As we noted in the review of the first volume (on Fourier Analysis) of the Princeton Lectures in Analysis, this series is a result of a radical rethinking of how to introduce graduate students to analysis. It is only after having discussed quite a bit of Fourier Analysis (without the Lebesgue integral!) and Complex Analysis (the topic of volume II), that the authors turn to topics which have often been placed at the beginning of graduate (and even advanced undergraduate) courses in Analysis: measure, integration, and Hilbert spaces.
The book opens with an introduction that seeks to motivate the ideas by discussing Fourier series, limits of continuous functions, the relationship between differentiation and integration, and finally "the problem of measure". It then has chapters on the Lebesgue measure, Lebesgue integral, and the connection between differentiation and integration. Hilbert spaces are introduced next, with L2 as the crucial example. The final three chapters deal with other examples of Hilbert spaces, abstract measure and integration, and the Hausdorff measure.
As one might expect, several of the main results of the first volume are re-examined in this one. This, of course, is one of the features of the authors' approach, which makes it easier to motivate the more abstract material in this book and also highlights the connections between different branches of analysis. The fourth volume, which is to cover functional analysis, distributions, and parts of probability theory, will complete the series.
This volume lives up to the high standard set up by the previous ones.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College, where he occasionally gets to teach some analysis.