In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral 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.
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.
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
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.