Choquet theory provides a unified approach to integral representations in potential theory, probability, function algebras, operator theory, group representations, and ergodic theory. There are some well-known monographs on the subject, we recall two: R. R. Phelps [Lectures on Choquet’s theorem. Lecture Notes in Mathematics 1757. Berlin: Springer (2001; Zbl 0997.46005)] and E. M. Alfsen [Compact convex sets and boundary integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete 57. Springer (1971; Zbl 0209.42601)].
The book under review differs substantially from all other books on the subject. On more than 700 pages the book provides a comprehensive treatment of integral representation theory and its applications. This book may be used as a textbook for students, as a monograph for researchers, and as a reference book for specialists. It is divided into 14 chapters. Each chapter is supplemented with a variety of exercises with hints and is finished by notes and comments with bibliographical references. Many open problems are formulated in the comments. The book is reader-friendly, self-contained and well-written.
Let us now consider the contents of the book (some part of the text we take from the Introduction). Chapter 1 “Prologue” is devoted to the Korovkin theorem. The authors give the definition of Korovkin closure and state two questions about the structure of this closure. Chapter 2 “Compact convex sets” begins with the traditional preliminary results: convex sets in finite-dimensional spaces, extreme points, Minkowski’s theorem, Carathéodory’s theorem, the Krein-Milman theorem. Then a detailed description of compact convex sets in a locally convex spaces is given.
Chapter 3 “Choquet theory of function spaces” contains the basic notions that determine the subsequent exposition style.
Definition 3.1 (Function spaces). By a function space \(\mathcal H\) on a compact topological space \(K\), we mean a (not necessarily closed) linear subspace of \(C(K)\) containing the constant functions and separating the points of \(K\).
Then the definitions of \(\mathcal H\)-representing measures, of the Choquet boundary \({\text{Ch}_{\mathcal H}(K)}\), of \(\mathcal H\)-exposed points, of \(\mathcal H\)-affine, \(\mathcal H\)-convex and \(\mathcal H\)-concave functions are formulated. The Choquet ordering is defined and considered. The basic integral representation theorems due to G. Choquet and E. Bishop and K. de Leeuw are presented. Measures maximal with respect to the Choquet ordering are investigated and Mokobodzki’s characterization of maximal measures is proved. These abstract results are then applied to a reexamination of Korovkin’s theorems.
The next chapter studies basic properties of affine functions on compact convex sets and characterizations of functions satisfying the barycentric formula. The aim of Chapter 5 is to introduce a hierarchy of Borel sets in topological spaces and to apply these facts to get a link between topological properties of function spaces and compact convex sets. Simplicial function spaces are studied in Chapter 6. In particular, Bauer and Markov simplicial function spaces are considered. Choquet simplices are presented at the end of the chapter. In Chapter 7, the basic concepts are generalized for function cones, the authors focus in particular on ordered compact convex sets.
Chapter 8 “Choquet-like sets” aims to transfer the concept of faces of compact convex sets to the framework of general function spaces. In the next chapter, families of \(\mathcal H\)-extremal and Choquet sets are considered. Topologies and corresponding continuous functions on the set of extreme points are studied. It is shown that maximal measures induce measures on sets of extreme points that are regular with respect to boundary topologies.
In Chapter 10, entitled “Deeper results on function spaces and compact convex sets”, the authors study Shilov and James boundaries, Lazar’s improvement of the Banach-Stone theorem, results on automatic boundedness of affine and convex functions, metrizability of compact convex sets and their open images, and some topological properties of the set of extreme points. Chapter 11 is devoted to continuous and Borel measurable selectors of multivalued mappings defined on compact convex sets. The main results are the Lazar selection theorem, the Michael selection theorem, Talagrand’s theorem on measurable selectors. Chapter 12 is concerned with two methods of constructing new function spaces: products and inverse limits.
In Chapter 13, general results from Choquet theory are applied to potential theory and several of its basic notions are investigated from this perspective. Important function cones and spaces appearing in potential theory are studied in detail, in particular, in connection with various solution methods for the Dirichlet problem. The functional analysis approach makes it possible to provide an interesting interpretation, for instance, of balayage and regular points in terms of representing measures and the Choquet boundary of suitable spaces and cones. The exposition covers potential theory for the Laplace equation and the heat equation as well as a more general setting. The final Chapter 14 presents several applications of the integral representation theorems, such as for doubly stochastic matrices, the Riesz-Herglotz theorem, the Lyapunov theorem on the range of a vector measure, the Stone-Weierstrass theorem, positive-definite functions, and invariant and ergodic measures.
The book has an appendix with the necessary notions and facts from functional analysis, measure theory, topology, descriptive set theory, resolvable sets and Baire-one functions, the Laplace and heat equations, and axiomatic potential theory.
Some questions of Choquet theory and its applications are omitted from the book. In particular, integral representation theorems of sets with the Radon-Nikodým property, Choquet theory in sets of measures, applications of integral representation theory in \(C^*\)-algebras are not considered. The authors give recommendations and references for interested readers.
We recommend this book to everyone interested in functional analysis and its applications.