Measure and Integration

In the Preface to the book under review Dietmar Salamon states that his book is based in part on Chapters 1–8 of Rudin’s Real and Complex Analysis, i.e. what we used to call “Green Rudin” when I was in school on this side of the Atlantic, back in the late 1970s and early 1980s (Salamon is at the ETH in Zürich). Green Rudin is to be differentiated, if you’ll pardon my language, from “Blue Rudin” or “Baby Rudin,” the latter being his Principles of Mathematical Analysis, one of the truly great books in analysis (for gifted undergraduates). Green Rudin used to qualify as probably the nastiest graduate analysis text around, despite its ubiquity and undeniable excellence — it’s fair to say that, at least based on anecdotal evidence, it was (is still?) the favorite graduate analysis text among analysts properly so-called. I think non-analysts might have a different opinion.

Well, besides Rudin, Salamon gives us various other slants on his subject, including material from, e.g., “the five volumes on measure theory by David H. Fremlin,” and various sets of lecture notes by other scholars and even a topology blog. Regarding Fremlin’s herculean effort, here is his own propaganda for the series.

So, to be sure, we’re not in Kansas anymore, and Salamon’s Measure and Integration reflects this. But it’s a very nice book, indeed. The layout is particularly effective: after a discussion of abstract measure theory (\(\sigma\)-algebras and so on), we get Lebesgue followed by Borel, and then a very effective treatment of \(L^p\) spaces, including a discussion of duality (shades of Rudin, to be sure).

Salamon follows this with a discussion of Radon-Nikodým, which “includes a generalized version … for signed measures.” After this he addresses differentiation (and I guess its location as just one chapter among eight illustrates what I tell my students from time to time, namely, that integration is more important than differentiation — at least from some perspectives). Then come product measures (with not only Fubini present, as is to be expected, but also Calderón-Zygmund, i.e. their famous inequality). He caps off the book with what I, as a number theorist, find particularly apt: a chapter (albeit brief) on Haar measure. There are three appendices: Urysohn’s Lemma, the product topology, and the inverse function theorem.

To be sure, much of the book is reminiscent of Green Rudin, and this is a virtue on several counts: it’s very good mathematical pedagogy, of course, and Salamon presents it very clearly. Indeed, his writing is clear and effective throughout, making the book easy to read in that everything is developed very nicely and explicitly. The proofs are laid out very well and are accessible to the careful reader.

This is notoriously difficult stuff to the novice in analysis (but note the recent MAA Press books, Varieties of Integration and The Lebesque Integral for Undergraduates) Salamon does a good job in covering this challenging and non-trivial material carefully and systematically. There are plenty of examples and remarks, as well as sets of exercises which are, when necessary, provided with good hints. The obvious point should be made, and emphasized: these exercises are crucial if one seeks to learn analysis well, and the reader who goes after them with commitment and perseverance will achieve this objective.

I think this is a very good book, and should serve well on this side of the Pond, too: it should make an excellent textbook for at least part of the standard introductory graduate analysis sequence.

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