Real Analysis for Graduate Student 2nd.pdf

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1、RealanalysisforgraduatestudentsSecondeditionRichardF.BassiicCopyright2013RichardF.BassAllrightsreserved.ISBN-13:978-1481869140ISBN-10:1481869140Firsteditionpublished2011.iiiTothememoryofmyparentsJayandBettyBassivContentsPrefacexiii1Preliminaries11.1Notationandterminology.........

2、....11.2Someundergraduatemathematics.........22Familiesofsets72.1Algebrasand-algebras..............72.2Themonotoneclasstheorem............102.3Exercises.......................113Measures133.1Denitionsandexamples..............133.2Exercises.......................164Constructiono

3、fmeasures194.1Outermeasures...................204.2Lebesgue-Stieltjesmeasures............244.3Examplesandrelatedresults............274.4Nonmeasurablesets.................304.5TheCaratheodoryextensiontheorem.......314.6Exercises.......................34vviCONTENTS5Measurablefunct

4、ions375.1Measurability....................375.2Approximationoffunctions............415.3Lusin'stheorem...................425.4Exercises.......................446TheLebesgueintegral476.1Denitions......................476.2Exercises.......................497Limittheorems517.1Monot

5、oneconvergencetheorem..........517.2Linearityoftheintegral...............527.3Fatou'slemma....................547.4Dominatedconvergencetheorem..........557.5Exercises.......................568PropertiesofLebesgueintegrals638.1Criteriaforafunctiontobezeroa.e........638.2Anapproxim

6、ationresult..............658.3Exercises.......................669Riemannintegrals699.1ComparisonwiththeLebesgueintegral......699.2Exercises.......................7110Typesofconvergence7510.1Denitionsandexamples..............7510.2Exercises.......................78CONTENTSvii11Pr

7、oductmeasures8111.1Product-algebras.................8111.2TheFubinitheorem.................8511.3Examples.......................8711.4Exercises.......................8812Signedmeasures9312.1Positiveandnegativesets.............9312.2Hahndecompositiontheorem...........9512.3Jordan

8、decompositiontheorem...........9712.4Exe