In mathematics, Choquet theory is an area of functional analysis and convex analysis created by the French mathematician Gustave Choquet. It concerns 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 topological vector space along lines similar to the finite-dimensional case. The name is for Gustave Choquet, whose main concerns 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

with

In the case of a simplex the weights w(e) are unique, but in other cases such as a ball that is not so. In any case the w(e) give a probability measure supported on a finite subset of E. The idea in Choquet theory is to relax the finite support condition, but retain the probability measure, by replacing the sum by an integral and w by a more general function. Implicit in this is the need to define integrals for vector-valued functions, taking values in V. This will only make good sense if V is at least a topological vector space, and in practice a complete metric space as well so that infinite sums can be made from Cauchy sequences.

Choquet's theorem states that for a compact convex subset C in a normed space V, any c in C is the barycentre of a probability measure supported on the set E of extreme points of C. In practice V will be a Banach space. The original Krein-Milman theorem is a corollary of Choquet's result.

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

[edit] References

• R. R. Phelps (1966, 2nd edition 2001) Lectures on Choquet's Theorem