歐氏空間上的勒貝格積分(修訂版)=Lebesgue Integration on Euclidean Space

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