromameasureonthesigma-,weuseapproximationfrombelowbymeansofsimpleFirstwemustde.Therepresentationofasimplefunctionasalinearcombinationofindicatorfunctionsisnotunique,buttheadditivitypropertiesofthemeasurewillletususeanyrepresentationtodenetheintegral.Forexample,if,then Chapter2:AmodicumofmeasuretheoryMoregenerally,ifhasanotherrepresentation,then.Proof?ThuswecanuniquelydeforasimplenetheincreasingfunctionalThatis,theintegraloffisasupremumofintegralsofnonnegativesimplefunctionslessthanFromtherepresentationofsimplefunctionsaslinearcombinationsofdisjointsetsin,itiseasytoshowthatforevery.Itisalsoeasytoshowthat0,andfornonnegativereal,andThelastinequality,whichisusuallyreferredtoasthesuperadditivityproperty,followsfromthefactthatif,andbotharesimple,thenOnlytheMonotoneConvergencepropertyandthecompaniontorequirerealwork.Hereyouwillseewhymeasurabilityisneeded.Proofofinequality.Letbeasimplefunction,andletbeasmallpositivenumber.Itisenoughtoconstructsimplefunctionssuchthat.Forthen,fromwhichthesubadditivityinequalityfollowsbytakingasupremumoversimplefunctionsthenlettingtendtozero.ForsimplicityofnotationIwillassumetobeverysimple:.Youcan f A u repeattheargumentforeachinarepresentationwithdisjointtogetthegeneralresult.Supposeforsomepositiveinteger.Write.DenesimplefunctionsThemeasurabilityof-measurabilityofallthesetsenteringintothedenitionsof.Forthe,noticethat1on,so.Finally,notethatthesimplefunctionswerechosensothatasdesired.ProofoftheMonotoneConvergenceproperty. f f n f .Suppose,withthesetsin0.Deneapproximatingsimplefunctions.Clearly.The 2.4Constructionofintegralsfrommeasuressimplefunctionisoneofthosethatentersintothesupremumde.ItfollowsthatOnthesetthefunctionsincreasemonotonelyto,whichis.Thesetsexpanduptothewholeof.Countableadditivityimpliesthatmeasuresofthosesetsincreaseto.ItfollowsthatlimsupTakeasupremumoversimplethenlettendtozerotocompletetheproof.$  edanintegralonasanincreasinglinearfunctionalwiththeMonotoneConvergenceproperty:if0Twodirectconsequencesofthislimitpropertyhaveimportantapplicationsthrough-outprobabilitytheory.TheFatousLemma,assertsaweakerlimitpropertyfornonnegativefunctionswhentheconvergenceandmonotonicityassumptionsaredropped.Thesecond,DominatedConvergence,dropsthemonotonicityandnonneg-ativitybutimposesanextradominationconditionontheconvergentsequenceIhaveslowlyrealizedovertheyearsthatmanysimpleprobabilisticresultscanbeestablishedbyDominatedConvergencearguments.TheDominatedConvergenceTheoremistheSwissArmyKnifeofprobabilitytheory.Itisimportantthatyouunderstandwhysomeconditionsareneededbeforewecaninterchangeintegration(whichisalimitingoperation)withanexplicitlimit,asin.Variationsonthefollowingexampleformthebasisformanycounterexamples.beLebesguemeasureonon1]andletbeasequenceofpositivenumbers.Thefunctionconvergestozero,pointwise,butitsintegralneednotconvergetozero.Forexample,gives;theintegralsdiverge.Andevengives6foreven3forTheintegralsoscillate.%&Foreverysequence(notnecessarilyconvergent),liminfliminfProof.Writeforliminf.Rememberwhataliminfmeans.De.Thenforeveryandthesequenceincreasesmonotonelytothefunction.ByMonotoneConvergence,.Bytheincreasingproperty,foreach,andhencelimliminf