Measure, Integration & Real Analysis

Sheldon Axler

This book seeks to provide students with a deep understanding of the definitions, examples, theorems, and proofs related to measure, integration, and real analysis. The content and level of this book fit well with the first-year graduate course on these topics at most American universities. This textbook features a reader-friendly style and format that will appeal to today's students.

Measure, Integration & Real Analysis was published in Springer's Graduate Texts in Mathematics series in 2020. This is an Open Access book. Thus the electronic version of the book is legally available without cost by clicking below.

The review of undergraduate real analysis freely available at the following link was written by the author of Measure, Integration & Real Analysis to help students review prerequisite material.

The print version of Measure, Integration & Real Analysis is available from Springer, Amazon, and Barnes&Noble.

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reviews of Measure, Integration & Real Analysis

Excerpt from zbMATH: The book is a perfect introduction to graduate students into the theory of measure and Lebesgue integration together with some topics in Real Analysis. ... The presentation is a gentle approach to serious mathematics with many examples and detailed proofs. ... The book will become an invaluable reference for graduate students and instructors.
full review in zbMATH (subscription to zbMATH is not needed to read this review)

review in MathSciNet

5-star user review on Amazon titled “Very clear exposition of important results in Measure Theory, Functional Analysis”: The author has done an excellent job of exposition and proofs making this a very readable book. Once started I had a hard time putting the book down. Key points are highlighted in yellow or blue, yellow for definitions and blue for propositions or theorems. The book is packed with proofs of results and in some cases these are counter examples of previous theorems where the hypotheses have been weakened. I already had some familiarity with measure theory but the book provides a reference for the results and the proofs. These results are of course very important in that they all have applications in mathematical physics. This book goes beyond the main results of measure theory including some results on Functional Analysis, Spectral Theory and Fourier Analysis. It does not attempt to go into topics too advanced and again this makes the book very readable as it means that the reader will get a feeling that he or she really understands the topic after reading the proofs. I recommend this book very highly for anyone interested in Measure Theory & Real Analysis.

5-star user review on Amazon titled “Great Reference and Textbook”: This is an excellent reference and textbook on measure theory and real analysis. The book is intended for someone with a background in undergraduate or first year graduate analysis such as Walter Rudin's mathematical analysis. The book can be read and understood by anyone with a good solid background in mathematics.

5-star user review on Amazon titled “Compact yet very readable”: Compact but very readable, rigorous but friendly. It's hard to find books on measure theory or spectral theory that don't require a lot of time. This book does a lot in 400 pages, but they're all quite readable and flow well. At the end you'll have done the spectral theory of compact operators with 100% rigour. It summarised nicely pretty much all the analysis of my undergrad maths degree, excluding complex analysis. It's not quite the pedagogical masterpiece of Axler's Linear Algebra Done Right, but it's an even, straight path to walk on.

5-star user review on Amazon titled “Axler Strikes Again: Another Classic Text”: Stay away from this book unless you want to get addicted to beautifully written math textbooks. This book is right up the alley of "Linear Algebra Done Right" -- beautifully written, amazing for self-study (I'm self-studying and sometimes it's hard to put the book down), and leaves you wanting for more. Don't get me wrong -- it's by no means an easy text. But it keeps everything interesting and motivated, the proofs are clear and very well written, and there are enough (at least for me) end of chapter exercises to keep everyone happy and drive home the key ideas from the associated chapters. All in all, among my favorites. Will be on the look out for more from Dr. Axler. (I hope he can write a text on functional analysis next!)

excerpts from reviews of the author's book Linear Algebra Done Right

Altogether, the text is a didactic masterpiece.

Axler demotes determinants (usually quite a central technique in the finite dimensional setting, though marginal in infinite dimensions) to a minor role. To so consistently do without determinants constitutes a tour de force in the service of simplicity and clarity; these are also well served by the general precision of Axler's prose... The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library.

The determinant-free proofs are elegant and intuitive.
American Mathematical Monthly

Clarity through examples is emphasized... the text is ideal for class exercises... I congratulate the author and the publisher for a well-produced textbook on linear algebra.
Mathematical Reviews

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