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 = 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 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
"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.