A Concise Introduction to Measure Theory

Preface......Page 5
Contents......Page 8
 1.1 The Riemann Integral Revisited......Page 10
 1.2 Improper Integrals......Page 19
 2.1 Measure and Measurability......Page 28
 2.2 Measurability and the Integral......Page 36
 2.3 The Monotone Convergence Theorem......Page 44
 3.1 Simple Functions......Page 47
 3.2 Other Measurable Functions......Page 55
 3.3 Subadditive Fuzzy Measures......Page 64
 4.1 Lebesgue Outer Measure......Page 72
 4.2 Measure from Outer Measure......Page 78
 4.3 Lebesgue Measure......Page 85
 4.4 Induced Measure and an Application......Page 92
 5.1 Interchanging the Order of Summation......Page 97
 5.2 Integration with the Counting Measure......Page 104
 6.1 Algebras and Monotone Classes of Sets......Page 109
 6.2 Defining a Product σ-Algebra......Page 111
 6.3 Sections of a Subset and of a Function......Page 114
 6.4 Defining a Product Measure......Page 117
 6.5 The Tonelli and Fubini Theorems......Page 122
 6.6 Borel Sets......Page 128
 7.1 Integrability and Step Functions......Page 133
 7.2 The Vitali Covering Theorem......Page 139
 7.3 Dini Derivatives......Page 144
 7.4 The Lebesgue Differentiability Theorem......Page 148
 7.5 The Derivative of the Integral......Page 154
 7.6 Bounded Variation......Page 157
 7.7 Absolute Continuity......Page 164
 7.8 The Integral of the Derivative......Page 168
 8.1 A Surjection from the Cantor Set to [0, 1]......Page 174
 8.2 A Bijection from the Cantor Set to  [0, 1] and the Cantor Function......Page 178
 1.1 The Riemann Integral Revisited......Page 184
  1.2 Improper Integrals......Page 189
  2.1 Measure and Measurability......Page 191
  2.2 Measurability and the Integral......Page 199
  2.3 The Monotone Convergence Theorem......Page 203
  3.1 Simple Functions......Page 205
  3.2 Other Measurable Functions......Page 211
  3.3 Subadditive Fuzzy Measures......Page 215
  4.1 Lebesgue Outer Measure......Page 220
  4.2 Measure from Outer Measure......Page 221
  4.3 Lebesgue Measure......Page 227
  4.4 Induced Measure and an Application......Page 232
  5.1 Interchanging the Order of Summation......Page 233
  5.2 Integration with the Counting Measure......Page 235
  6.1 Algebras and Monotone Classes of sets......Page 238
  6.2 Defining a Product σ-algebra......Page 240
  6.3 Sections of a Subset and of a Function......Page 241
  6.4 Defining a Product Measure......Page 242
  6.5 The Tonelli and Fubini Theorems......Page 243
  6.6 Borel Sets......Page 246
  7.1 Integrability and Step Functions......Page 247
  7.3 Dini Derivatives......Page 250
  7.4 The Lebesgue Differentiability Theorem......Page 251
  7.6 Bounded Variation......Page 255
  7.7 Absolute Continuity......Page 259
  7.8 The Integral of the Derivative......Page 262
  8.1 A Surjection from the Cantor Set to [0, 1]......Page 264
  8.2 A Bijection from the Cantor Set to [0, 1] and the Cantor Function......Page 267
References......Page 271
Index......Page 272
Untitled......Page 1