Barycenter of a measure on a Banach space ($b(\mu )=\int xd\mu (x)$)

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 $\phi (b(\mu ))=\int \phi (x)d\mu (x)$ 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.