This is a remarkably comprehensive treatise on modern, as well as classical, measure theory and integration. Volume 1 covers constructions and extensions of measures, the (abstract) Lebesgue integral, Lp-spaces, signed measures, product measures (including infinite products), change of variables and connections between the integral and derivative (covering theorems, the maximal function, functions of bounded variation and absolutely continuous functions). Of course the Radon-Nikodym theorem, convolution and basic facts on the Fourier transform are included. Also, less traditional topics are discussed: uniform integrability, strong convergence of measures and a concise introduction to the Henstock-Kurzweil integral. The core material of volume 1 (500 pages in total) is divided into five chapters and the exposition is presented on about 170 pages.
What makes both volumes exceptional, interesting and extremely valuable are the sections “Supplement and exercises” attached to each chapter. These sections provide important additional material. Let us mention just a few subjects: set-theoretic problems in measure theory, invariant extensions of Lebesgue measure, Whitney’s decomposition, Steiner’s symmetrization, Hausdorff measures, the Brunn-Minkowski inequality, mixed volumes, weak compactness in L1, Hellinger’s integral, additive set functions, density of point sets, differentiation of measures, BMO, the area and coarea formulas, surface measures and the Calderon-Zygmund decomposition. Some of the exercises are marked as problems accessible for individual work of students while others extend the basic exposition and include plenty of material with hints and references. The concluding part “Bibliographical and Historical Comments” offers a rich, detailed and competent picture of the development and the present state of measure and integration theory.
Volume 2 is organised analogously with five chapters, each accompanied by “Supplement and exercises”. The selection of material reflects the research orientation of the author and is written for analysts as well as for probabilists. The arrangement is not necessarily designed for linear reading; individual chapters are high quality detailed surveys on important parts of modern measure theory: Borel, Baire and Souslin sets, topological measure theory, weak convergence of measures, transformation of measures and isomorphisms and conditional measures and conditional expectations. There is again a wealth of material, which cannot be described here in detail. However, what should be mentioned is the collection of 2038 references also representing large contributions from the Russian mathematical school. This is an excellent and impressive monograph, which I can strongly recommended to researchers in analysis and probability, to university teachers as well as to students. I am convinced that this two volume treatise cannot be missing from university libraries and the shelves of mathematicians interested in measure and integration.