Let $V$ be a Banach space considered with its Borel $\sigma$-algebra $\mathcal{B}(V).$ Let $V^*$the dual of $V.$ A probability measure $\mu$ on $(V, \mathcal{B}(V))$ is said to have a barycenter if there exists a point $y \in V,$ denoted by $r(\mu),$ such that $$ L(y)= \int_V L(x) \, \mu(dx) $$ for all $L \in V^* .$ Let $\mathcal{P}(V)$ be the collection of probability measures on $(V, \mathcal{B}(V))$ with the weak* topology. Let $\mathcal{M}(V)$ be the collection those having barycenters.
Is it known whether $\mathcal{M}(V) $ is closed or measurable in $\mathcal{P}(V)?$ Also under what assumptions do we guarantee continuity or measurability of the map $r \colon \mathcal{M}(V) \to V?$