World Library  
Flag as Inappropriate
Email this Article

Choquet theory

Article Id: WHEBN0003655598
Reproduction Date:

Title: Choquet theory  
Author: World Heritage Encyclopedia
Language: English
Subject: Krein–Milman theorem, List of convexity topics, List of mathematical theories, State (functional analysis), Extreme point
Collection: Convex Hulls, Functional Analysis, Integral Representations
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Choquet theory

In mathematics, Choquet theory is an area of functional analysis and convex analysis created by Gustave Choquet. It is concerned with measures with support on the extreme points of a convex set C. Roughly speaking, all vectors of C should appear as 'averages' of extreme points, a concept made more precise by the idea of convex combinations replaced by integrals taken over the set E of extreme points. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics.

The two ends of a line segment determine the points in between: in vector terms the segment from v to w consists of the λv + (1 − λ)w with 0 ≤ λ ≤ 1. The classical result of Hermann Minkowski says that in Euclidean space, a bounded, closed convex set C is the convex hull of its extreme point set E, so that any c in C is a (finite) convex combination of points e of E. Here E may be a finite or an infinite set. In vector terms, by assigning non-negative weights w(e) to the e in E, almost all 0, we can represent any c in C as

c = \sum_{e\in E} w(e) e\

with

\sum_{e\in E} w(e) = 1.\

In any case the w(e) give a probability measure supported on a finite subset of E. For any affine function f on C, its value at the point c is

f (c) = \int f(e) d w(e).

In the infinite dimensional setting, one would like to make a similar statement.

Choquet's theorem states that for a compact convex subset C of a normed space V, given c in C there exists a probability measure w supported on the set E of extreme points of C such that, for any affine function f on C,

f (c) = \int f(e) d w(e).

In practice V will be a Banach space. The original Krein–Milman theorem follows from Choquet's result. Another corollary is the Riesz representation theorem for states on the continuous functions on a metrizable compact Hausdorff space.

More generally, for V a locally convex topological vector space, the Choquet-Bishop-de Leeuw theorem[1] gives the same formal statement.

In addition to the existence of a probability measure supported on the extreme boundary that represent a given point c, one might also consider the uniqueness of such measures. It is easy to see that uniqueness does not hold even in the finite dimensional setting. One can take, for counterexamples, the convex set to be a cube or a ball in R3. Uniqueness does hold, however, when the convex set is a finite dimensional simplex. So that the weights w(e) are unique. A finite dimensional simplex is a special case of a Choquet simplex. Any point in a Choquet simplex is represented by a unique probability measure on the extreme points.

See also

Notes

  1. ^ Errett Bishop; Karl de Leeuw. "The representations of linear functionals by measures on sets of extreme points". Annales de l'institut Fourier, 9 (1959), p. 305-331.

References

  • Asimow, L.; Ellis, A. J. (1980). Convexity theory and its applications in functional analysis. London Mathematical Society Monographs 16. London-New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+266.  
  • Bourgin, Richard D. (1983). Geometric aspects of convex sets with the Radon-Nikodým property. Lecture Notes in Mathematics 993. Berlin: Springer-Verlag. pp. xii+474.  
  •  
  • Hazewinkel, Michiel, ed. (2001), "Choquet simplex",  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.