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。

Preface

Chapter 1 Constructions and extensions of measures

 1.1 Measurement of length: introductory remarks

 1.2 Algebras and σ-algebras

 1.3 Additivity and countable additivity of measures

 1.4 Compact classes and countable additivity

 1.5 Outer measure and the Lebesgue extension of measures

 1.6 Infinite and a-finite measures

 1.7 Lebesgue measure

 1.8 Lebesgue-Stieltjes measures

 1.9 Monotone and σ-additive classes of sets

 1.10 Souslin sets and the A-operation

 1.11 Caratheodory outer measures

 1.12 Supplements and exercises

Set 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 integral

 2.1 Measurable functions

 2.2 Convergence in measure and almost everywhere

 2.3 The integral for simple functions

 2.4 The general definition of the Lebesgue integral

 2.5 Basic properties of the integral

 2.6 Integration with respect to infinite measures

 2.7 The completeness of the space L1

 2.8 Convergence theorems

 2.9 Criteria of integrability

 2.10 Connections with the Riemann integral

 2.11 The HSlder and Minkowski inequalities

 2.12 Supplements and exercises

The a-algebra generated by a class of functions (143) Borel mappings on IRn (145) The functional monotone class theorem (146) Baire classes of functions (148) Mean value theorems (150) The Lebesgue-Stieltjes integral (152) Integral inequalities (153) Exercises (156)

Chapter 3 Operations on measures and functions

 3.1 Decomposition of signed measures

 3.2 The Radon-Nikodym theorem

 3.3 Products of measure spaces

 3.4 Fubini's theorem

 3.5 Infinite products of measures

 3.6 Images of measures under mappings

 3.7 Change of variables in IRn

 3.8 The Fourier transform

 3.9 Convolution

 3.10 Supplements and exercises

On Fubini's theorem and products of σ-algebras (209) Steiner's symmetrization (212) Hausdorff measures (215) Decompositions of set 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 measures

 4.1 The spaces Lp

 4.2 Approximations in Lp

 4.3 The Hilbert space L2

 4.4 Duality of the spaces Lp

 4.5 Uniform integrability

 4.6 Convergence of measures

 4.7 Supplements and exercises

The spaces Lp and the space of measures as structures (277) The weak topology in LP(280) Uniform convexity of LP(283) Uniform integrability and 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 and ellinger'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 line

 5.2 Functions of bounded variation

 5.3 Absolutely continuous functions

 5.4 The Newton-Leibniz formula

 5.5 Covering theorems

 5.6 The maximal function

 5.7 The Henstock-Kurzweil integral

 5.8 Supplements and exercises

Covering theorems (361) Density points and Lebesgue points (366).Differentiation of measures on IRn (367) The approximate continuity (369) Derivates and the approximate differentiability (370).The class BMO (373) Weighted inequalities (374) Measures with the doubling property (375) Sobolev derivatives (376) The area and coarea formulas and change of variables (379) Surface measures (383).The Calder6n-Zygmund decomposition (385) Exercises (386)

Bibliographical and Historical Comments

References

Author Index

Subject Index