##
**Lectures on Choquet’s theorem.**
*(English)*
Zbl 0997.46005

Lecture Notes in Mathematics. 1757. Berlin: Springer. 124 p. (2001).

One of the masterpieces in the literature of mathematics has been reprinted. Since its appearance in 1966 this book has served as the introduction for many students as a semester course to the part of infinite-dimensional convexity theory that deals with representation of points in a compact convex set as the resultants of probability measures.

It supposes a basic knowledge of locally convex spaces including the various versions of the Hahn-Banach theorems, the Riesz representation theorem and the Krein-Milman theorem.

It starts by seeing the Krein-Milman theorem in the light of representing measures and applies it to characterize the completely monotonic functions on \((0,\propto)\) as the integrals with respect to a positive measure of the functions \(a\to\exp(-ax)\).

The Choquet theorem is proved in the metrizable case. In this case the set of extreme points of a compact convex set form a Borel set, and each point can be represented as the resultant of a probability measure carried by the extreme points. The generalization to the non-metrizable case and its subtleties is clearly presented. Also the Choquet boundary of different function spaces is applied to resolvents, to ergodic and invariant measures, to uniform algebras and a new chapter on approximation theory. The uniqueness problem for representing measures and the theory of simplices are treated. Different types of representations of points are also discussed.

The final chapter is mainly devoted to a non-technical description of various applications and of newer results among which the most astonishing is the result by Lindenstrauss, Olsen and Sternfeld that there is essentially only one metrizable simplex with a dense set of extreme points namely the Poulsen simplex discovered in 1960.

It was and is a pleasure to read this elegant presentation of a beautiful theory.

It supposes a basic knowledge of locally convex spaces including the various versions of the Hahn-Banach theorems, the Riesz representation theorem and the Krein-Milman theorem.

It starts by seeing the Krein-Milman theorem in the light of representing measures and applies it to characterize the completely monotonic functions on \((0,\propto)\) as the integrals with respect to a positive measure of the functions \(a\to\exp(-ax)\).

The Choquet theorem is proved in the metrizable case. In this case the set of extreme points of a compact convex set form a Borel set, and each point can be represented as the resultant of a probability measure carried by the extreme points. The generalization to the non-metrizable case and its subtleties is clearly presented. Also the Choquet boundary of different function spaces is applied to resolvents, to ergodic and invariant measures, to uniform algebras and a new chapter on approximation theory. The uniqueness problem for representing measures and the theory of simplices are treated. Different types of representations of points are also discussed.

The final chapter is mainly devoted to a non-technical description of various applications and of newer results among which the most astonishing is the result by Lindenstrauss, Olsen and Sternfeld that there is essentially only one metrizable simplex with a dense set of extreme points namely the Poulsen simplex discovered in 1960.

It was and is a pleasure to read this elegant presentation of a beautiful theory.

Reviewer: Tage Bai Andersen (Aarhus)

### MathOverflow Questions:

Closed convex hull in infinite dimensions vs. continuous convex combinations### MSC:

46A55 | Convex sets in topological linear spaces; Choquet theory |

46J20 | Ideals, maximal ideals, boundaries |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |