Customer Reviews

A Radical Approach to Lebesgue's Theory of Integration (Mathematical Association of America Textbooks)

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4 star:		(1)
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1 star:		(1)

 

 

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!

This is an extremely enjoyable book on integration theory, contrasting with the "usual" teaching. Normally, after two years of college, the student is introduced to measure theory, the Borel coverings and its application to Lebesgues integration as a generalisation of Riemann integrals . The whole subject can be covered in 4 or 5 dry lessons. This is like going to McDonald to eat.

This book on the other hand is like going to a three star Michelin Restaurant where each small dish is prepared in order to fulfill the senses.

The book begins with Riemann integration and step by step introduces the difficulties people ran into trying to handle more complex functions, how they tried to overcome these issues, the advances and the dead ends. Theorems and definitions are introduced when needed during this journey and are always put into perspective. The book begins (as often) With Riemann around 1850, and soon the reader is faced with the intricacy of the Real numbers ; set theory, Cantor's work on cardinals, the continuum hypothesis are "discovered". Delicate functions escaping the intuition such as the SCVn or the Volterra function are studied in order to feel all the difficulties posed by Riemann Integrals. The author then brings us to the final steps, the Weirestrauss, Jordan, Borel and Lebesgues works which put a beautiful end to the revolutionary era covering 1850-1900. This journey takes the first ~ 200 pages of the book, the remaining is devoted to extensions and deepening of the concepts.

This is not a book on history of science, but one where the history of science is used as a pedagogical tool. I recommend it to people who are not impatient and have time for a real meal.

My review is limited to what I have read so far as I am currently going through this book. Although I am a graduate student in mathematics I felt that I still had not fully grasped the whole Riemann integration - Lebesgue integration stuff so I bought this book. I am not disappointed. As I am reading and working through the book, I am finding my understanding of stuff that I already knew enhanced. I think this is a supplementary book for those who have studied the theory elsewhere but want to enrich their understanding.

If you had calculus in high school or college then you learned about Newton, Leibnitz, and Riemann but probably did not encounter Lebesgue (pronounced le-bek). At the University of Alabama Huntsville learning about Lebesgue integration is key to advancing into graduate studies in mathematics.

I purchased this book, after reading about it in the Mathematical Association of America (MAA). For autodidacts like myself, it is a good first introduction to the topic.

This is an utterly useless book!

I bought it.

Read it.

This book does not contain anything new that is not contained in any other book on Measure Theory. The title of this book contains "Radical Approach". There is nothing "radical here".

All the contents are standard bromide. No clarity is added.

Besides, this book contains plenty of mathematical errors.