Measure, Integration & Real Analysis

Reviewed by Stephen Deterding, Assistant Professor, Marshall University on 4/4/23

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

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.

Relevance/Longevity rating: 5

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.

Clarity rating: 5

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.

Consistency rating: 5

The notation and definitions are consistent throughout the book.

Modularity rating: 5

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.

Organization/Structure/Flow rating: 4

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.

Interface rating: 5

I didn't find any interface issues.

Grammatical Errors rating: 5

I couldn't find any grammatical errors.

Cultural Relevance rating: 5

The text is not culturally insensitive or offensive.