A Radical Approach to Lebesgue's Theory of Integration (Mathematical Association of America Textbooks)

This lively introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author stresses the original purpose of the definitions and theorems, highlighting the difficulties mathematicians encountered as these ideas were refined. The story begins with Riemann's definition of the integral, and then follows the efforts of those who wrestled with the difficulties inherent in it, until Lebesgue finally broke with Riemann's definition. With his new way of understanding integration, Lebesgue opened the door to fresh and productive approaches to the previously intractable problems of analysis. Cover......Page 1 Semi-Title......Page 2 A RADICAL APPROACH TO LEBESGUE'S THEORYOF INTEGRATION......Page 4 Editorial Board......Page 6 Title......Page 8 Copyright......Page 9 Dedication......Page 10 Contents......Page 12 Preface......Page 14 1 Introduction......Page 18 Fourier Series......Page 19 Integration......Page 21 Cauchy and Riemann Integrals......Page 24 The Fundamental Theorem of Calculus......Page 25 Continuity and Differentiability......Page 29 Term-by-term Integration......Page 30 Notation......Page 32 Definitions......Page 33 Theorems......Page 35 2 The Riemann Integral......Page 40 2.1 Existence......Page 41 The Darboux Integrals......Page 45 Improper Integrals......Page 47 2.2 Nondifferentiable Integrals......Page 50 Darboux's Observation......Page 53 Summary......Page 55 2.3 The Class of 1870......Page 57 Hankel's Innovations......Page 59 Hankel's Types of Discontinuity......Page 61 Cantor's 1872 Paper......Page 63 3.1 Geometry of R......Page 68 Implications of the Bolzano-Weierstrass Theorem......Page 76 Completeness......Page 78 Harnack's Mistake......Page 80 Borel's Series......Page 81 Compactness......Page 82 Two Corollaries......Page 83 How Heine's Name Got Attached to This Theorem......Page 84 An Infinite Extension......Page 69 Topology of R......Page 70 More Definitions......Page 72 Cardinality......Page 88 The Continuum Hypothesis......Page 92 Power Sets......Page 93 4 Nowhere Dense Sets and the Problem with theFundamental Theorem of Calculus......Page 98 4.1 The Smith-Volterra-Cantor Sets......Page 99 The Cantor Ternary Set......Page 100 The Devil's Staircase......Page 102 4.2 Volterra's Function......Page 106 SVC(4)......Page 107 Perfect, Nowhere Dense Sets......Page 111 SVC(n)......Page 113 4.3 Term-by-Term Integration......Page 115 What Can Happen......Page 117 Preserving Some Uniformity......Page 120 Is Boundedness Sufficient?......Page 121 The Arzela-Osgood Theorem......Page 123 4.4 The Haire Category Theorem......Page 126 Applications of Haire's Theorem......Page 129 Baire's Big Theorem......Page 131 Lebesgue's Proof of Theorem 4.11......Page 132 Discontinuities of Derivatives......Page 133 5 The Development of Measure Theory......Page 137 5.1 Peano, Jordan, and Borel......Page 139 Jordan Measure......Page 141 Borel Measure......Page 143 Borel Sets......Page 144 The Limitations of Borel Measure......Page 145 5.2 Lebesgue Measure......Page 148 Improving on Borel......Page 151 Alternate Definition of Lebesgue Measure......Page 154 5.3 Caratheodory's Condition......Page 157 5.4 Nonmeasurable Sets......Page 167 Difficulties......Page 168 Pursuing the Axiom of Choice......Page 171 Do Nonmeasurable Sets Exist?......Page 172 6.1 Measurable Functions......Page 176 Limits of Measurable Functions......Page 178 Farewell to the Riemann Integral......Page 181 6.2 Integration......Page 186 Integration of Measurable Functions......Page 188 The Monotone Convergence Theorem......Page 190 Uniform Convergence......Page 200 Example 4.7 from Section 4.3......Page 201 Sufficient but Not Necessary......Page 203 Fatou's Lemma......Page 204 Proof of the Dominated Convergence Theorem......Page 205 6.4 Egorov's Theorem......Page 208 Convergence in Measure......Page 211 Limits of Step Functions......Page 213 Luzin's Theorem......Page 215 7 The Fundamental Theorem of Calculus......Page 220 7.1 The Dini Derivatives......Page 221 Bounded Variation......Page 223 7.2 Monotonicity Implies Differentiability Almost Everywhere......Page 229 Outlining the Proof......Page 230 The Proof of Theorem 7.4......Page 232 The Faber-Chisholm-Young Theorem......Page 235 7.3 Absolute Continuity......Page 240 The Evaluation Part......Page 241 Lebesgue Integral and Absolute Continuity......Page 243 A Hierarchy of Functions......Page 245 Absolute Continuity and Monotonicity......Page 246 7.4 Lebesgue's FTC......Page 248 8 Fourier Series......Page 258 8.1 Pointwise Convergence......Page 259 Cesaro Convergence......Page 262 8.2 Metric Spaces......Page 268 LP Spaces......Page 270 Convergence......Page 274 Ordering L P Spaces......Page 277 8.3 Banach Spaces......Page 280 The Riesz-Fischer Theorem......Page 284 8.4 Hilbert Spaces......Page 288 Complete Orthogonal Set......Page 290 Complete Orthonormal Sets......Page 294 Completing the Proof of the Riesz-Fischer Theorem......Page 296 9 Epilogue......Page 299 A.l The Cardinality of the Collection of Borel Sets......Page 304 Connection to Haire......Page 307 A.2 The Generalized Riemann Integral......Page 308 The Fundamental Theorem of Calculus......Page 310 Comparison with the Lebesgue Integral......Page 311 Final Thoughts......Page 313 AppendixB: Hints to Selected Exercises......Page 316 Bibliography......Page 334 Index......Page 340 Ideal for advanced undergraduate and graduate students in mathematics, this introduction to measure theory and Lebesgue integration is rooted in and motivated by the historical questions that led to its development

An introduction to measure theory and Lebesgue integration rooted in the historical questions that led to its development.