Choquet's Theorem

Theorem
Let $X$ be a locally convex vector space over $\R$.

Let $K$ be a none-empty metrizable compact convex subset of $X$.

Let $K_e$ be the set of extreme points of $K$.

Then each $u \in K$ can be represented in the weak sense as:
 * $\ds u = \int_{K_e} e \rd \map {m_u} e$

where $m_u$ is a Borel probability measure on $K_e$.

That is:
 * $\ds \forall \ell \in X' : \map \ell u = \int_{K_e} \map \ell e \rd \map {m_u} e$

where $X'$ is the dual of $X$.