A Radical Approach to Lebesgue's Theory of Integration
| Date: 04 May 2011, 07:50
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A Radical Approach to Lebesgue's Theory of Integration (Mathematical Association of America Textbooks)
By David M. Bressoud
* Publisher: Cambridge University Press
* Number Of Pages: 344
* Publication Date: 2008-01-14
* ISBN-10 / ASIN: 0521711835
* ISBN-13 / EAN: 9780521711838
* Binding: Paperback
Product Description:
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.
Summary: The best book on integrating integration theory
Rating: 5
The history of integration theory is a tortuous path of subtle nuances in our understanding of how to generalise apparently simple concepts. Most students go through life without understanding the turmoil that was going on at the foundations of the edifice. David Bressoud is well known for a number of books but this one is a real tour de force. This is a very difficult conceptual area and what he has done is to link the historical development to a rigorous analytical context. This helicopter view of the sweep of history is one of the hardest things for a student to obtain. Bressoud has done a magnificent job in pulling all the strands together in a way which could be understood by a student who has made a serious study of analysis. There are many examples which walk you through important foundational results eg Darboux's proof that a certain function was continuous everywhere but differentiable nowhere. The exercises are supplemented by some hints but I think for such a difficult area it would be a good idea to have a website with full solutions or perhaps a separate problem based book.
Analytic concepts such as continuity, compactness, uniform convergence, closed and open sets etc all figure in their historical context and Bressoud demonstrates from the historical record why, for instance, Riemann integration was supplanted by Lebesgue integration (at least at the theoretical level). Notwithstanding that Lebesgue theory is over a century old it is still not usually taught at undergraduate level. It is generally extremely poorly explained even though it is absolutely fundamental to the techniques of modern finance theory.
Very simply Bressoud takes you on a journey that seeks to explain why Lebesgue integration theory holds sway at the theoretical level. Along the way you get to meet your old friends: Riemann, Bolzano, Weierstrass, Borel, Heine, Cantor etc etc .
Books such as this provide invaluable insights for those who really want to understand the subject. The pieces do fall together and one's understanding of the subject is enriched.
Brilliant!
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