A (Terse) Introduction to Lebesgue Integration

This is a joint review of

• Measure and Integration: A Concise Introduction to Real Analysis, by Leonard F. Richardson
• A Primer of Lebesgue Integration, 2nd edition, by H. S. Bear
• A (Terse) Introduction to Lebesgue Integration, by John Franks

I shall refer to these books as Concise, Primer, and Terse, respectively. For reference:

• concise, adjective: giving a lot of information clearly and in a few words; brief but comprehensive.
• primer, noun: an elementary textbook that serves as an introduction to a subject of study.
• terse, adjective: sparing in the use of words; abrupt.

These short treatments are suitable for different audiences. Bear’s Primer is ideal for beginners, especially for those studying on their own. Franks’ Terse is a very nice introduction for readers with some mathematical sophistication. Both Terse and Richardson’s Concise would be good texts for a semester graduate course, and Terse would be especially useful for someone needing a refresher or an overview of the subject.

In the August-September 2009 Monthly, Neil Falkner gave an excellent review of two longer analysis textbooks, authored by Anthony W. Knapp and published in 2005. I highly recommend this incisive review to anyone planning on teaching graduate-level real analysis.

Here are some observations about the three books under review.

Bear’s Primer is a friendly introduction that could be read by a motivated undergraduate. It begins with the basic definitions of the Riemann integral. A simple, easily motivated, definition of measurable set is given on page 27 that is later shown to be equivalent to Carathéodory’s “slick, even brilliant’ but unintuitive definition. Halfway through the book, Lebesgue measure on the plane is developed. After this, the book tackles general measures on general measure spaces. The final chapter introduces L²(X).

Richardson’s Concise starts out focusing on the “deficiencies” of the Riemann integral. But then Chapter 2 jumps right into abstract measure theory. The reader or student needs to be quite comfortable with abstraction to survive this concise treatment. Then Chapter 3 develops Lebesgue measure. The transition from Chapter 2 to Chapter 3 is very efficient, but very abstract. The book doesn’t get to integrals again until Chapter 5. The last chapters include introductions to related topics from the Riesz representation theorem for bounded linear functionals on C(X) to closed invariant subspaces of L(R).

I like Franks’ Terse book, though I’m shocked that it doesn’t even mention Fubini’s theorem. Like the other two books, it starts out with a brief treatment of the Riemann integral that motivates the need for a more general Lebesgue integral. This book defers the construction of Lebesgue measure to Appendix B, but clearly describes measures, the basic idea of null sets, σ-algebras, and finally general measurable sets, all in Chapter 2. This device works, and the book goes on to develop integration theory nicely. Incidentally, Appendix A provides the background material needed and, 2 unlike some such appendices, includes some proofs and is not too terse! The second half of the book covers Hilbert space, motivated by a study of L²[a,b], does some classical Fourier series, and discusses two important ergodic transformations.

Each of these three books give different developments of Lebesgue’s Dominated Covergence Theorem (LDCT), and Richardson’s Concise uses Egoroff’s theorem! I still think the best and best-motivated proof is accomplished by first proving the Monotone Convergence Theorem, then Fatou’s Lemma as a corollary, and then the LDCT. See Chapter 2 of Folland’s fine Real Analysis: Modern Techniques and Their Applications (2nd edition, Wiley, 1999).

I am surprised that Richardson’s Concise and Bear’s Primer both state and prove Vitali’s classical covering lemma, because Vitali’s covering lemma is not needed to prove Lebesgue’s density theorem. Its quite technical proof was a major stumbling block when I was first studying analysis over 50 years ago. All that is needed is a simplified version of Wiener’s covering lemma, which is also sufficient to prove a Hardy-Littlewood maximal theorem. See, for example, Lemma 3.15 in Folland’s book.

Richardson’s Concise and Bear’s Primer include proofs of versions of the Radon-Nikodym theorem and of Fubini’s theorem. Franks’ Terse omits Fubini’s theorem; it motivates and states the Radon-Nikodym, but omits the proof.

Kenneth A. Ross (ross@math.uoregon.edu) taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).