Measure and Integration: A First Course

M. Thamban Nair’s Measure and Integration: A First Course is a text that arose from the notes the author developed teaching the subject matter over thirty years. As such, the text claims to take a student-friendly approach that is full of insights and queries that help develop the subject’s narrative. In this, the text is reasonably successful (the reviewer’s caveat is that he has found no text on measure theory to be truly “student-friendly”), even as it approaches the material with a relative level of generality. It should be considered for a first graduate-level course in measure theory, especially when the subject matter is meant to be broadly applicable (with a focus on general measurable spaces, not just Euclidean measurable spaces).

 

 The author makes it clear in the preface that ``the attempt in this book is to introduce the students to this modern subject in a simple and natural manner so that they can pursue the subject further with confidence, and also apply the concepts to other branches of mathematics.’’ The first chapter serves to (re)introduce the Riemann Integral, and to highlight some advantages and – more importantly – some disadvantages that this integral brings to the table. Thus prompted, the student is introduced to Lebesgue measure in the second chapter (where they are initially introduced to Lebesgue outer measure before moving on to Lebesgue measurable sets on \( \mathbb{R} \)). The heart of the text is in the next three chapters, which introduce measures (on more abstract spaces) and measurable functions; integral of positive measurable functions; and integration of complex measurable functions. The text concludes with two final chapters: one on integration over product spaces, and one (serving to highlight the value of measure theory) on an introduction to the Fourier Transform. The entire text features a good number of worked-out examples, as well as appropriate exercises at the end of each chapter.

 

In reading Nair’s text, the reviewer cannot help but compare it to the text Measure and Integral: An Introduction to Real Analysis by Wheeden and Zygmund. Both texts are written for a first-year graduate course audience, with a similar scope and aim. Where the two differ, primarily, is in their treatment of more abstract (non-Euclidean) metric spaces. Wheeden and Zygmund’s text focuses primarily on Euclidean measure spaces, venturing into the abstract integration only after the main theory has been covered (approximately 2/3 of the way through the book). Nair’s text, however, introduces abstract measure spaces at the front of chapter 3, and this allows Nair to cover all fundamental integration theory from this abstract viewpoint. While this is preferable for those who want to use measure theory outside of a purely Euclidean framework (for example, a probabilist), it does ratchet up the level of abstraction that is introduced before integration.

 

In summary: Measure and Integration: A First Course is a concise text that serves as an introduction to graduate-level integration theory. It covers the essentials of the subject, and does a fair job providing motivation for new concepts (typically using boxed queries). There are a large number of examples worked out in the text, and a good number of exercises are asked at the end of each chapter. The primary difference between this text and others like it (specifically Wheeden and Zygmund’s text) is the early introduction of abstract measure theory, which makes the text more challenging but also more general from an early moment. The text should be fairly considered for those who are interested in learning very general measure theory.

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John Ross is an assistant professor of mathematics at Southwestern University