A gentle introduction to Measure Theory .pdf

《A gentle introduction to Measure Theory .pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、AgentleintroductiontoMeasureTheoryGauravChandaliaDepartmentofComputerScienceandEngineeringSUNY-UniversityatBuffalo,Buffalo,NYgsc4@buffalo.eduMarch12,2007AbstractThisnoteintroducesthebasicconceptsanddefinitionsofmeasuretheoryrelevanttoprobabilitytheory.Itismeanttobeasimplifiedtuto

2、rialonmeasuretheory.1IntroductionTheMethodofExhaustionisatechniquetofindouttheareaofashape(forexample,circle)byapproximating(lowerbound)itbyinscribingitusingapolygon.Theideaisthatasthenumberofsidesofthepolygontendstoinfinitytheareaofthepolygonwillgetarbitrarilyclosetotheareaofthe

3、circle.ThismethodwasusedbyArchimedestofindoutthevalueof.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).IwillclearlydefineaspacesothatthereissomethingtomeasureinSections2.3and2.4;thenIwillintroducethenotionofameasureonthisspaceinSection3andshowhowmeasuresareusedtodefineanintegralinSection3.2.2BasicsBeforelookingatmeasuretheorymoreformally,Iwillreviewsomeofthebasicconceptsinclud

7、ingsomeideasfromabstractalgebrathatdefinetheunderlyingmathematicalstructuresonwhichmeasuresaredefined.12.1LimitofasequenceAsequencex1;x2;:::xnissaidtoconvergetothepointorhaslimitxifforevery>0thereisanaturalnumberNsuchthattheneighborhoodofO(x;)(acirclewithcenterxandradius)hasa

8、llthepointsxnwithn>N.Intuitively,itmeansthattheelemen