测度论（第一卷 影印版）

出版时间：2010-7  出版社：高等教育出版社  作者：Vladimir Bogachev  页数：500  
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前言

为了更好地借鉴国外数学教育与研究的成功经验，促进我国数学教育与研究事业的发展，提高高等学校数学教育教学质量，本着“为我国热爱数学的青年创造一个较好的学习数学的环境”这一宗旨，天元基金赞助出版“天元基金影印数学丛书”。该丛书主要包含国外反映近代数学发展的纯数学与应用数学方面的优秀书籍，天元基金邀请国内各个方向的知名数学家参与选题的工作，经专家遴选、推荐，由高等教育出版社影印出版。为了提高我国数学研究生教学的水平，暂把选书的目标确定在研究生教材上。当然，有的书也可作为高年级本科生教材或参考书，有的书则介于研究生教材与专著之间。欢迎各方专家、读者对本丛书的选题、印刷、销售等工作提出批评和建议。

内容概要

　　《测度论（第1卷）（影印版）》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，L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度，Brunn-Minkowski不等式，Hellinger积分与Hellinger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。

作者简介

作者：（俄罗斯）博根切维（V.l.Bogachev）

书籍目录

PrefaceChapter 1.Constructions and extensions of measures1.1.Measurement of length: introductory remarks1.2.Algebras and c-algebras1.3.Additivity and countable additivity of measures1.4.Compact classes and countable additivity1.5.Outer measure and the Lebesgue extension of measures1.6.Infinite and a-finite measures1.7. Lebesgue measure1.8.Lebesgue-Stieltjes measures1.9.Monotone and a-additive classes of sets1.10.Souslin sets and the A-operation1.11.Carath~odory outer measures1.12. Supplements and exercisesSet operations (48).Compact classes (50).Metric Boolean algebra (53).Measurable envelope, measurable kernel and inner measure (56).Extensions of measures (58).Some interesting sets (61).Additive, but not countably additive measures (67).Abstract inner measures (70).Measures on lattices of sets (75).Set-theoretic problems in measure theory (77).Invariant extensions of Lebesgue measure (80).Whitney's decomposition (82).Exercises (83).Chapter 2.The Lebesgue integral2.1.Measurable functions2.2.Convergence in measure and almost everywhere2.3. The integral for simple functions2.4.The general definition of the Lebesgue integral2.5.Basic properties of the integral2.6.Integration with respect to infinite measures2.7.The completeness of the space L12.8.Convergence theorems2.9.Criteria of integrability2.10.Connections with the Riemann integral2.11.The HSlder and Minkowski inequalities2.12.Supplements and exercisesThe a-Mgebra generated by a class of functions (143).The functional monotone class theorem (146).Balre classes of functions (148).Mean value theorems (150).The LebesgueStieltjes integral (152).Integral inequalities (153).Exercises (156).Chapter 3. Operations on measures and functions3.1.Decomposition of signed measures3.2.The Radon-Nikodym theorem3.3.Products of measure spaces3.4.F~abini's theorem3.5.Infinite products of measures3.6. Images of measures under mappings3.7.Change of variables in 3.8.The Fourier transform3.9.Convolution3.10. Supplements and exercisesOn Fubini's theorem and products of a-algebras (209).Steiner's symmetrization (212).Hausdorff measures (215).Decompositions ofset functions (218).Properties of positive definite functions (220).The Brunn-Minkowski inequality and its generalizations (222).Mixed volumes (226).The Radon transform (227).Exercises (228).Chapter 4.The spaces Lp and spaces of measures4.1.The spaces Lp4.2.Approximations in Lp4.3.The Hilbert space L24.4.Duality of the spaces Lp4.5.Uniform integrability 4.6.Convergence of measures4.7.Supplements and exercisesThe spaces Lp and the space of measures as structures (277).The weaktopology in LP(280).Uniform convexity of LP(283).Uniform integrabilityand weak compactness in L1 (285).The topology of setwise convergence of measures (291).Norm compactness and approximations in Lp (294).Certain conditions of convergence in LP (298). Hellinger's integral andHellinger's distance (299).Additive set functions (302).Exercises (303).Chapter 5. Connections between the integral and derivative.5.1.Differentiability of functions on the real line5.2.Functions of bounded variation5.3.Absolutely continuous functions5.4.The Newton-Leibniz formula5.5.Covering theorems5.6.The maximal function5.7.The Henstock-Kurzweil integral5.8.Supplements and exercisesCovering theorems (361).Density points and Lebesgue points (366).Differentiation of measures on ]Rn (367). The approximatecontinuity (369). Derivates and the approximate differentiability (370).The class BMO (373). Weighted inequalities (374). Measures withthe doubling property (375). Sobolev derivatives (376). The area and coarea formulas and change of variables (379). Surface measures (383).The CalderSn-Zygmund decomposition (385).Exercises (386).Bibliographical and Historical CommentsReferencesAuthor IndexSubject Index

章节摘录

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《测度论(第1卷)(影印版)》：天元基金影印数学丛书。

用户评论 (总计7条)

•   对数学研究很有帮助。可作为工具书。
•   这套书我拿来当字典，当课本的话不行，东西多，初学者没必要。
•   测度论中写得全的书了,一些习题有提示
•   这学期学了测度论，课本选用的是严加安的测度论讲义，和这本没法比。这本书条理清晰，内容详实，尤其是每章的supplements and exercises，补充了很多一般书上没有的，习题也非常多，质量也很高。英文版的，坚持看下来，会收获颇丰。
•   这么厚的测度实在少见
•   测度论(第1卷)(影印版) 不错的一本书
•   测度论很好的参考书，感觉比传说中的halmos那本还要强一些