A Radical Approach to Lebesgue's Theory of Integration

by David M. Bressoud

• ISBN13: 9780521884747
• ISBN10: 0521884748
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-01-21
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	p. xi
Introduction	p. 1
The Five Big Questions	p. 2
Presumptions	p. 15
The Riemann Integral	p. 23
Existence	p. 24
Nondifferentiable Integrals	p. 33
The Class of 1870	p. 40
Explorations of R	p. 51
Geometry of R	p. 51
Accommodating Algebra	p. 59
Set Theory	p. 71
Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus	p. 81
The Smith-Volterra-Cantor Sets	p. 82
Volterra's Function	p. 89
Term-by-Term Integration	p. 98
The Baire Category Theorem	p. 109
The Development of Measure Theory	p. 120
Peano, Jordan, and Borel	p. 122
Lebesgue Measure	p. 131
Carathéodory's Condition	p. 140
Nonmeasurable Sets	p. 150
The Lebesgue Integral	p. 159
Measurable Functions	p. 159
Integration	p. 169
Lebesgue's Dominated Convergence Theorem	p. 183
Egorov's Theorem	p. 191
The Fundamental Theorem of Calculus	p. 203
The Dini Derivatives	p. 204
Monotonicity Implies Differentiability Almost Everywhere	p. 212
Absolute Continuity	p. 223
Lebesgue's FTC	p. 231
Fourier Series	p. 241
Pointwise Convergence	p. 242
Metric Spaces	p. 251
Banach Spaces	p. 263
Hilbert Spaces	p. 271
Epilogue	p. 282
Appendixes	p. 287
Other Directions	p. 287
The Cardinality of the Collection of Borel Sets	p. 287
The Generalized Riemann Integral	p. 291
Hints to Selected Exercises	p. 299
Bibliography	p. 317
Index	p. 323
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