  We first develop a theory of conditional expectations for random variables
with values in a complete metric space $M$ equipped with a contractive
barycentric map $\beta$, and then give convergence theorems for martingales of
$\beta$-conditional expectations. We give the Birkhoff ergodic theorem for
$\beta$-values of ergodic empirical measures and provide a description of the
ergodic limit function in terms of the $\beta$-conditional expectation.
Moreover, we prove the continuity property of the ergodic limit function by
finding a complete metric between contractive barycentric maps on the
Wasserstein space of Borel probability measures on $M$. Finally, the large
derivation property of $\beta$-values of i.i.d. empirical measures is obtained
by applying the Sanov large deviation principle.
