本书简明、详细地介绍勒贝格测度和Rn上的积分。全书共分十六章，主要内容包括Rn勒贝格测度；勒贝格积分的不变性；一些有趣的集合；集合代数和可测函数；积分；Rn勒贝格积分；Rn的Fubini定理；Gamma函数；Lp空间；抽象测度的乘积；单变量傅里叶积分等。

本书可供数学专业的学生、老师和相关的科研人员参考。

Preface

Bibliography

Acknowledgments

1 Introduction to Rn

 A Sets

 B Countable Sets

 C Topology

 D Compact Sets

 E Continuity

 F The Distance Function

2 Lebesgue Measure on Rn

 A Construction

 B Properties of Lebesgue Measure

 C Appendix: Proof of P1 and P2

3 Invariance of Lebesgue Measure

 A Some Linear Algebra

 B Translation and Dilation

 C Orthogonal Matrices

 D The General Matrix

4 Some Interesting Sets

 A A Nonmeasurable Set

 B A Bevy of Cantor Sets

 C The Lebesgue Function

 D Appendix: The Modulus of Continuity of the Lebesgue Functions

5 Algebras of Sets and Measurable Functions

 A Algebras and a-Algebras

 B Borel Sets

 C A Measurable Set which Is Not a Borel Set

 D Measurable Functions

 E Simple Functions

6 Integration

 A Nonnegative Functions

 B General Measurable Functions

 C Almost Everywhere

 D Integration Over Subsets of Rn

 E Generalization: Measure Spaces

 F Some Calculations

 G Miscellany

7 Lebesgue Integral on Rn

 A Riemann Integral

 B Linear Change of Variables

 C Approximation of Functions in L1

 D Continuity of Translation in L1

8 Fubini's Theorem for Rn

9 The Gamma Function

 A Definition and Simple Properties

 B Generalization

 C The Measure of Balls

 D Further Properties of the Gamma Function

 E Stirling's Formula

 F The Gamma Function on R

10 LP Spaces

 A Definition and Basic Inequalities

 B Metric Spaces and Normed Spaces

 C Completeness of Lp

 D The Case p=∞

 E Relations between Lp Spaces

 F Approximation by C∞c (Rn)

 G Miscellaneous Problems

 H The Case 0＜p＜1

11 Products of Abstract Measures

 A Products of 5-Algebras

 B Monotone Classes

 C Construction of the Product Measure

 D The Fubini Theorem

 E The Generalized Minkowski Inequality

12 Convolutions

 A Formal Properties

 B Basic Inequalities

 C Approximate Identities

13 Fourier Transform on Rn

 A Fourier Transform of Functions in L1 (Rn)

 B The Inversion Theorem

 C The Schwartz Class

 D The Fourier-Plancherel Transform

 E Hilbert Space

 F Formal Application to Differential Equations

 G Bessel Functions

 H Special Results for n =1

 I Hermite Polynomials

14 Fourier Series in One Variable

 A Periodic Functions

 B Trigonometric Series

 C Fourier Coefficients

 D Convergence of Fourier Series

 E Summability of Fourier Series

 F A Counterexample

 G Parseval's Identity

 H Poisson Summation Formula

 I A Special Class of Sine Series

15 Differentiation

 A The Vitali Covering Theorem

 B The Hardy-Littlewood Maximal Function

 C Lebesgue's Differentiation Theorem

 D The Lebesgue Set of a Function

 E Points of Density

 F Applications

 G The Vitali Covering Theorem (Again)

 H The Besicovitch Covering Theorem

 I The Lebesgue Set of Order p

 J Change of Variables

 K Noninvertible Mappings

16 Differentiation for Functions on R

 A Monotone Functions

 B Jump Functions

 C Another Theorem of Fubini

 D Bounded Variation

 E Absolute Continuity

 F Further Discussion of Absolute Continuity

 G Arc Length

 H Nowhere Differentiable Functions

 I Convex Functions

Index

Symbol Index