# Measure, Integration & Real Analysis

Sheldon Axler, San Francisco State University

Copyright Year: 2020

Last Update: 2021

Publisher: Sheldon Axler

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial

CC BY-NC

## Reviews

The book covers all core concepts plus lots of additional material. Chapters 1-5 would make for a good one semester course on measure theory and Lebesgue integration while Chapters 6-10 would make for a good follow up course primarily focused on... read more

The book covers all core concepts plus lots of additional material. Chapters 1-5 would make for a good one semester course on measure theory and Lebesgue integration while Chapters 6-10 would make for a good follow up course primarily focused on function spaces. Chapters 11 and 12 on Fourier analysis and Probability can be used as additional topics.

The definition given for the Riemann integral is actually that of the Darboux integral, but since these are equivalent it doesn't affect the presentation, and in a side bar the book acknowledges that the given definition is due to Darboux.

I was unable to find any other errors.

The content is up-to-date with the modern presentation of measure theory and integration. It should be usable for a long period of time.

The prose is clear and easy to follow. The book does a good job of motivating results throughout. For instance, it begins with a discussion of the problems with the Riemann integral and why a different definition of the integral is required in many situations. In the text theorems tend to be preceded by a short verbal summary of their content. These things help make the book very readable.

The notation and definitions are consistent throughout the book.

The text is divided into 12 chapters which are further divided into about 2-5 sections, which are in turn divided into 2-5 subsections of about 3 pages each. Each subsection is small enough to be reasonably assigned as a reading assignment.

The first 5 chapters cover measures and Lebesgue integration in a clear, logical manner. Chapters 6-8 cover normed vector spaces. Banach spaces are covered in Chapter 6, L^p spaces in Chapter 7, and Hilbert spaces in Chapter 8. Chapter 10 covers additional topics on Hilbert spaces, but Chapter 9 on real and complex measures sticks out since it is awkwardly positioned between the two Hilbert space chapters. I would prefer to see the order of Chapters 9 and 10 switched for this reason and since none of the material from Chapter 9 is needed for Chapter 10.

I didn't find any interface issues.

I couldn't find any grammatical errors.

The text is not culturally insensitive or offensive.

## Table of Contents

- 1 Riemann Integration
- 2 Measures
- 3 Integration
- 4 Differentiation
- 5 Product Measures
- 6 Banach Spaces
- 7 Lp Spaces
- 8 Hilbert Spaces
- 9 Real and Complex Measures
- 10 Linear Maps on Hilbert Spaces
- 11 Fourier Analysis
- 12 Probability Measures

## Ancillary Material

## About the Book

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.

## About the Contributors

### Author

**Sheldon Axler** was valedictorian of his high school in Miami, Florida. He received his AB from Princeton University with highest honors, followed by a PhD in Mathematics from the University of California at Berkeley. As a postdoctoral Moore Instructor at MIT, Axler received a university-wide teaching award. He was then an assistant professor, associate professor, and professor at Michigan State University, where he received the first J. Sutherland Frame Teaching Award and the Distinguished Faculty Award.

Axler received the Lester R. Ford Award for expository writing from the Mathematical Association of America in 1996. In addition to publishing numerous research papers, he is the author of six mathematics textbooks, ranging from freshman to graduate level. His book Linear Algebra Done Right has been adopted as a textbook at over 300 universities and colleges.

Axler has served as Editor-in-Chief of the Mathematical Intelligencer and Associate Editor of the American Mathematical Monthly. He has been a member of the Council of the American Mathematical Society and a member of the Board of Trustees of the Mathematical Sciences Research Institute. He has also served on the editorial board of Springer’s series Undergraduate Texts in Mathematics, Graduate Texts in Mathematics, Universitext, and Springer Monographs in Mathematics.

He has been honored by appointments as a Fellow of the American Mathematical Society and as a Senior Fellow of the California Council on Science and Technology. Axler joined San Francisco State University as Chair of the Mathematics Department in 1997. In 2002, he became Dean of the College of Science & Engineering at San Francisco State University. After serving as Dean for thirteen years, he returned to a regular faculty appointment as a professor in the Mathematics Department.