Choquet's Theorem

Theorem

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

Let $K$ be a non-empty metrizable compact convex subspace of $X$.

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

Then each $x \in K$ is a barycenter of a Borel probability measure $m_x$ on $K$ such that:

$\map {m_x} {K_e} = 1$

That is:

$\ds \forall \ell \in X^\ast : \map \ell x = \int_{K_e} \map \ell u \rd \map {m_x} u$

where $X^\ast$ is the dual of $X$.

Proof

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Source of Name

This entry was named for Gustave Alfred Arthur Choquet.

Sources

• 2002: Peter D. Lax: Functional Analysis: $13.4$: Choquet's Theorem
• 2017: Manfred Einsiedler and Thomas Ward: Functional Analysis, Spectral Theory, and Applications $8.6.2$: Choquet's Theorem