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.
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 but readable. There are heaps of exercises, and they are quite accessible. I know of no other source with such a wealth of information about the genesis of the modern integral concept. This book will be valuable for mathematicians, for scholars of mathematical history, and certainly for students."
Steven G. Krantz, American Institute of Mathematics for The UMAP Journal
"A new and noteworthy title from Cambridge University Press! An outstanding book meant to advance undergraduate and beginning graduate students in mathematics."
B. Crstici, Mathematical Reviews
"I find it difficult to think of a better introduction to this corner store of modern mathematics and highly recommend the book to a very broad readership of students and researchers alike."
Paul Embrechts, ETH Zurich for the Journal of the American Statistical Association