Barycenter of a measure on a Banach space
= Barycenter of a measure on a Banach space
{title2=$b(\mu)=\int x\,d\mu(x)$}
For a <probability measure> whose identity map is strongly measurable with integrable norm, its barycenter is the <Bochner integral> of that map. It is characterized by $\varphi(b(\mu))=\int\varphi(x)\,d\mu(x)$ for every <continuous linear functional>. For a measure on a <weakly compact set> in a <separable Banach space>, its barycenter belongs to the norm-closed <convex hull> of that set.