  We study barycenters in the space of probability measures on a Riemannian
manifold, equipped with the Wasserstein metric. Under reasonable assumptions,
we establish absolute continuity of the barycenter of general measures $\Omega
\in P(P(M))$ on Wasserstein space, extending on one hand, results in the
Euclidean case (for barycenters between finitely many measures) of Agueh and
Carlier \cite{ac} to the Riemannian setting, and on the other hand, results in
the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschl\"ager
\cite{c-ems} for barycenters between two measures to the multi-marginal
setting. Our work also extends these results to the case where $\Omega$ is not
finitely supported. As applications, we prove versions of Jensen's inequality
on Wasserstein space and a generalized Brunn-Minkowski inequality for a random
measurable set on a Riemannian manifold.
