  We introduce a weak notion of barycenter of a probability measure $\mu$ on a
metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference
measure ${\bf m}$. Under the assumption that optimal transport plans are given
by mappings, we prove that our barycenter $B(\mu)$ is well defined; it is a
probability measure on $X$ supported on the set of the usual metric barycenter
points of the given measure $\mu$. The definition uses the canonical embedding
of the metric space $X$ into its Wasserstein space $P(X)$, pushing a given
measure $\mu$ forward to a measure on $P(X)$. We then regularize the measure by
the Wasserstein distance to the reference measure ${\bf m}$, and obtain a
uniquely defined measure on $X$ supported on the barycentric points of $\mu$.
We investigate various properties of $B(\mu)$
