I wonder if it’s still the case that graduate analysis means Rudin or Royden. Probably not. So many books have flooded the field in recent decades, including a number of fine efforts aiming at serving the changing needs of beginning graduate students in mathematics, that professors’ choices are no longer as clear-cut as they once were. Under these present circumstances there is a choice to be made whether to jump on the bandwagon and go for recent creations by in-the-trenches colleagues, or to do something reactionary and reach back across the years, perhaps not to a Rudin or Royden, but to something somewhat more off the beaten track: these are different times, after all. Sterling Berberian’s Measure and Integration is such a choice in the area of graduate real analysis, and it’s a fine one.
The book under review is an AMS Chelsea reprint of the 1965 work, and is accordingly eminently affordable and well-made (I much prefer hardcover books to softcovers, for one thing, and the print is exceptionally clear). It is distinguished from today’s efforts as a matter of historical context: 1965 was a different world, after all. And it is distinguished from Rudin and Royden for other reasons, which I think are more important. Briefly put, Berberian’s approach is consciously and explicitly in step with Halmos; says Berberian: “I am deeply indebted to Paul R. Halmos, from whose textbook I first studied measure theory; I hope that these pages may reflect their debt to this book without seeming to be almost everywhere equal to it.” (A nicely placed pun, eh? Bodes well …)
It is indeed the case that leafing through the pages of Berberian’s Measure and Integration one is unmistakably reminded of any and all of Halmos’ fine, fine texts, particularly his Measure Theory. Not only is the layout the same, but the authors’ styles and approaches jive. As a random test, for example, one finds that Berberian’s rendering of Radon-Nikodym on his p. 167 is very close in language to that of Halmos on his p.128 (well, it’s Radon-Nikodym, so there can’t be a huge variation, of course). But when one looks closer he finds a substantial difference in presentations: Berberian states immediately before said theorem that it’s all because of Jordan-Hahn decomposition and a “preliminary Radon-Nikodym theorem” (three sections back), and then, two pages later, we hit the Riesz Representation. Halmos doesn’t give us such a motivational prelude and doesn’t say a thing about Riesz anywhere in his (biggish) book.
Well, what does this mean? I think the answer is very simple and very important: Berberian aims at a more holistic coverage of the material and shoots for pedagogical connections and consequences, while Halmos’ text is more along the lines of a monograph on a serious part of mathematics, set in the graduate curriculum in a different location, really: Halmos hints that this work of his is more along the lines of a springboard to creative work than something meant for a younger graduate student. So, I would not start off with Halmos’ book, but would certainly cover Berberian’s book very thoroughly as both a preliminary and, at least in some ways, an alternative. In other words, when it comes to rookie grad students in analysis, Berberian’s book is far more user-friendly, and is so obviously by the author’s intention.
Thus, even in today’s market, the book under review has an awful lot going for it. Its pages are not covered with essays, but, instead, are chock-full of definitions, theorems, proofs, corollaries, etc.; however, the author presents a lot of motivational snippets as he hits something novel or sticky: a crisp and to-the-point approach I like a great deal. Berberian advertises another aspect of his approach thus: “Throughout the book, I have indulged my fondness for order theoretic arguments by casting the Monotone Convergence Theorem in the leading role.” Additionally he notes that “none of the theorems require the underlying space itself to be a measurable set; that is, we deal exclusively with σ-rings rather than σ-algebras (…) Measurability of the underlying space is dispensed with by requiring each measurable function to vanish outside some measurable set, and by systematically employing locally measurable sets, that is, sets whose intersection with every measurable set is measurable.” (Would’ve made Carathéodory happy, no?)
Beyond this, Measure and Integration naturally (and intentionally) splits into two parts: the first seven chapters (Part I) “deals with the general theory of measure and integration over abstract measure spaces,” and the last two chapters (Part II) deal with “locally compact topological spaces and topological groups,” taking the material from Part I and “put[ting] it to work.” It is also the case that Berberian structured his textbook to be “self-contained, and independent of the exercises.” Thus the book can be read in (at least) two ways: in a more or less cursory manner, or in a thorough fashion with all the i’s dotted and t’s crossed (and the exercises done). The author even included “[s]tarred exercises contain[ing] questions I asked myself and am wiling to confess I could not answer.” (He goes on to say: “…some of these are probably trivial, but others may be interesting questions for research.” It makes one wonder what has transpired in the interim since 1965.)
I think this is a really good book, wonderfully useful today. As some one whose graduate school experience with real analysis could have been a lot less painful, I wish that I had had this book in my possession at that time.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.