欧氏空间上的勒贝格积分

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 Fubinis 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 Stirlings 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 = i

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 Parsevals 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 Lebesgues 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

《欧氏空间上的勒贝格积分(修订版)(英文版)》简明、详细地介绍勒贝格测度和Rn上的积分。《欧氏空间上的勒贝格积分(英文版)》的基本目的有四个，介绍勒贝格积分；从一开始引入n维空间；彻底介绍傅里叶积分；深入讲述实分析。贯穿全书的大量练习可以增强读者对知识的理解。目次：Rn导论；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；卷积；Rn+上的傅里叶变换；单变量傅里叶积分；微分；R上函数的微分。
　　读者对象：《欧氏空间上的勒贝格积分(修订版)(英文版)》适用于数学专业的学生、老师和相关的科研人员。