= Barycenter
{title2=$a(b_\mu)=\int a\,d\mu\quad(a\text{ continuous affine})$}
= Barycentre
{synonym}
The <barycenter> of a <probability measure> on a compact convex set is the point satisfying the displayed affine integral identities. It exists by approximating the <measure> by finitely supported <measures> and using compactness of their finite convex combinations. It is unique because continuous affine <functions> separate points. In a Banach-space setting it agrees with the appropriate vector integral when that integral is defined.
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