A User-Friendly Introduction to Lebesgue Measure and Integration

Gail S. Nelson’s A User-Friendly Introduction to Lebesgue Measure and Integration covers those aspects of the theory of integration typically associated with the name of Lebesgue, plus some related topics. But as a quick search shows, there is no shortage of books that cover the theory of measure and integration as commonly attributed to the French mathematician Henri Lebesgue. Thus, it is reasonable to ask how does this book stand among the growing list of other books on the subject?

A User-Friendly Introduction to Lebesgue Measure and Integration is based on course notes written specifically for, and as the author explains in the preface, by undergraduate students. It should be noted that it assumes familiarity with sequences, series, limits, continuity, and compactness at the level of a first undergraduate course in real analysis, so it is not a self-contained text. In addition, it is short (approximately 200 pages) and focuses entirely on measure theory, integration, and \(L^{p}\) spaces. Many texts that cover the Lebesgue theory also cover a great deal more analysis or discuss measure and integration in greater generality.

Another feature of A User-Friendly Introduction to Lebesgue Measure and Integration is the level of detail and clarity of the proofs of the theorems covered. The entire book is very carefully and clearly written. For instance, the author provides many illuminating examples and helps the reader develop intuition for many of the proofs. I believe that an average student with the appropriate background can read and digest the book without too much assistance. There are also excellent sets of exercises accompanying each chapter. I would encourage any student planning to enter a mathematics graduate program to spend the summer between their senior year of undergraduate studies and first semester of graduate studies reading and working the exercises in Nelson’s book.

After an initial chapter summarizing the Riemann integral by way of what is sometimes called the Darboux approach, the author defines the notion of Lebesgue measure for subsets of \(\mathbb{R}^{n}\). With students in mind, Nelson does not use the Caratheodory approach to defining measurability of a set; instead one finds the “open set” definition of a measurable subset of \(\mathbb{R}^{n}\): a set \(E\subset \mathbb{R}^{n}\) is Lebesgue measurable if for every \(\varepsilon > 0\) there is an open subset \(G\) of \(\mathbb{R}^{n}\) with \(E \subset G\) and \(m^{\ast}(G\backslash E)<\varepsilon\), where \(m^{\ast}\) denotes the outer (exterior) measure of a set. The author then proceeds to carefully and in detail prove that the measurable subsets of \(\mathbb{R}^{n}\) form a \(\sigma\)-algebra. Note that the open set definition is also used in Measure and Integral by Wheeden and Zygmund, which is referenced in A User-Friendly Introduction to Lebesgue Measure and Integration.

Following the treatment of measure theory is Chapter 2 on integration. The author takes an approach which is motivated by the definition of the Riemann integral summarized in Chapter 0. It is noteworthy that Nelson first proves the Lebesgue dominated convergence theorem directly and then uses this to derive Fatou’s lemma and the monotone convergence theorem. Chapter 3 covers the basics of the function spaces \(L^{p}[a,b]\) and Fourier series from the \(L^{2}[a,b]\) perspective. Chapter 4 concludes the main discussion with an introduction to the more abstract approach to measure theory and integration common in graduate level texts. Finally, there is a nice section, “Ideas for Projects,” which is a great resource for possible topics for student presentations that build on the material that forms the main focus of A User-Friendly Introduction to Lebesgue Measure and Integration.

In summary, Nelson has written a very readable account of Lebesgue measure and integration that should be accessible to students who have successfully completed a, more or less, standard first course in real analysis at the undergraduate level. A User-Friendly Introduction to Lebesgue Measure and Integration really is a lovely book, and can be profitably read by advanced undergraduate or beginning graduate students as either a primary or supplementary text for learning the basics of Lebesgue measure and integration on \(\mathbb{R}^{n}\). Any student who masters the contents of this book and is able to work most of the exercises is very well prepared to study any subject that relies heavily on measure theory or the Lebesgue approach to integration.

The following are some brief remarks regarding certain details in A User-Friendly Introduction to Lebesgue Measure and Integration. In Example 2.2.2, the given collection of sets is not a measurable partition according to Definitoin 2.2.1, since the nonoverlapping condition fails. However, I believe that the authors intent is to define a measurable partition to be one satisfying Definition 2.2.1 parts (i)&(ii) and then to define a nonoverlapping measurable partition to be a collection satisfying Definition 2.2.1 parts (i)–(iii).

As a final note, A Primer of Real Functions by Boas is perfect as background or a refresher for reading this book, and Measure and Integral by Wheeden and Zygmund would make excellent followup reading. Both of these texts are referenced in Nelson’s book.

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Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.