Royden_Real_Analysis_Solutions .pdf

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1、RealAnalysisbyH.L.RoydenContents1SetTheory11.1Introduction............................................11.2Functions.............................................11.3Unions,intersectionsandcomplements.............................11.4Algebrasofsets.................................

2、.........21.5Theaxiomofchoiceandinnitedirectproducts.......................21.6Countablesets..........................................31.7Relationsandequivalences....................................31.8Partialorderingsandthemaximalprinciple..........................31.9Wellord

3、eringandthecountableordinals............................32TheRealNumberSystem52.1Axiomsfortherealnumbers...................................52.2ThenaturalandrationalnumbersassubsetsofR.......................52.3Theextendedrealnumbers...................................52.4Sequenc

4、esofrealnumbers....................................52.5Openandclosedsetsofrealnumbers..............................72.6Continuousfunctions.......................................92.7Borelsets.............................................133LebesgueMeasure133.1Introduction........

5、....................................133.2Outermeasure..........................................143.3MeasurablesetsandLebesguemeasure.............................143.4Anonmeasurableset.......................................153.5Measurablefunctions................................

6、.......153.6Littlewood'sthreeprinciples...................................174TheLebesgueIntegral184.1TheRiemannintegral......................................184.2TheLebesgueintegralofaboundedfunctionoverasetofnitemeasure..........184.3Theintegralofanonnegativefunction.........

7、....................194.4ThegeneralLebesgueintegral..................................194.5Convergenceinmeasure.....................................215DierentiationandIntegration225.1Dierentiationofmonotonefunctions..............................225.2Functionsofboundedvariation

8、.................................235.3Dierentiationof