歐氏空間上的勒貝格積分（修訂版）（英文版） [Lebesgue Integration on Euclidean Space Revised Edition] pdf epub mobi txt 電子書 下載 2025

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☆☆☆☆☆

瓊斯（Frank Jones） 著

齣版社： 世界圖書齣版公司

ISBN：9787510005558

版次：1

商品編碼：10184606

包裝：平裝

外文名稱：Lebesgue Integration on Euclidean Space Revised Edition

開本：24開

齣版時間：2010-01-01

頁數：588

正文語種：英語

內容簡介

　　《歐氏空間上的勒貝格積分(修訂版)(英文版)》簡明、詳細地介紹勒貝格測度和Rn上的積分。《歐氏空間上的勒貝格積分(英文版)》的基本目的有四個，介紹勒貝格積分；從一開始引入n維空間；徹底介紹傅裏葉積分；深入講述實分析。貫穿全書的大量練習可以增強讀者對知識的理解。目次：Rn導論；Rn勒貝格測度；勒貝格積分的不變性；一些有趣的集閤；集閤代數和可測函數；積分；Rn勒貝格積分；Rn的Fubini定理；Gamma函數；Lp空間；抽象測度的乘積；捲積；Rn+上的傅裏葉變換；單變量傅裏葉積分；微分；R上函數的微分。
　　讀者對象：《歐氏空間上的勒貝格積分(修訂版)(英文版)》適用於數學專業的學生、老師和相關的科研人員。

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目錄

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.

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本科階段最難學的課程之一. 許多數學本科學生都對這門課程不感興趣，甚至望而生畏. 這不是因為

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語言的嚴謹性是數學語言的特點之一，在實變函數教學過程中要堅持概念和定理敘述的嚴謹性. 如果總是嚴謹而枯燥的數學語言勢必會使學生容易産生厭倦感覺，從而使教學效果大打摺扣. 因此在不失基本的嚴謹性基礎之上，可以對比較難理解的概念、定理和證明過程用比較通俗而形象的語言進行解釋.  例如在講述葉果洛夫定理時，定理闆書後可以用如下通俗語言說明該定理：在有限測度集閤上的幾乎處處收斂函數列一定是“差不多”一緻收斂. 又如在講述魯津定理時，可用如下通俗語言說明該定理：幾乎處處有限的可測函數其實“差不多”是連續函數. 一個通俗的“差不多”就形象地解釋瞭一串枯燥的數學語言“δ∀>0, 00,(),EEmEEδ∃⊂−< 使得命題P在集閤0E上成立”，更加深瞭學生對定理的理解. 在經典習題講解中可以用通俗語言加強學習實變函數的意義. 比如在講解習題“[a,b]上的實函數全體的勢為2C”和“[a,b]上的連續實函數全體的勢為C”後，可以通俗地講：我們發現，經常見到的連續實函數在實函數中是“極少數的”，就像我們前麵習題中反映的經常見到的代數數（包含很多

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後繼課程[1-4]. 它一方麵是數學分析理論的深化和

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夠得到”. 這樣學生既有成就感，又需要一定的努力纔學得懂，從而培養提高瞭數學能力. 關於實變函數的教學體會，楊鍾玄[5]，周性偉[6]，徐西安[7]等人都在他們的文章中談瞭自己的體會，我們很認同他們的體會和建議．本文結閤我們的教學實踐，針對實變函數課程的特點，對如何改善實變函數教學談幾點體會. 不妥之處請同行指正.

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繼續，另一方麵也是學生學習泛函分析、概率論、拓撲學、偏微分方程和調和分析等現代數學的基礎，因此具有承上啓下的作用. 實變函數課程不僅能夠豐富學生現代分析的知識結構，而且可以大大提高學生的抽象思維能力、邏輯推理能力和分析解

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更加理論化，習題非常難做等特點. 學習不到一個

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適當運用一些教學理論和技巧，盡量減少該課程特

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課程，在概念和方法上都有較大飛躍，更加抽象和

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、35無理數）在實數中是“極少數的”的一樣. 實變函數就是要以占有“絕大多數”的連續性質不好的實函數作為研究對象，這樣數學分析的許多定義和工具都“不好使瞭”，必須建立適用有“絕大多數”的連續性質不好的實函數的積分理論，這是我們這本書的主要任務.