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.
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Editorial Reviews
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"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
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
Book Description
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.
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Most Helpful Customer Reviews
60 of 62 people found the following review helpful:
5.0 out of 5 stars
The best book on integrating integration theory,
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This review is from: A Radical Approach to Lebesgue's Theory of Integration (Mathematical Association of America Textbooks) (Paperback)
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!
8 of 8 people found the following review helpful:
5.0 out of 5 stars
Enjoyable,
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This review is from: A Radical Approach to Lebesgue's Theory of Integration (Mathematical Association of America Textbooks) (Paperback)
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.
14 of 16 people found the following review helpful:
5.0 out of 5 stars
EXCELLENT PEDAGOGICAL TOOL - FILLS IN THE MISSING GAPS,
Amazon Verified Purchase(What's this?)
This review is from: A Radical Approach to Lebesgue's Theory of Integration (Mathematical Association of America Textbooks) (Paperback)
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.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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nowhere dense sets, absolute continuity, outer content zero, positive outer content, nested interval principle, antiderivative part, pointwise discontinuous, trigonometric series converges, pairwise disjoint open intervals, positive outer measure, finite outer measure, ternary set, differentiability almost everywhere, finite trigonometric series, lim area, countable open cover, pairwise disjoint intervals, real number line, piecewise monotonic, disjoint measurable sets, nonmeasurable sets, intermediate value property, inner measure, countable union, derived set Key Phrases - Capitalized Phrases (CAPs): (learn more)
The Development of Measure Theory, The Lebesgue Integral, Paul Du Bois-Reymond, Egorov's Theorem, University of Berlin, Hilbert Spaces, Metric Spaces, Accommodating Algebra, The Baire Category Theorem, Real Analysis, Henri Lebesgue, Monotonicity Implies Differentiability Almost Everywhere, Radical Approach, Camille Jordan, Jordan's Cours, Lebesgue's Dominated Convergence Theorem, Nondifferentiable Integrals, Georg Cantor, David Hilbert, Moscow University, Banach Spaces, The Five Big Questions, Axel Hamack, The Smith-Volterra-Cantor Sets, Richard Dedekind Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
nowhere dense sets, absolute continuity, outer content zero, positive outer content, nested interval principle, antiderivative part, pointwise discontinuous, trigonometric series converges, pairwise disjoint open intervals, positive outer measure, finite outer measure, ternary set, differentiability almost everywhere, finite trigonometric series, lim area, countable open cover, pairwise disjoint intervals, real number line, piecewise monotonic, disjoint measurable sets, nonmeasurable sets, intermediate value property, inner measure, countable union, derived set Key Phrases - Capitalized Phrases (CAPs): (learn more)
The Development of Measure Theory, The Lebesgue Integral, Paul Du Bois-Reymond, Egorov's Theorem, University of Berlin, Hilbert Spaces, Metric Spaces, Accommodating Algebra, The Baire Category Theorem, Real Analysis, Henri Lebesgue, Monotonicity Implies Differentiability Almost Everywhere, Radical Approach, Camille Jordan, Jordan's Cours, Lebesgue's Dominated Convergence Theorem, Nondifferentiable Integrals, Georg Cantor, David Hilbert, Moscow University, Banach Spaces, The Five Big Questions, Axel Hamack, The Smith-Volterra-Cantor Sets, Richard Dedekind Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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