A Radical Approach to Lebesgue's Theory of Integration [Hardcover]

An introduction to measure theory and Lebesgue integration rooted in the historical questions that led to its development.Meant for advanced undergraduate and graduate students in mathematics, this introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author tells the story of the mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work of Jordan, Borel, and Lebesgue.Meant for advanced undergraduate and graduate students in mathematics, this introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author tells the story of the mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work of Jordan, Borel, and Lebesgue.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 Riemanns definition of the integral, and then follows the efforts of those who wrestled with the difficulties inherent in it, until Lebesgue finally broke with Riemanns definition. With his new way of understanding integration, Lebesgue opened the door to fresh and productive approaches to the previously intractable problems of analysis.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. Bressoud is an insightful writer, and he presents this material in an enchanting fashion. The writing is scholarly but inviting, rigorous bul£|