Learning measure theory in the context of ergodic theory seems like a pretty good idea. When the measure theory starts, as it does in Invitation to Ergodic Theory, on the real line and then is extended in a natural way to multidimensional Euclidean space, then the measure-preserving transformations of ergodic theory seem very natural indeed. A student seeing measure theory for the first time this way, with the standard theorems, gets a far more down-to-earth view than those of us who learned, à la Rudin, in locally compact Hausdorff spaces. It might amount to the same thing, but context really does make a difference.
This book aims to introduce the basic topic of ergodic theory — recurrence, ergodicity, the ergodic theorem, mixing and weak mixing — without assuming any background in measure theory. The author provides a detailed development of Lebesgue measure and an introduction to measure spaces that goes as far as the Carathéodory extension theorem. He also carefully develops the Lebesgue theory of integration up to the dominated convergence theorem and Lp spaces.
Topics in the book alternate between Lebesgue theory and ergodic theory. We begin with measure theory, go on to recurrence and ergodicity, return to take up the Lebesgue integral, and then move on to prove a version of the ergodic theorem and explore various notions of mixing. There are a number of good examples of dynamical systems, many of them standard: the baker’s transformation, irrational rotations, the Bernoulli doubling map, the Kakutani-von Neumann dyadic odometer, as well as the Gauss and Chacón transformations.
The writing is crisp and clear. Proofs are written carefully with adequate levels of detail. Exercises are plentiful and well-integrated with the text. This is a book written with undergraduates in mind, for use in a capstone or special topics course. The primary prerequisite is a real analysis course that includes basic topology on the real line including compactness properties and the notions of open and closed sets.
Nonetheless, this would be a challenging text for most undergraduates; it demands maturity and tenacity as well as the skills to follow detailed analytical arguments and then to create such arguments on one’s own.
Although the author declares it “out of scope,” I would have wished to see more of the context and history of ergodic theory presented to the student. In this book, as in many others of the Student Mathematical Library series, there is a tendency to immerse the student immediately in technical details without much in the way of historical background or context. This is unfortunate. In the case of this book, ergodic theory has a rich history and deep connections to physics via statistical mechanics. Surely, that proud history is worth a little bit of attention.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.