General Theory of Functions and Integrations

At the risk of shedding all pretense that I am anything but a curmudgeon, let me start off this review of Angus E. Taylor’s General Theory of Functions and Integration by saying that “they don’t make ’em like that anymore!” In fact, they weren’t even making them like that when I was a UCLA student in the 1970s, given that Angus Taylor, a UCLA stalwart and classic, was no longer in evidence there: he had chaired the department at UCLA in the 1960s and by the time I got there had moved on to his chancellorship at UC Santa Cruz. He had been a major force in building UCLA’s department up in the preceding decades, and there was no doubt that analysis was very powerfully represented.

By the time I got there as an undergraduate, the baton had been passed to the next generation. Arens and Johnny Green were still going strong, but the seminars were being run by Garnett and Koosis: what Taylor and his fellows had started and nourished had undergone at least one evolutionary jump. The analysis texts used in the advanced courses were at this stage characterized by corresponding evolutionary features that could probably be traced to the post-Sputnik boon in mathematics education that had just swept the country in the preceding decade or so. More substantial prerequisites could be assumed as gifted kids swarmed into mathematics departments with measurably better preparation than even their older siblings had had a decade earlier.

Taylor’s General Theory of Functions and Integration is an “older sibling book,” then, in the sense that it reads as very old-fashioned: the material between the book’s covers goes back to lectures Taylor gave in the early 1950s and the book was written in 1960–61, after all. But I claim that it’s precisely this old-fashioned style and perhaps somewhat anachronistic approach that makes this book exceptionally valuable today, for curmudgeonly reasons.

The obvious trouble we face in higher mathematical education today is that our charges (are we on generation Z yet?) are given to communicate and, possibly, to think and reason in ways that don’t exactly jive with ours. Yes, to be sure, the evidence of this disconnect is largely anecdotal. But there are so many anecdotes at the disposal of all of us of a certain age who stand in front of a class on a regular basis that to deny the unpleasant reality is just plain impossible. Today’s mathematical youths are largely unable to deal with Rudin, blue or green, baby or papa, partly because their pre-Rudin training has been staccato: undergraduate real analysis is now largely advanced calculus with some epsilons and deltas thrown in, and the serious results that should provide continuity and analytic ability, if I may be forgiven an egregious pun, are given in sound bites. To be blunt, what took a semester in 1975 now takes more than a year, and the dozen or so upper division students whining in today’s classroom would have been utterly reduced to tears in their first lower division course back in the old days… except of course for the (very) proper subset of truly talented kids for whom it’s all candy, just as it was in those old halcyon days.

All right, then, with my spleen appropriately vented, we still have a population of majors and graduate students to educate, different beasties than we were, so the question of how best to serve them, as far as a good real analysis and measure theory textbook is concerned, is crucial. Indeed, we owe something to especially the ones who really do want to go all the way, the kids who’d be in the desks next to us in the old days, replete with long hair, tie-dyed T-shirts, jeans and sandals, and pencil and paper (no infernal machines like the PC I’m using to write this and no cell-phones like the one on my desk next to me). It’s this niche Taylor’s book fills to a tee.

The book’s first three chapters contain, roughly speaking, all the analysis situs and topology of the reals (or Cartesian products of such) needed for advanced undergraduates and beginning graduate students, done in a thoroughgoing but effective style. Taylor provides motivation, gives examples, and adds a lot of problems (of the right degrees of difficulty) to his chapters, and the prose is all very readable: if the reader sits down and works through the material, success — a true knowledge of real analysis — will result. In fact, Taylor’s coverage is so thorough that what is now generally postponed to topology proper (most often occurring first in graduate school) is dealt with very effectively in the third chapter of General Theory of Functions and Integration.

The centrally important third chapter closes with a very suggestive quartette of sections forming a conduit to the fourth chapter and therewith the book’s main thrust. Specifically, with his coverage of vector spaces, normed linear spaces, Hilbert spaces, and spaces of continuous functions, Taylor opens the door to a very, very careful and ramified treatment of the theory of integration, spread over some 240 pages. Both Lebesgue and Daniell integration are covered in great detail, and the book’s closing chapters deal with iteration of integrals, signed measures, and, as what is surely Taylor’s pitch to future hard analysts, a careful closing discussion of “functions of one real variable.” In this last chapter the Stieltjes integral makes an appearance.

I think this book is a great introduction to serious (real) analysis for serious students, and its reappearance on the mathematical scene is particularly well-timed given what we’re dealing with today in the way of hugely disparate levels of preparation among incoming graduate students.

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