Definition:Barycenter (Locally Convex Space)

Definition

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $K \subseteq X$ be a compact convex subset.

Suppose that $K$ is a metrizable subspace of $X$.

Let $\mu$ be a Borel probability measure on $K$.

Then $x \in K$ is the barycenter of $\mu$ if and only if:

$\ds \forall \ell \in X^\ast : \map \ell x = \int_K \map \ell u \rd \map \mu u$

where:

$X^\ast$ is the dual space of $X$

It is also said that $\mu$ represents $x$ (in the weak sense).

Sources

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