Every one should know the marvels of the Lebesgue integral, and the earlier the better. I don’t think any college or university professor would disagree with this statement, provided that “every one” is taken to mean “all true mathematics majors,” particularly those bound for graduate school, and by “earlier” we mean, “as soon as the kid’s ready.” Typically, such a kid would be ready for Lebesgue after finding out that, its intrinsic clarity notwithstanding, the Riemann integral doesn’t spread its net wide enough (or, to push a pun, the net doesn’t have a fine enough mesh) to capture, for example (and what an example it is!), the characteristic function of the rationals in the unit interval. In my own most recent experience, my students hit this point in the middle of their second semester of Real Analysis, and I spent quite a bit of time not only dissecting the example just mentioned (due to Dirichlet, I recall) but giving an informal motivation of why this function’s integral should be 0. Of course, at this point the stage is set: enter Lebesgue, stage right…
I had been developing Carathéodory outer measure rather systematically, leading up to the obvious dénouement, when my top student started making noises that caused me to make an executive decision which left something to be desired: he asked about inner measure, their to-be-hoped-for-coincidence, and thus, in essence, Lebesgue measure. Well, I fielded his question, but realized that with only days of class left, I had better cut to the chase quickly and tie everything together — you all know the routine: sigma-algebras, countable subadditivity, &c.
All in all it was a net win, but wouldn’t it have been nice to use my student’s excellent and insightful question as a springboard to spend a much larger chunk of time on the intricacies of measure theory per se, carefully tracing the evolution from Carthéodory’s approach to its crystallization, as it were, in Borel’s formulation? And only then, building on this measure theoretic scaffolding, to go to Lebesgue’s development of his integral, culminating in the dominated and bounded convergence theorems? Of course the answer is a resounding yes!
Well, this is precisely what is available in just the first 40 pages of J. C. Burkill’s The Lebesgue Integral. This beautiful little of a book is a treasure trove of hard analysis for the beginner: Burkill, in true Cambridge style, notes that “[t]he groundwork in analysis and calculus with which the reader is assumed to be acquainted is, roughly, what is in Hardy’s A Course of Pure Mathematics.” Of course, the latter is another classic of the genre, and any kid going this route and sticking to it will soon find himself, well, doing real analysis! This despite Burkill’s opening claim in his preface that “[his] aim is to give an account of the theory of integration due to Lebesgue in a form which may appeal to those who have no wish to plumb the depths of the theory of real functions.”
Then, in telepathic response to my earlier misgivings about finishing my course the way I did, Burkill goes on to say in his preface that “[s]ince Lebesgue’s original exposition [which B. follows pretty closely] a number of different approaches to the theory have been discovered, some of them having attractions of simplicity or generality. It is possible to arrive quickly at the integral without any stress on the idea of measure. I believe, however, that there is an ultimate gain in the knowing the outline of the theory of measure, and I have developed this first in as intuitive a way as possible.”
So there it is. After 40 pages leading the reader through the construction of the indicated measures and then the Lebesgue integral, Burkill spends the next half (about another 40 pages) of the book on the themes of differentiation vis à vis integration, with the important theme of functions of bounded variation appearing on p. 50; special themes concerning integral, such as Fubini’s theorem, Lp-spaces, &c.; and finally the Lebesgue-Stieltjes integral.
The Lebesgue Integral is also equipped with wonderful exercise sets at the end of each chapter. They vary from pretty and accessible to pretty darn sporty. I have vivid memories of cutting my philosophy class so as to continue working on one of Burkill’s problems, having just gotten a glimpse of the right approach. Finding the proof was one of the highlights of my year, and then some: I have never forgotten the problem, which I could only solve after restating it a couple of times and then going at it indirectly. The challenge was irresistible and obviously left me with a deep pedagogical impression. For Burkill’s book this is the rule, rather than the exception!
One criticism (and one only): the book is dated in that Burkill uses + and • for, respectively, set theoretic union and intersection, which takes a little getting used to. But otherwise, as I have tried to make abundantly clear already, the book is fantastic. A semester, devoted to teaching gifted undergraduates from this text, would be a true pedagogical treat; or maybe a senior seminar, or a reading course: there are definite possibilities. But, whatever the case may be, for the right undergraduate, this book cannot be surpassed.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.