Lebesgue Integration on Euclidean Space

Front cover......Page 1
Various Integration Formulas......Page 3
Various Special Integrals and Sums......Page 4
Title......Page 5
Series......Page 6
Title page......Page 7
Date-line......Page 8
Contents......Page 9
Preface......Page 13
Bibliography......Page 15
Acknowledgments......Page 17
Dedication......Page 19
A Sets......Page 21
B Countable Sets......Page 24
C Topology......Page 25
D Compact Sets......Page 30
E Continuity......Page 35
F The Distance Function......Page 40
A Construction......Page 45
B Properties of Lebesgue Measure......Page 69
C Appendix: Proof of P1 and P2......Page 80
3 Invariance of Lebesgue Measure......Page 85
A Some Linear Algebra......Page 86
B Translation and Dilation......Page 91
C Orthogonal Matrices......Page 93
D The General Matrix......Page 95
A A Nonmeasurable Set......Page 101
B A Bevy of Cantor Sets......Page 103
C The Lebesgue Function......Page 106
D Appendix: The Modulus of Continuity of the Lebesgue Functions......Page 115
A Algebras and $\\sigma$-Algebras......Page 123
B Borel Sets......Page 127
C A Measurable Set which Is Not a Borel Set......Page 130
D Measurable Functions......Page 132
E Simple Functions......Page 137
A Nonnegative Functions......Page 141
B General Measurable Functions......Page 150
C Almost Everywhere......Page 155
D Integration Over Subsets of $\\mathbb{R}^n$......Page 159
E Generalization: Measure Spaces......Page 162
F Some Calculations......Page 167
G Miscellany......Page 172
A Riemann Integral......Page 177
B Linear Change of Variables......Page 190
C Approximation of Functions in $L^1$......Page 191
D Continuity of Translation in $L^1$......Page 200
8 Fubini\'s Theorem for $\\mathbb{R}^n$......Page 201
A Definition and Simple Properties......Page 219
B Generalization......Page 222
C The Measure of Balls......Page 225
D Further Properties of the Gamma Function......Page 229
E Stirling\'s Formula......Page 232
F The Gamma Function on $\\mathbb{R}$......Page 236
A Definition and Basic Inequalities......Page 241
B Metric Spaces and Normed Spaces......Page 247
C Completeness of $L^p$......Page 251
D The Case $p=\\infty$......Page 255
E Relations between $L^p$ Spaces......Page 258
F Approximation by $C_c^\\infty(\\mathbb{R}^n)$......Page 264
G Miscellaneous Problems......Page 266
H The Case $0A Products of $\\sigma$-Algebras......Page 275
B Monotone Classes......Page 278
C Construction of the Product Measure......Page 281
D The Fubini Theorem......Page 288
E The Generalized Minkowski Inequality......Page 291
A Formal Properties......Page 297
B Basic Inequalities......Page 300
C Approximate Identities......Page 304
A Fourier Transform of Functions in $L^1(\\mathbb{R}^n)$......Page 313
B The Inversion Theorem......Page 328
C The Schwartz Class......Page 340
D The Fourier-Plancherel Transform......Page 343
E Hilbert Space......Page 354
F Formal Application to Differential Equations......Page 359
G Bessel Functions......Page 364
H Special Results for $n=1$......Page 372
I Hermite Polynomials......Page 376
A Periodic Functions......Page 387
B Trigonometric Series......Page 393
C Fourier Coefficients......Page 412
D Convergence of Fourier Series......Page 420
E Summability of Fourier Series......Page 430
F A Counterexample......Page 438
G Parseval\'s Identity......Page 441
H Poisson Summation Formula......Page 448
I A Special Class of Sine Series......Page 456
15 Differentiation......Page 467
A The Vitali Covering Theorem......Page 468
B The Hardy-Littlewood Maximal Function......Page 470
C Lebesgue\'s Differentiation Theorem......Page 476
D The Lebesgue Set of a Function......Page 478
E Points of Density......Page 483
F Applications......Page 486
G The Vitali Covering Theorem (Again)......Page 498
H The Besicovitch Covering Theorem......Page 502
I The Lebesgue Set of Order $p$......Page 511
J Change of Variables......Page 514
K Noninvertible Mappings......Page 525
A Monotone Functions......Page 531
B Jump Functions......Page 541
C Another Theorem of Fubini......Page 547
D Bounded Variation......Page 550
E Absolute Continuity......Page 564
F Further Discussion of Absolute Continuity......Page 573
G Arc Length......Page 583
H Nowhere Differentiate Functions......Page 590
I Convex Functions......Page 596
Index......Page 601
Symbol Index......Page 607
Assorted Facts......Page 609
Fourier Transform Table......Page 610
Back cover......Page 611