欧氏空间上的勒贝格积分（修订版）（英文版） [Lebesgue Integration on Euclidean Space Revised Edition] 下载 mobi epub pdf 电子书 2024

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　　《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。

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目录

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

前言/序言

　　"Though of real knowledge there be little， yet of books there are plenty" -Herman Melville， Moby Dick， Chapter XXXI.
　　The treatment of integration developed by the French mathematician Henri Lebesgue （1875-1944） almost a century ago has proved to be indispensable in many areas of mathematics. Lebesgues theory is of such extreme importance because on the one hand it has rendered previous theories of integration virtually obsolete， and on the other hand it has not been replaced with a significantly different， better theory. Most subsequent important investigations of integration theory have extended or illuminated Lebesgues work.
　　In fact， as is so often the case in a new field of mathematics， many of the best consequences were given by the originator. For example，Lebesgues dominated convergence theorem， Lebesgues increasing convergence theorem， the theory of the Lebesgue function of the Cantor ternary set， and Lebesgues theory of differentiation of indefinite integrals.
　　Naturally， many splendid textbooks have been produced in this area.I shall list some of these below. They axe quite varied in their approach to the subject. My aims in the present book are as follows.
　　1. To present a slow introduction to Lebesgue integration Most books nowadays take the opposite tack. I have no argument with their approach， except that I feel that many students who see only a very rapid approach tend to lack strong intuition about measure and integration. That is why I have made Chapter 2， "Lebesgue measure on Rn，"so lengthy and have restricted it to Euclidean space， and why I have （somewhat inconveniently） placed Chapter 3， "Invaxiance of Lebesgue measure，" before Pubinis theorem. In my approach I have omitted much important material， for the sake of concreteness. As the title of the book signifies， I restrict attention almost entirely to Euclidean space.
　　2. To deal with n-dimensional spaces from the outset. I believe this is preferable to one standard approach to the theory which first thoroughly treats integration on the real line and then generalizes. There are several reasons for this belief. One is quite simply that significant figures are frequently easier to sketch in IRe than in R1！ Another is that some things in IR1 are so special that the generalization to Rn is not clear; for example， the structure of the most general open set in R1 is essentially trivial —— it must be a disjoint union of open intervals （see Problem 2.6）. A third is that coping with the n-dimensional case from the outset causes the learner to realize that it is not significantly more difficult than the one-dimensional case as far as many aspects of integration are concerned.
　　3. To provide a thorough treatment of Fourier analysis. One of the triumphs of Lebesgue integration is the fact that it provides definitive answers to many questions of Fourier analysis. I feel that without a thorough study of this topic the student is simply not well educated in integration theory. Chapter 13 is a very long one on the Fourier transform in several variables， and Chapter 14 also a very long one on Fourier series in one variable.

欧氏空间上的勒贝格积分（修订版）（英文版） [Lebesgue Integration on Euclidean Space Revised Edition] 下载 mobi epub pdf txt 电子书 格式

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老师推荐的教材，学习勒贝格积分的，很喜欢！

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很不错的书，内容很详细，还会继续关注的！

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如果一个教“实变函数”的教师多从学生角度出发，

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学专业的重要专业课程，该课程是数学分析的一门

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如果一个教“实变函数”的教师多从学生角度出发，

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、35无理数）在实数中是“极少数的”的一样. 实变函数就是要以占有“绝大多数”的连续性质不好的实函数作为研究对象，这样数学分析的许多定义和工具都“不好使了”，必须建立适用有“绝大多数”的连续性质不好的实函数的积分理论，这是我们这本书的主要任务.

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教，才能够把握给学生讲解到如何细致的程度教学

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学生不努力，而是因为该课程相对数学分析等基础

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语言的严谨性是数学语言的特点之一，在实变函数教学过程中要坚持概念和定理叙述的严谨性. 如果总是严谨而枯燥的数学语言势必会使学生容易产生厌倦感觉，从而使教学效果大打折扣. 因此在不失基本的严谨性基础之上，可以对比较难理解的概念、定理和证明过程用比较通俗而形象的语言进行解释.  例如在讲述叶果洛夫定理时，定理板书后可以用如下通俗语言说明该定理：在有限测度集合上的几乎处处收敛函数列一定是“差不多”一致收敛. 又如在讲述鲁津定理时，可用如下通俗语言说明该定理：几乎处处有限的可测函数其实“差不多”是连续函数. 一个通俗的“差不多”就形象地解释了一串枯燥的数学语言“δ∀>0, 00,(),EEmEEδ∃⊂−< 使得命题P在集合0E上成立”，更加深了学生对定理的理解. 在经典习题讲解中可以用通俗语言加强学习实变函数的意义. 比如在讲解习题“[a,b]上的实函数全体的势为2C”和“[a,b]上的连续实函数全体的势为C”后，可以通俗地讲：我们发现，经常见到的连续实函数在实函数中是“极少数的”，就像我们前面习题中反映的经常见到的代数数（包含很多

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