Measure and Integration

Preface......Page 6
Contents......Page 11
 1.1 Sets and Functions......Page 13
 1.2 Countable and Uncountable Sets......Page 19
 1.3 Metric Spaces......Page 20
 1.4 Sequences and Series of Extended Real Numbers......Page 30
 1.5 Sequences and Series of Functions......Page 32
 1.6 Derivatives......Page 34
 1.7 Riemann Integration......Page 36
 1.8 Decimals and the Cantor Set......Page 40
 2.1 Introduction......Page 55
 2.2 Lebesgue Outer Measure......Page 58
 2.3 Measurable Sets and Lebesgue Measure......Page 68
 2.4 Measurable Functions......Page 83
 2.5 Extended Real-Valued Functions......Page 89
 2.6 Egorov’s and Luzin’s Theorems......Page 99
 2.7 Lebesgue Outer Measure in {\\mathbb{R}}^{{\\varvec n}}......Page 105
 2.8 Measurable Sets and Lebesgue Measure in {\\mathbb{R}}^{{\\varvec n}}......Page 112
 3.1 Integrals of Simple Functions......Page 121
 3.2 Integrals of Measurable Functions......Page 131
 3.3 Lp Spaces......Page 150
 3.4 Dense Subsets of Lp......Page 169
 4.1 Introduction......Page 175
 4.2 The Convergence Problem......Page 182
 4.3 Cesàro Summability of Fourier Series......Page 197
 4.4 Even and Odd Functions......Page 203
 4.5 Orthonormal Expansions......Page 209
 Appendix......Page 217
 5.1 Background......Page 222
 5.2 Monotone Functions and Continuity......Page 223
 5.3 Monotone Functions and Differentiability (A)......Page 228
 5.4 Monotone Functions and Differentiability (B)......Page 239
 5.5 Integral of the Derivative......Page 251
 5.6 Total Variation......Page 263
 5.7 Absolute Continuity......Page 277
 5.8 Differentiation and the Integral......Page 288
 5.9 Signed Measures......Page 305
 5.10 The Radon–Nikodým Theorem......Page 316
 5.11 The Lebesgue–Stieljes Measure......Page 327
 6.1 The Spaces Lp as Normed Linear Spaces......Page 348
 6.2 Modes of Convergence......Page 366
 7.1 Product Measure......Page 386
 7.2 The Completion of a Measure......Page 403
8 Hints......Page 416
References......Page 602
Index......Page 604