The Lebesgue Integral for Undergraduates

It was the title of this book, rather than the subject matter, that led me to request it for review. Like, I suspect, most of my colleagues, I learned the details of the Lebesgue integral as a first-year graduate student; I may have heard about the existence of this other integral as an undergraduate, but certainly did not see anything of a systematic development of it back then. When I finally did encounter measure theory as a graduate student, I was decidedly underwhelmed; the whole subject struck me as a lot of technical fussing with no real major payoff at the end. Measure theory and real analysis also struck me as a distinctly graduate-level enterprise.

So when I noticed the title of this book, I was curious to see if this subject actually could be made comprehensible to an undergraduate. It turns out that it really can be, via a path to the Lebesgue integral that is different from the one I took as a graduate student. The approach used here is one of Daniell and Riesz, and avoids the development of a lot of technical measure theory.

Essentially, the idea behind the text’s definition of the Lebesgue integral is as follows: first, the integral of a step function is defined in the natural way, and then one extends the integral to a broader class of functions by considering limits of nondecreasing sequences of step functions. This approach to the Lebesgue integral is equivalent to the more traditional measure-theoretic definition often learned in graduate school, but that equivalence is not established in this text.

The process of defining the integral and proving that everything works takes a certain amount of care; one must, for example, show that the integral is well-defined in the sense that it is independent of the choice of sequence. Even defining the class of Lebesgue-integrable functions requires a little effort. However, all of these issues are quite adroitly dealt with by the author in a clear, step-by-step manner, with background material introduced as necessary. This constitutes chapter 1 of the text.

There are four chapters that follow this: in the next chapter, the Riemann integral is defined and compared and contrasted with the Lebesgue integral, with an emphasis on showing that the latter has some nice features (notably behavior with respect to limits) that the Riemann integral lacks. Chapter 3 is on \(L^p\) spaces; Banach and Hilbert spaces are also defined here, and it is shown that the case \(p=2\) produces a Hilbert space.

Measure theory, a brief introduction to which was given in chapter 1, is discussed in more detail in chapter 4. This is certainly a good idea, because the notion of a measure space has undeniable importance in its own right; consider, for example, people who are interested in probability. The measure of a set is defined here in terms of the integral; it is simply the integral of the characteristic function of that set. This reverses the traditional graduate-school approach of defining the integral in terms of the measure, but it provides for a relatively accessible introduction to the idea of measure, and one that might well motivate further developments down the road.

The final chapter is on linear operators on Hilbert space, particularly \(L^2\) spaces. The chapter culminates in a statement (without proof) of a version of a spectral theorem for bounded self-adjoint operators on a Hilbert space. Concrete examples are looked at in detail to help foster intuition and connections are made to linear algebra via a statement (also without proof) of the Jordan canonical form theorem.

The author asserts in the Preface that the text is “accessible to anyone who has mastered the single-variable calculus concepts of limits, derivatives and series.” Of course, claims of broad accessibility such as this are frequently made and often exaggerated, but here the author is, if you add “integration” to the list of calculus-related concepts that are necessary, being honest: most of this book really is pretty accessible to the audience described, modulo of course the usual and not surprising caveat that because the ideas in this book can be reasonably subtle, and because this book does employ a theorem-proof format, a certain amount of mathematical maturity and familiarity with reading and writing proofs is necessary for full understanding.

The author achieves this level of accessibility by employing a number of useful pedagogical features in the text. First and foremost is a very reader-friendly writing style that I found to be slow, careful, conversational, well-motivated and clear. A careful selection of homework exercises also enhances the pedagogical value of the text; in fact, the author provides various different kinds of exercises. Some, for example, are integrated right into the text, with the intention of having the student answer them as encountered. Solutions to these appear at the end of each chapter.

In addition to these, there are “Reading Questions”; these are simple end-of-section problems (e.g., “Give examples of three countably infinite sets”) that are designed to quickly test the reader’s memory and understanding of the concepts just learned. Solutions to these do not appear in the book. (The idea of inserting exercises like this at the end of a section is a simple idea, and a very good one, but not one I have seen in many other books.) Finally, there are more traditional exercises, some denominated “Advanced Exercises”; these are more conceptual and generally call for proofs. Solutions to just about all odd-numbered exercises (except for many of the advanced exercises) appear at the end of the book, but these solutions are often quite terse and amount to little more than hints.

In addition to this variety of exercises, each chapter closes with an endnote, about three or four pages in length, giving further historical/biographical/bibliographical material. Historical comments are also sprinkled throughout the text, as are photographs.

The last numbered section of each chapter (except the first) is devoted to an application of that chapter’s material; these applications are Fourier series, quantum mechanics, probability and the spectral theorem. These give the reader a concrete “payoff” to each chapter that should enhance student interest.

Background material on real analysis (e.g., convergence of sequences of numbers and functions) is, as noted earlier, developed as needed, so a course in real analysis is not technically a prerequisite for this book. However, for purposes of deriving maximum benefit from this text, I think such a course is probably advisable. For one thing, it will have introduced a student to the concept of proof. Proofs appear repeatedly throughout this text, and are frequently requested in the exercises; since proof techniques and basic logic are not discussed in detail here, some prior background in this area is almost essential, and obviously there is some added benefit in having seen such proofs in the context of analysis.

In addition, such a course has the advantage of getting students to think about the assumptions that underlie analysis — why we need the real numbers rather than the rational numbers to do calculus, for example. It should also get them thinking about just what the Riemann integral is and why, in some respects, it doesn’t behave as nicely as one might want. Therefore, such a course provides not only the kind of mathematical maturity necessary to really appreciate the subject of this text, but also a kind of motivation for the subject as well. The author has, I hasten to point out, not neglected motivation in this text, but there is only so much that one can do in a text that is largely devoted to other topics. Although comparisons with the Riemann integral are indeed made in this text, they do not appear until after the Lebesgue integral has been defined, and some prior theoretical exposure to the Riemann integral might provide better motivation for the definition of a different integral.

Chapter 5 of this book, in particular, strikes me as quite demanding for a person without some prior background in real analysis. Topics like compact operators and Hardy spaces are simply not common currency for students just getting out of a calculus course, and based on my own teaching experiences at Iowa State, I think this material could potentially cause difficulty even for experienced undergraduate mathematics majors here.

A natural question to ask is: can this book be used as a substitute for a traditional undergraduate real analysis class? Many of the “usual suspects” in such a course do appear here: cardinality, sup and inf of a set, convergence of sequences of numbers and functions, function limits and continuity, the Riemann integral. However, the discussion of these topics seems a bit more compact than in a typical real analysis course; they are introduced here primarily as tools for the development of the Lebesgue integral. Additionally, there are some topics that are often presented in a standard real analysis course (examples: the nature of the real numbers, differentiation, basic topology on the real line) that are not covered here. So, I don’t really see this text as replacing the standard analysis course. Therefore, notwithstanding the author’s considerable efforts to make this text accessible to a student right after calculus II, it seems to me that its optimal use would be as a follow-up to a traditional real analysis course.

Of course, most universities do not offer undergraduate courses in Lebesgue theory at all. Perhaps this text might inspire some to do so. Even in the absence of such a course, however, this text could find productive use in a senior seminar, or in a second course in real analysis. Iowa State University, for example, offers two semesters of analysis; historically the second semester seems to have been used to discuss topics like uniform convergence of sequences and series and multi-variable analysis, but some recent instructors have used the course to discuss other topics, such as elementary differential geometry or calculus on manifolds. A course in the Lebesgue integral would be an interesting thing to try as well.

To summarize and conclude: I like books that try something new, offer a different perspective on things, and are carefully and clearly written. This one qualifies on all counts. This is a book, I think, that students will actually read, and, even better, enjoy.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.