Lectures on Choquet's theorem
type: Item
instance of: scholarly article
title: Lectures on Choquet's theorem
zbMATH Open document ID: 0997.46005
author: R. R. Phelps
published in: Lecture Notes in Mathematics
publication date: 2001-05-30T00:00:00Z
full work available at URL: t1757.htm
review text: 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.
reviewed by: Tage Bai Andersen
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 46-02
zbMATH DE Number: 1601012
zbMATH Keywords: infinite-dimensional convexity
zbMATH Keywords: Hahn-Banach theorems
zbMATH Keywords: Riesz representation theorem
zbMATH Keywords: Krein-Milman theorem
zbMATH Keywords: Choquet theorem
zbMATH Keywords: extreme points of a compact convex
zbMATH Keywords: Borel set
zbMATH Keywords: Choquet boundary
zbMATH Keywords: resolvents
zbMATH Keywords: ergodic and invariant measures
zbMATH Keywords: uniform algebras
zbMATH Keywords: approximation theory
zbMATH Keywords: Poulsen simplex
MaRDI profile type: Publication
OpenAlex ID: W1561681248
Wikidata QID: Q56602891
DOI: 10.1007/B76887
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description: scientific article; zbMATH DE number 1601012