测度论
出版社：高等教育出版社
年代：2010
定价：39.5
书籍简介:
本书是天元基金影印数学丛书之一，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。全面丰富地阐述了现代侧度论的基本理论，不要求读者从头到尾系统阅读，特别，补充内容几乎与各章节彼此无关，主要针对那些对测度论有兴趣深入了解的研究生、侧度论和积分理论课程的教师，以及数学各领域的研究人员。全书共分两卷。第一卷包括通常侧度论教材中的内容：侧度的构造与延拓，Lebesgue积分的定义及基本性质，Jordan分解，Radon-Nikodym定理，Fourier变换，卷积，L（p）空间，侧度空间，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习题（篇幅占全书的一半），其中包含许多有用的知识，例如：Whitney分解，Lebesgue-Stieltjes积分，Hausdorff侧度，Brunn-Minkowski不等式，Hellinger积分与Hellinger距离，BMO类，Calderon-Zygmund分解等。另外，书的最后有详尽的参考文献及历史注记。北京大学陈天权教授评价此书“这是一本很好的研究生教材和教学参考书”。本书可作为高等学校数学类专业本科高年级和研究生的教材或预习课程的材料，也可供相关科学工作者参考。
书籍目录:
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
内容摘要:
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。
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