Berlin: Springer (ISBN 978-3-540-34513-8/hbk). xvii, 500 p./v.1; xiii, 575 p./v.2. EUR 119.95/net; SFR 203.00; $ 159.00; £ 92.50 (2007).

One of the key aspects of graduate instruction in `Real Analysis’ is to produce a $\sigma$-additive set function, or measure, quickly to define and then establish the main Lebesgue-Vitali limit theorems as they are of enormous interest in many applications. The most efficient method now available is the one devised by C. Carathéodory, who produced it after analyzing Lebesgue’s ideas and results. This important and very general approach needs some reflection, which prompted E. Hewitt to express: “how Carathéodory came to think of this definition seems mysterious… (but it) has many useful implications.” However, Dunford-Schwartz and many others make this the central approach to the subject, after putting some light do dispel the `mystery’ by way of outlining Lebesgue’s prior successful efforts in introducing the concept of modern measure and integral as distinct from the Riemann-Darboux approach. The main thrust of the very detailed account of this subject by Bogachev in two volumes making up of approximately 1100 pages, with 2038 references listed, is a scholarly rendering of its many sided view, some highlights of which will be commented on here. The two volumes are nearly evenly divided each having five (long) chapters. The first volume concentrates on the basic construction of measures and the (Lebesgue) integral, the fundamental limit theorems, the Radon-Nikodym and Fubini-Tonelli theorems as well as the $L^p$, $p\ge 1$, spaces and their duality. The last chapter of the first volume contains some specialized material including (the weak or Sobolev-L. Schwartz) differentiation, change of variables, weighted (Hardy-Littlewood type) inequalities and BMO spaces (but not the ($H^1$, BMO) duality). Excluding the last chapter, the material contains a standard graduate course coverage of the subject in the U.S. universities, and is detailed. The author includes many extensions and sidelights in a long section called `Supplements and exercises’ for each chapter. The readers are encouraged to glance through the material, if there is no time, to get at least a feeling on how the use of measure and integral enables new insights to both classical and contemporary applications. There are also brief but useful historical accounts of many of the contributors of the subject. A few glaring omissions (or oversights) are the inadequate emphasis on Carathéodory’s construction of outer measures which would explain the significance as well as a clarification of the concept and its power. Also, the crucial aspect of Fubini’s theorem, amplified by M. H. Stone, now called the Fubini-Stone theorem, does not appear here. It has a significant part to play in product measure and integral construction without $\sigma$-finite restriction. On the other hand, the Lebesgue integral being an absolute one and the Riemann’s nonabsolute, the gap between these two was bridged by the (independent) construction of Henstock and Kurzweil around 1960, explains the subject better and the author has included only a brief account of this matter in Chapter 5. Thus Volume 1 contains an essentially complete treatment of the basic subject that every student should know in starting a possible research career in analysis.
Turning to Volume 2, it may be observed that its five chapters concentrate on the interplay of measure and topology, as well as topological measures. This naturally covers a vast area of mathematical analysis. Thus Chapter 6 is devoted to systematic study of Baire, Borel, analytic or Souslin classes of sets, and includes some measurable selection theorems. With topology one can consider regularity properties of measures, and particularly those which can be determined from their values on compact subsets of given sets. These are Radon measures. Their tightness properties, and the projective systems (and limits) prompted by Kolmogorov, and Fubini-Jessen theory on products of the spaces are treated in Chapter 7. Here probability theory enters crucially. Alternatively, one can consider the integral as a linear functional on a space of bounded (scalar) functions and start from such functionals and then produce measures which is Daniell’s method. The author indicated this also, although this could have been deduced efficiently and quickly from Choquet’s capacity theory which was sketched, but this possibility was not considered. Only in the supplements, Carathéodory metric outer measures, as well as capacity were briefly discussed. Then Chapter 8 takes up weak convergence of sequences or sets of measures, related to weak$^*$-convergence of Banach spaces, which plays an important part in probability theory, and this chapter is devoted to various aspects of the subject. The whole theory of function spaces such as $L^p$ can be replaced by spaces of set functions since measurability questions play a lesser role, and some of this is discussed. Also Bochner-Kolmogorov-Prokhorov-Sazonov theory of projective systems fits in here, and its discussion with some of its analysis is included. Chapter 9 deals with (point) transformations on spaces and their (induced) set mappings leading to translation invariance when the space is a locally compact group (Haar measures) and specializations such as `quasi-invariance’. There are several supplements. In studying regularity of the mappings and measures, conditional probability functions appear significantly, and they are discussed in the final chapter. Since a conditional measure is really a vector measure, the regularity becomes important so as to treat them as ordinary measures in the form of `transition probability functions’. From here on the material leads to more advanced parts of probability theory, including liftings and martingales. These are outlined, to make the discussion understandable, as the true discussion needs many more pages. This description shows that a study of `Real Analysis’ leads to most parts of mathematical analysis, and the material grows without bounds so that an author has to restrain the impulse and cover some parts in depth and some superficially. As the late Prof. Mark Kac used to say that Probability is a major customer of Measure theory, and many new results are motivated by it. If an author looks on probability favorably, as the author does, having already written a good sized volume on “Gaussian measures” (1997;

Zbl 0883.60032)], then measure theory gets a better than a fair treatment in this sense. The general point of view is somewhat along the lines of the reviewer’s book on the subject [“Measure theory and integration” (1987;

Zbl 0619.28001)], second edition (2004;

Zbl 1108.28001), but the author covers more topics. There is a good set of references having a balance of Russian works and the western contributions, a fact which unfortunately is not always the case in books by many authors. The treatment is reader friendly and I would recommend that each graduate real analysis student own both volumes, at least volume 1 for study (the publisher should allow this), and they make a good reference set to keep on ones shelf. Each chapter starts with an appealing quote from a stalwart (a mathematician, a poet, or a literary person). It is therefore appropriate to end this review with another quote from a literary giant (R. Tagore): “Where the words come out of the depths of truth; where the mind is led forward by thee into ever-widening thought and action; into that heaven of freedom”; shall we aspire in this subject.