Bass - Real Analysis.pdf

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1、Realanalysisforgraduatestudents:measureandintegrationtheoryRichardF.BassiicCopyright2011iiiTothememoryofmyparentsJayandBettyBassivContentsPrefacexi1Preliminaries11.1Notationandterminology....................11.2Someundergraduatemathematics................22Familiesofsets72

2、.1Algebrasandσ-algebras.....................72.2Themonotoneclasstheorem...................102.3Exercises..............................113Measures133.1Definitionsandexamples.....................133.2Exercises..............................154Constructionofmeasures174.1Outerme

3、asures..........................184.2Lebesgue-Stieltjesmeasures...................224.3Examplesandrelatedresults...................254.4Nonmeasurablesets........................284.5TheCarath´eodoryextensiontheorem..............29vviCONTENTS4.6Exercises....................

4、..........315Measurablefunctions355.1Measurability...........................355.2Approximationoffunctions...................395.3Lusin’stheorem..........................395.4Exercises..............................416TheLebesgueintegral436.1Definitions......................

5、.......436.2Exercises..............................457Limittheorems477.1Monotoneconvergencetheorem.................477.2Linearityoftheintegral.....................487.3Fatou’slemma...........................517.4Dominatedconvergencetheorem.................517.5Exercises...

6、...........................528PropertiesoftheLebesgueintegral598.1Criteriaforafunctiontobezeroa.e...............598.2Anapproximationresult.....................618.3Exercises..............................629Riemannintegrals659.1ComparisonwiththeLebesgueintegral.............

7、659.2Exercises..............................6710Typesofconvergence7110.1Definitionsandexamples.....................71CONTENTSvii10.2Exercises..............................7411Productmeasures7711.1Productσ-algebras........................7711.2TheFubinitheorem...............

8、.........8111.3Examples.............................8211.4Exercises......................