测度论

introductoryremarks

1.2Algebrasandσ-algebras

1.3Additivityandcountableadditivityofmeasures

1.4Compactclassesandcountableadditivity

1.5OutermeasureandtheLebesgueextensionofmeasures

1.6Infiniteanda-finitemeasures

1.7Lebesguemeasure

1.8Lebesgue-Stieltjesmeasures

1.9Monotoneandσ-additiveclassesofsets

1.10SouslinsetsandtheA-operation

1.11Caratheodoryoutermeasures

1.12Supplementsandexercises

Setoperations(48)Compactclasses(50)MetricBooleanalgebra(53).Measurableenvelope,measurablekernelandinnermeasure(56).Extensionsofmeasures(58)Someinterestingsets(61)Additive,butnotcountablyadditivemeasures(67)Abstractinnermeasures(70).Measuresonlatticesofsets(75)Set-theoreticproblemsinmeasuretheory(77)InvariantextensionsofLebesguemeasure(80)Whitney'sdecomposition(82)Exercises(83)

Chapter2TheLebesgueintegral

2.1Measurablefunctions

2.2Convergenceinmeasureandalmosteverywhere

2.3Theintegralforsimplefunctions

2.4ThegeneraldefinitionoftheLebesgueintegral

2.5Basicpropertiesoftheintegral

2.6Integrationwithrespecttoinfinitemeasures

2.7ThecompletenessofthespaceL1

2.8Convergencetheorems

2.9Criteriaofintegrability

2.10ConnectionswiththeRiemannintegral

2.11TheHSlderandMinkowskiinequalities

2.12Supplementsandexercises

Thea-algebrageneratedbyaclassoffunctions(143)BorelmappingsonIRn(145)Thefunctionalmonotoneclasstheorem(146)Baireclassesoffunctions(148)Meanvaluetheorems(150)TheLebesgue-Stieltjesintegral(152)Integralinequalities(153)Exercises(156)

Chapter3Operationsonmeasuresandfunctions

3.1Decompositionofsignedmeasures

3.2TheRadon-Nikodymtheorem

3.3Productsofmeasurespaces

3.4Fubini'stheorem

3.5Infiniteproductsofmeasures

3.6Imagesofmeasuresundermappings

3.7ChangeofvariablesinIRn

3.8TheFouriertransform

3.9Convolution

3.10Supplementsandexercises

OnFubini'stheoremandproductsofσ-algebras(209)Steiner'ssymmetrization(212)Hausdorffmeasures(215)Decompositionsofsetfunctions(218)Propertiesofpositivedefinitefunctions(220).TheBrunn-Minkowskiinequalityanditsgeneralizations(222).Mixedvolumes(226)TheRadontransform(227)Exercises(228)

Chapter4ThespacesLpandspacesofmeasures

4.1ThespacesLp

4.2ApproximationsinLp

4.3TheHilbertspaceL2

4.4DualityofthespacesLp

4.5Uniformintegrability

4.6Convergenceofmeasures

4.7Supplementsandexercises

ThespacesLpandthespaceofmeasuresasstructures(277)TheweaktopologyinLP(280)UniformconvexityofLP(283)UniformintegrabilityandweakcompactnessinL1(285)Thetopologyofsetwiseconvergenceofmeasures(291)NormcompactnessandapproximationsinLp(294).CertainconditionsofconvergenceinLp(298)Hellinger'sintegralandellinger'sdistance(299)Additivesetfunctions(302)Exercises(303)

Chapter5Connectionsbetweentheintegralandderivative

5.1Differentiabilityoffunctionsontherealline

5.2Functionsofboundedvariation

5.3Absolutelycontinuousfunctions

5.4TheNewton-Leibnizformula

5.5Coveringtheorems

5.6Themaximalfunction

5.7TheHenstock-Kurzweilintegral

5.8Supplementsandexercises

Coveringtheorems(361)DensitypointsandLebesguepoints(366).DifferentiationofmeasuresonIRn(367)Theapproximatecontinuity(369)Derivatesandtheapproximatedifferentiability(370).TheclassBMO(373)Weightedinequalities(374)Measureswiththedoublingproperty(375)Sobolevderivatives(376)Theareaandcoareaformulasandchangeofvariables(379)Surfacemeasures(383).TheCalder6n-Zygmunddecomposition(385)Exercises(386)

BibliographicalandHistoricalComments

References

AuthorIndex

SubjectIndex

　　本书是作者在莫斯科国立大学数学力学系的讲稿基础上编写而成的：第一卷包括了通常测度论教材中的内容：测度的构造与延拓，Lebesgue积分的定义及基本性质，Jordan分解，Radon-Nikodym定理，Fourier变换，卷积，Lp空间，测度空间，Newton-Leibniz公式，极大函数，Henstock-Kurzweil；积分等。每章最后都附有非常丰富的补充与习题，其中包含许多有用的知识，例如：Whitney分解，Lebesgue-Stieltjes积分，Hausdorff测度，Brunn-Minkowski不等式，Hellinger积分与Heltinger距离，BMO类，Calderon-Zygmund分解等。书的最后有详尽的参考文献及历史注记。这是一本很好的研究生教材和教学参考书。