Lebesgue Integration on Euclidean Space (Revised Ed.)

Rating:
Summary: High Praise for Jones
Review: "Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0 < p < 1.

The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.

Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.

While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!

As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.

Rating:
Summary: High Praise for Jones
Review: "Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0 < p < 1.

The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.

Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.

While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!

As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.

Rating:
Summary: an excellent introductory text
Review: As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.

The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.

For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library.

Rating:
Summary: treasure trove of mathematical technique
Review: This book is a treasure trove of mathematical technique. It covers topics that are relevant to many broad areas of real and functional analysis including signal processing and approximation theory. The author takes the time not only to prove the results, but also to construct the proofs so that the technique is made explicit to the reader. The author also motivates definitions by breaking them into the successively more complicated pieces so as to build intuition in the reader.

I especially recommend this book to anyone who lacks formal training in mathematics or wishes to develop mathematical technique in the areas of real and functional analysis.

Rating:
Summary: treasure trove of mathematical technique
Review: This book is a treasure trove of mathematical technique. It covers topics that are relevant to many broad areas of real and functional analysis including signal processing and approximation theory. The author takes the time not only to prove the results, but also to construct the proofs so that the technique is made explicit to the reader. The author also motivates definitions by breaking them into the successively more complicated pieces so as to build intuition in the reader.

I especially recommend this book to anyone who lacks formal training in mathematics or wishes to develop mathematical technique in the areas of real and functional analysis.