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

Meant for advanced undergraduate and graduate students in mathematics, this lively introduction to measure theory and Lebesgue integration is rooted in and motivated by the historical questions that led to its development. The author stresses the original purpose of the definitions and theorems and highlights some of the difficulties that were encountered as these ideas were refined. The story begins with Riemann's definition of the integral, a definition created so that he could understand how broadly one could define a function and yet have it be integrable. The reader then follows the efforts of many mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work in the late 19th and early 20th centuries of Jordan, Borel, and Lebesgue, who finally broke with Riemann's definition. Ushering in a new way of understanding integration, they opened the door to fresh and productive approaches to many of the previously intractable problems of analysis. Features • Exercises at the end of each section, allowing students to explore their understanding • Hints to help students get started on challenging problems • Boxed definitions make it easier to identify key definitions Table of Contents 1. Introduction 2. The Riemann integral 3. Explorations of R 4. Nowhere dense sets and the problem with the fundamental theorem of calculus 5. The development of measure theory 6. The Lebesgue integral 7. The fundamental theorem of calculus 8. Fourier series 9. Epilogue: A. Other directions B. Hints to selected exercises. Cover......Page 1 About the Book and the Author......Page 2 Editors and List of Publications......Page 4 A RADICAL APPROACH TO LEBESGUE'S THEORY OF INTEGRATION......Page 6 QA312.B67 2008 5 15'.42—dc22......Page 7 Dedication......Page 8 Contents......Page 10 Preface......Page 12 1 Introduction......Page 16 Fourier Series......Page 17 Integration......Page 19 Cauchy and Riemann Integrals......Page 22 The Fundamental Theorem of Calculus......Page 23 A Brief History of Theorems 1.1 and 1.2^2......Page 24 Continuity and Differentiability......Page 27 Exercises......Page 28 Notation......Page 30 Definitions......Page 31 Theorems......Page 33 Exercises......Page 35 2 The Riemann Integral......Page 38 2.1 Existence......Page 39 The Darboux Integrals......Page 43 Improper Integrals......Page 45 Exercises......Page 46 2.2 Nondifferentiable Integrals......Page 48 Exercises......Page 53 2.3 The Class of 1870......Page 55 Hankel's Innovations......Page 57 Hankel's Types of Discontinuity......Page 59 Cantor's 1872 Paper......Page 61 Exercises......Page 63 3.1 Geometry of R......Page 66 An Infinite Extension......Page 67 Topology of R......Page 68 More Definitions......Page 70 Exercises......Page 72 Implications of the Bolzano—Weierstrass Theorem......Page 74 Completeness......Page 76 Harnack's Mistake......Page 78 Borel's Series......Page 79 Compactness......Page 80 Two Corollaries......Page 81 How Heine's Name Got Attached to This Theorem......Page 82 Exercises......Page 84 Cardinality......Page 86 The Continuum Hypothesis......Page 90 Power Sets......Page 91 Exercises......Page 93 4 Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus......Page 96 4.1 The Smith—Volterra—Cantor Sets......Page 97 The Cantor Ternary Set......Page 98 The Devil's Staircase......Page 100 Exercises......Page 103 4.2 Volterra's Function......Page 104 SVC(4)......Page 105 Perfect, Nowhere Dense Sets......Page 109 Exercises......Page 111 4.3 Term-by-Term Integration......Page 113 What Can Happen......Page 115 Preserving Some Uniformity......Page 118 Is Boundedness Sufficient?......Page 119 The Arzelà—Osgood Theorem......Page 121 Exercises......Page 123 4.4 The Baire Category Theorem......Page 124 Applications of Baire's Theorem......Page 127 Baire's Big Theorem......Page 129 Lebesgue's Proof of Theorem 4.11......Page 130 Discontinuities of Derivatives......Page 131 Exercises......Page 132 5 The Development of Measure Theory......Page 135 5.1 Peano, Jordan, and Borel......Page 137 Jordan Measure......Page 139 Borel Measure......Page 141 Borel Sets......Page 142 The Limitations of Borel Measure......Page 143 Exercises......Page 144 5.2 Lebesgue Measure......Page 146 Improving on Borel......Page 149 Alternate Definition of Lebesgue Measure......Page 152 Exercises......Page 153 5.3 Carathéodory's Condition......Page 155 Exercises......Page 163 5.4 Nonmeasurable Sets......Page 165 Difficulties......Page 166 Pursuing the Axiom of Choice......Page 169 Do Nonmeasurable Sets Exist?......Page 170 Exercises......Page 172 6.1 Measurable Functions......Page 174 Limits of Measurable Functions......Page 176 Farewell to the Riemann Integral......Page 179 Exercises......Page 182 6.2 Integration......Page 184 Integration of Measurable Functions......Page 186 The Monotone Convergence Theorem......Page 188 Exercises......Page 195 Uniform Convergence......Page 198 Example 4.7 from Section 4.3......Page 199 Sufficient but Not Necessary......Page 201 Fatou's Lemma......Page 202 Proof of the Dominated Convergence Theorem......Page 203 Exercises......Page 204 6.4 Egorov's Theorem......Page 206 Convergence in Measure......Page 209 Limits of Step Functions......Page 211 Luzin's Theorem......Page 213 Exercises......Page 214 7 The Fundamental Theorem of Calculus......Page 218 7.1 The Dini Derivatives......Page 219 Bounded Variation......Page 221 Exercises......Page 225 7.2 Monotonicity Implies Differentiability Almost Everywhere......Page 227 Outlining the Proof......Page 228 The Proof of Theorem 7.4......Page 230 The Faber—Chisholm--Young Theorem......Page 233 Exercises......Page 236 7.3 Absolute Continuity......Page 238 The Evaluation Part......Page 239 Lebesgue Integral and Absolute Continuity......Page 241 A Hierarchy of Functions......Page 243 Absolute Continuity and Monotonicity......Page 244 Exercises......Page 245 7.4 Lebesgue's FTC......Page 246 Exercises......Page 253 8 Fourier Series......Page 256 8.1 Pointwise Convergence......Page 257 Cesàro Convergence......Page 260 Exercises......Page 263 8.2 Metric Spaces......Page 266 L^p Spaces......Page 268 Convergence......Page 272 Exercises......Page 275 8.3 Banach Spaces......Page 278 The Riesz—Fischer Theorem......Page 282 Exercises......Page 285 8.4 Hubert Spaces......Page 286 Complete Orthogonal Set......Page 288 Complete Orthonormal Sets......Page 292 Completing the Proof of the Riesz—Fischer Theorem......Page 294 Exercises......Page 295 9 Epilogue......Page 297 A.1 The Cardinality of the Collection of Borel Sets......Page 302 Exercises......Page 305 A.2 The Generalized Riemann Integral......Page 306 The Fundamental Theorem of Calculus......Page 308 Comparison with the Lebesgue Integral......Page 309 Final Thoughts......Page 311 Exercises......Page 313 Appendix B: Hints to Selected Exercises......Page 314 Bibliography......Page 332 Index......Page 338 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. 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.