Elementary Introduction to the Lebesgue Integral

Steve Krantz writes good mathematics books: he has written many and they do the job well. In the past I’ve used his books in a number of courses I have taught at the junior and senior undergraduate level at my university, and they have always done justice to the material, as well as standing out for their readability. Krantz (actually, I’ve known him for over forty years, so I’ll just call him what I always call him, Steve) is an excellent teacher and knows his audience, and, by transitivity, his books can aid us in at least trying to reach our own audiences in the same way. I am very happy that he’s written a book on the Lebesgue integral.

As Steve makes clear in the Preface to this compact (yeah, yeah, I know…) book, he is interested in presenting “a text on the Lebesgue theory that is accessible to bright undergraduates,” and he goes on to recommend that this book be used after undergraduate analysis. This is right on target. A few years back I taught a course in undergraduate analysis, with a number of very strong students (including some now in graduate school) in the mix. Toward the end, I showed them some things about the Lebesgue integral, e.g. the gem that while the characteristic function of the rationals in the unit interval defies the Riemann integral, it is no match for the Lebesgue; I would have loved to have been able to continue in that vein. But logistics prevailed against such a hope: the natural course for this material, at least at my university, would be the senior seminar, which, coincidentally, I taught soon afterward, but the Lebesgue integral didn’t cross my mind. Had I known about Steve’s book, it might have been a different story. (As it turned out I chose elliptic modular forms as my topic for the seminar, and… well, let’s say I won’t do that again.)

Parenthetically, the book I looked at (pretty carefully, in fact, but not carefully enough) in my own distant undergraduate days was Burkill’s The Lebesgue Integral, and I do have happy memories of working from it. But I think it’s fair to say that it is dated style-wise, and Steve’s book is doubtless a better choice, and for a number of reasons I’ll get into presently.

A few words are in order about what is actually contained in this Elementary Introduction to the Lebesgue Integral. He certainly covers all the bases very effectively, as one expects from an expert analyst like Steve. I guess it is fair to say that the early focus indeed falls on the integral as opposed to the (Lebesgue) measure per se. What I mean by this characterization is that, unlike what Burkill does (whom I’ll choose as my foil for the moment), Steve starts in on Borel sets — going hard — and then develops the Lebesgue integral accordingly, saving the business of (Carathéodory) outer measure till the seventh chapter. I think this is an excellent way to go, perhaps because I do recall chafing at the relative tedium of all the stuff Burkill does with outer measure before he gets to the yoga of the actual Lebesgue integral. Steve’s orchestration is better, as is his pacing.

On the other hand, measurable sets and the notion of measurability are of course a (very) big deal, and after chapter seven it’s off to the races for Steve, too. He indeed does a fabulous job with this material, hitting such topics as Radon-Nikodým, Riesz representation, and Borel-Stieltjes measure. It’s also marvelous to find (on p. 100) a section devoted explicitly to “A Lebesgue measurable set that is not Borel measurable,” where such a beast’s existence is demonstrated by messing about with Cantor’s cardinalities. Later, however, cf. p. 131, the penultimate chapter sports a section that deals with the “Existence of a measurable set that is not Borel,” where, by means of playing with the Cantor-Lebesgue function, this fauna is actually constructed. Obviously any one (e.g. a very lucky undergraduate) who covers this book thoroughly will come away knowing a lot of serious analysis, including minutiae of measure theory — a pretty sticky wicket at times, and, I think it’s fair to say, like spicy food, an acquired taste.

There are exercises galore, and good ones: accessible, and in concert with the ambient material. Selected exercises’ solutions are appended. The book also comes with a useful glossary, and a table of notations. Finally, the book’s final chapter concerns applications to harmonic analysis: very pretty stuff.

It is wonderful to encounter Steve’s style again, and, now, to have this book in the game: I can’t wait until I get another shot at teaching our senior seminar.

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