欧氏空间上的勒贝格积分(修订版)(英文版)

　　本书简明、详细地介绍勒贝格测度和rn上的积分。本书的基本目的有四个，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：本书适用于数学专业的学生、老师和相关的科研人员。

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

　　"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. Lebesgue's 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 Lebesgue's 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,Lebesgue's dominated convergence theorem, Lebesgue's increasing convergence theorem, the theory of the Lebesgue function of the Cantor ternary set, and Lebesgue's 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 Pubini's 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.
　　4. To prepare students to become "workers" in real analysis.I do not mean that they should become researchers, but instead that they be able to apply to other areas of interest to them the things they have seen in this book. As a certain sort of analyst myself, I have chosen to include those topics which ! have found to be of primary importance in my own research. This purpose partially explains the inclusion of the two long Chapters 15 and 16 on differentiation theory. They are also here because of their beauty and depth.
　　This last aim seems to be ever growing in its importance, as we mathematicians are seeing more and more students from other disciplines taking our advanced courses. It is now commonplace to find engineering graduate students, for example, taking courses in integration theory,differential geometry, etc.
　　I have written this book under the assumption that the student either is already familiar with certain basic concepts or has a teacher. Thus,the introductory chapter on the basic facts about Rn is extremely brief,except that I have tried to give a fairly careful account of compactness(in Rn). (I have done so because compactness is a serious stumbling block for many students.)
　　I confess that I am proud of the problems in this book. There are 600 of them, and I think most of them are interesting and neither trivial nor impossibly difficult. There are a few that are "challenging," and this is another reason for the utility of having a teacher. I have chosen to spread the problems throughout the text, in order to encourage the students and teacher to use them as an integral part of their study. Thus,when a problem appears .as a subject is being developed, the indication to the students is that they are now ready for this exercise to check their knowledge and to strengthen their understanding of what is being discussed.
　　Bibliography
　　I am placing this in such a prominent position in order to acknowledge some debts and to encourage the reader to engage in further reading. Paul R. Halmos, Measure Theory, Springer~Verlag, 1988.
　　H.L. Royden, Real Analysis, third edition, Macmillan, 1988.The two books just cited axe standard texts. They concentrate on theoretical aspects, especially the Caratheodory construction of measurable sets. They contain important topics which are not discussed at all in my book. For example, the Raxion-Nikodym theorem, Egorov's theorem,the dual space of Lp,
　　Richard L. Wheeden and Antoni Zygmund, Measure and Integral,Dekker, 1977.
　　This excellent book has similar aims to mine (it is quite concrete in its approach), but also concentrates on many technical aspects which I have not included.
　　Walter Rudin, Real and Complex Analysis, third edition, McGrawHill, 1987.
　　Herbert Federer, Geometric Measure Theory, Springer-Verlag, I969.Federer's book is perhaps the ultimate text on this subject. It seems to contain everything but Fourier analysis, including a'complete course on measure theory in Chapter Two, "General measure theory." The really serious student should find great benefit in working through this chapter.
　　de la ValiSe Poussin, Integrales de Lebesgue, Gauthier-Villars, 1950.When I first learned this subject from Professor Bray in 1958, this was the text he used. A beautiful source for the generalization of the Fourier transform to distributions is
　　L. HSrmander, The Analysis of Linear Partial Differential Operators I, Springer Study Edition, Springer-Verlag, 1990.