A Radical Approach to Lebesgue's Theory of Integration

ISBN-13: 9780521711838 / Angielski / Miękka / 2008 / 344 str.

David M. Bressoud

Opis

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."

Recenzje

'I find it difficult to think of a better introduction to this cornerstone of modern mathematics and highly recommend the book to a very broad readership of students and researchers alike.' Journal of the American Statistical Association

Spis treści

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.