Integration — A Functional Approach

The Lebesgue integral is of course both beautiful and useful: its utility is clear once the reasonably advanced undergraduate meets Dirichlet’s slick Gegenbeispiel of the characteristic function of the rationals in the unit interval in the context of the theory of the Riemann integral: surely this function’s integral must be 0, what with the irrationals flexing their muscles in no uncertain terms, but what fun we instructors have in demonstrating in a few lines that it simply isn’t so… in point of fact the integral doesn’t exist. But clearly it should, and presently Lebesgue rides to the rescue: in his sense the integral in question is nothing else than the measure of the rationals in the interval, and here we recover our much needed 0. Beautiful.

Down the line, such theorems as the dominated convergence theorem go on to add to one’s appreciation of Lebesgue’s theory as being eminently useful, and it quickly dawns on the student there are good reasons why Lebesgue integration is simply presupposed as being the order of the day in everything from functional analysis to Fourier analysis and PDE.

But the pedagogical sticky wicket in all this, at least at the early stages, is the business of introducing measure theory, which can be somewhat austere to the novice. I’m afraid that when I teach undergraduate real analysis I can’t resist this temptation myself, and so my lectures do go into this material to a small degree, but it is an unquestionably risky undertaking as far as resonance with the audience goes: suddenly we’re talking about Carathéodory outer measure, we’re doing strange constructions, and soon integration, just recently made transparent courtesy of Riemann’s definitions, grows opaque again. It takes persistence, and I think some maturity, to own that in due course the subject not only grows more transparent than ever before, but there is a lot of intrinsic elegance in Lebesgue’s approach. And usually this light doesn’t dawn until graduate real analysis (at the earliest).

For what it’s worth, I well recall first reading about the Lebesgue integral in Burkill’s wonderful book, whose elegance I utterly failed to appreciate at the time. Inner and outer measure, criteria for their coincidence, and other such minutiae left me pretty much cold. To boot, being anything but a natural analyst, I eventually developed an attitude which treated the more arcane elements of the subject as bitter pills to be swallowed along the way to completing one’s education — in my case in number theory. O, to be young and stupid: with age came at least the wisdom that I was very wrong in my youthful prejudice, and I am fortunate to be able to atone for it now by teaching the material with the respect it deserves.

So the issue on the table is that of teaching Lebesgue integration to rookies or at least relative rookies. Admitting one should, how should one? Well, I think a good answer is provided by the book under review, Integration — A Functional Approach, by Klaus Bichteler. The book first appeared in 1998 and the present edition, in Birkhäuser’s “Modern Classics” series, attests to the established success of the text.

Bichteler’s principal pedagogical gambit is to downplay the aforementioned minutiae of measure theory in favor of what amounts to Daniell’s approach. I think I first came across this way of doing things in another fine book on the subject: Integral, Measure, and Derivative: A Unified Approach, by Shilov (yes: the guy who gave us the Shilov boundary) and Gurevich. The back cover of Bichteler book provides the following telling propaganda: “From this point of view Lebesgue’s integral can be had as a rather straightforward, even simplistic, extension of Riemann’s integral… The notion of measurability… is suggested by Littlewood’s observations rather than being conveyed authoritatively through definitions of Σ-algebras [sic: the back cover’s upper case sigma should be a lower case sigma: the text does it right on the inside] and good-cut conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner…” (It’s déjà vu all over again …) Additionally, the claim is made that “[this] approach… provides the additional benefit of cutting the labor in half.”

So it is that Integration — A Functional Approach weighs in at a modest 193 pages in whose orbit Bichteler takes us from a review of the Riemann integral to its extension not quite in the usual style of Daniell but certainly achieving the Lebesgue integral, then to measurability à la Littlewood (as indicated above), and the classical Banach spaces, including, of course, the L^p spaces (with L² properly featured as a Hilbert space). The subsequent last chapter is particularly evocative: its title is “Operations on Measures” and this is where we find Fubini’s Theorem, a discussion of signed measures, the Radon-Nykodym Theorem, and, at the very end of the story, nothing less than “Differentiation.”

Regarding Bichteler’s heterodoxy vis à vis Daniell’s formalism, let him speak for himself: “Instead of arguing with the upper and lower integrals of Daniell, analogs of Lebesgue’s outer and inner measure, we observe that already in the case of the Riemann integral a simple absolute-value sign under the upper integral simplifies matters considerably, by rendering superfluous the study of the lower integral and providing a seminorm, the Daniell mean… [T]he problem is to extend the elementary integral from step functions to larger classes, and the Jordan and Daniell means are perfectly suitable seminorms toward that goal…”

Finally, Integration — A Functional Approach is very engagingly written. Bichteler has a good style: his very first phrase in the book proper is: “Riemann’s integral is a major achievement of civilization.” It is brimming with exercises (which, says the author “are an integral [!] part of the material … [and] those marked with an asterisk * are used later on and thus must be done…”), and has the extra distinction of having been field-tested by the author himself: “This text originated as notes for the introductory graduate Real Analysis course at the University of Texas at Austin. They were intended for students who have had a first course in Real Analysis … and know the topology of the real line, in particular the notion of compactness, and the Riemann integral.”

Happily these prerequisites describe perfectly what I try to do with my students in undergraduate analysis, so they should be ready for Integration — A Functional Approach when they hit graduate school. It would be a wonderful thing indeed if they were assigned this fine book at that point.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.