Every teacher of mathematical analysis has seen many books on real analysis, first as a student and later as a lecturer. I am not sure whether it would be completely fair to say that the book under review is the best book on real analysis I have ever seen, but it is certainly a good candidate for this position. I am sure I would like to have this book in my suitcase in case I would have to spend several years on a deserted island. The book covers virtually everything (in real analysis) that a teacher can dream of or that a gifted undergraduate or PhD. student might need for his/her studies and further research. The text is a deep self-contained exposition of all important features of real analysis involving just about the right amount of necessary abstraction and side-trips to various fields of application, such as functional analysis, harmonic analysis, function spaces, Sobolev embeddings, interpolation theory, PDEs, potential theory, etc. Moreover, it covers a number of topics, which appear rarely in introductory textbooks but which are absolutely indispensable in modern studies of mathematical analysis. To name just a few, let me mention covering theorems, the Hausdorff measure, the non-increasing rearrangement of a function, the Marcinkiewicz interpolation theorem, Radon measures, the Rademacher theorem, the Calderón-Zygmund decomposition, the Fefferman-Stein inequality, etc. The material is presented in a truly delightful way. Sufficient motivation for the investigation is given, and theorems are illustrated with a plenty of examples throughout the text. Each chapter is endowed with a `Problems and Complements' section, which give the reader plenty of further opportunities to exercise and to notice tiny links between different subjects.

Reviewer:

lp