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1、10APPUNTILECTURENOTESLuigiAmbrosio,GiuseppeDaPratoandAndreaMennucciScuolaNormaleSuperiorePiazzadeiCavalieri,756126Pisa,ItalyIntroductiontoMeasureTheoryandIntegrationLuigiAmbrosio,GiuseppeDaPratoandAndreaMennucciIntroductiontoMeasureTheoryandIntegrationc2011ScuolaNormaleSuperiore
2、PisaISBN:978-88-7642-385-7e-ISBN:978-88-7642-386-4ContentsPrefaceixIntroductionxi1Measurespaces11.1Notationandpreliminaries................11.2Rings,algebrasandσ–algebras..............21.3Additiveandσ–additivefunctions............41.4Measurablespacesandmeasurespaces..........71.5
3、Thebasicextensiontheorem...............81.5.1Dynkinsystems..................91.5.2Theoutermeasure................111.6TheLebesguemeasureinR...............141.7Innerandouterregularityofmeasuresonmetricspaces.182Integration232.1Inverseimageofafunction................232.2Measurable
4、andBorelfunctions.............242.3Partitionsandsimplefunctions..............252.4IntegralofanonnegativeE–measurablefunction....272.4.1Integralofsimplefunctions...........272.4.2Therepartitionfunction.............282.4.3Thearchimedeanintegral.............312.4.4Integralofanonnega
5、tivemeasurablefunction..322.5Integraloffunctionswithavariablesign.........352.6Convergenceofintegrals.................362.6.1UniformintegrabilityandVitaliconvergencetheorem......................382.7AcharacterizationofRiemannintegrablefunctions...39viLuigiAmbrosio,GiuseppeDaPrato
6、andAndreaMennucci3Spacesofintegrablefunctions453.1SpacesLp(X,E,μ)andLp(X,E,μ)..........453.2TheLpnorm.......................473.2.1HolderandMinkowskiinequalities.......¨483.3ConvergenceinLp(X,E,μ)andcompleteness.....493.4ThespaceL∞(X,E,μ)..................523.5DensesubsetsofLp(X,
7、E,μ)..............564Hilbertspaces614.1Scalarproducts,pre-HilbertandHilbertspaces.....614.2Theprojectiontheorem..................634.3Linearcontinuousfunctionals...............664.4Besselinequality,Parsevalidentityandorthonormalsystems..........................674.5Hilbertspaceso
8、nC....................705Fourierseries73
