Lectures on Choquet’s Theorem

Robert R. Phelps (eds.)3540418342, 9783540418344

A well written, readable and easily accessible introduction to “Choquet theory”, which treats the representation of elements of a compact convex set as integral averages over extreme points of the set. The interest in this material arises both from its appealing geometrical nature as well as its extraordinarily wide range of application to areas ranging from approximation theory to ergodic theory. Many of these applications are treated in this book. This second edition is an expanded and updated version of what has become a classic basic reference in the subject.

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Table of contents :
Introduction. The Krein-Milman theorem as an integral representation theorem….Pages 1-8
Application of the Krein-Milman theorem to completely monotonic functions….Pages 9-12
Choquet’s theorem: The metrizable case…..Pages 13-16
The Choquet-Bishop-de Leeuw existence theorem….Pages 17-23
Applications to Rainwater’s and Haydon’s theorems….Pages 25-26
A new setting: The Choquet boundary….Pages 27-33
Applications of the Choquet boundary to resolvents….Pages 35-38
The Choquet boundary for uniform algebras….Pages 39-45
The Choquet boundary and approximation theory….Pages 47-49
Uniqueness of representing measures…..Pages 51-63
Properties of the resultant map….Pages 65-71
Application to invariant and ergodic measures….Pages 73-78
A method for extending the representation theorems: Caps….Pages 79-87
A different method for extending the representation theorems….Pages 88-91
Orderings and dilations of measures….Pages 93-99
Additional Topics….Pages 101-113