Real analysis for graduate.pdf

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1、Realanalysisforgraduatestudents:measureandintegrationtheoryRichardF.BassiicCopyright2011RichardF.BassAllrightsreserved.ISBN-13:978-1466391574ISBN-10:146639157XiiiTothememoryofmyparentsJayandBettyBassivContentsPrefaceix1Preliminaries11.1Notationandterminology.....

2、..........11.2Someundergraduatemathematics..........22Familiesofsets72.1Algebrasand-algebras................72.2Themonotoneclasstheorem.............102.3Exercises........................113Measures133.1Denitionsandexamples................133.2Exercises.........

3、...............164Constructionofmeasures194.1Outermeasures.....................204.2Lebesgue-Stieltjesmeasures..............244.3Examplesandrelatedresults.............274.4Nonmeasurablesets..................304.5TheCaratheodoryextensiontheorem.........314.6Exerc

4、ises........................34vviCONTENTS5Measurablefunctions375.1Measurability......................375.2Approximationoffunctions..............415.3Lusin'stheorem.....................415.4Exercises........................446TheLebesgueintegral476.1Denitions....

5、....................476.2Exercises........................497Limittheorems517.1Monotoneconvergencetheorem............517.2Linearityoftheintegral................527.3Fatou'slemma.....................547.4Dominatedconvergencetheorem...........557.5Exercises........

6、................568PropertiesofLebesgueintegrals638.1Criteriaforafunctiontobezeroa.e..........638.2Anapproximationresult................658.3Exercises........................669Riemannintegrals699.1ComparisonwiththeLebesgueintegral........699.2Exercises..........

7、..............7110Typesofconvergence7510.1Denitionsandexamples................7510.2Exercises........................7811Productmeasures81CONTENTSvii11.1Product-algebras...................8111.2TheFubinitheorem..................8511.3Examples...................

8、.....8711.4Exercises........................8812Signedmeasures9312.1Positiveandnegativesets...............9312.2Hahndecompositiontheorem.............9512.3Jord