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1、AgentleintroductiontoMeasureTheoryGauravChandaliaDepartmentofComputerScienceandEngineeringSUNY-UniversityatBuffalo,Buffalo,NYgsc4@buffalo.eduMarch12,2007AbstractThisnoteintroducesthebasicconceptsandde?nitionsofmeasuretheoryrelevanttoprobabilitytheory.Itismeanttobeasimpli?edtuto
2、rialonmeasuretheory.1IntroductionTheMethodofExhaustionisatechniqueto?ndouttheareaofashape(forexample,circle)byapproximating(lowerbound)itbyinscribingitusingapolygon.Theideaisthatasthenumberofsidesofthepolygontendstoin?nitytheareaofthepolygonwillgetarbitrarilyclosetotheareaofthe
3、circle.ThismethodwasusedbyArchimedesto?ndoutthevalueof.Thus,byexhaustingthevaluesoftheareaofthepolygon,theareaofthecirclecanbeobtained.Infact,theRiemannintegralisanapplicationofthemethodofexhaustion.MeasuretheoryisconcernedwithgeneralizingthenotionsofareaonarbitrarysetsofEucli
4、deanspacesandnotionsoflengthofsubsetsofR.Essentially,itisacommongroundforanalysisofrealfunctionsandsettheory.Forthenotionofarea,therearecertainpropertieslikenon-negativityandadditivitythatholdtrue.Similarly,inmeasuretheorythereisastrongerassumptionofcountableadditivity.Measuret
5、heoryinvolvesstudyof-algebras(abstractmathematicalstructures),measures,measurablefunctionsandintegrals.Integrationinthecontextofmeasuretheoryinvolvesanalogoussumsandisbasedonfunctionsconstantonsetsofsome-algebraandnotonfunctionsconstantonintervals(asisdoneinthetraditionalmann
6、er).Iwillclearlyde?neaspacesothatthereissomethingtomeasureinSections2.3and2.4;thenIwillintroducethenotionofameasureonthisspaceinSection3andshowhowmeasuresareusedtode?neanintegralinSection3.2.2BasicsBeforelookingatmeasuretheorymoreformally,Iwillreviewsomeofthebasicconceptsinclud
7、ingsomeideasfromabstractalgebrathatde?netheunderlyingmathematicalstructuresonwhichmeasuresarede?ned.12.1LimitofasequenceAsequencex1;x2;:::xnissaidtoconvergetothepointorhaslimitxifforevery>0thereisanaturalnumberNsuchthattheneighborhoodofO(x;)(acirclewithcenterxandradius)hasa
8、llthepointsxnwithn>N.Intuitively,itmeansthattheelemen
