  We consider the space $\mathcal{P}(X)$ of probability measures on arbitrary
Radon space $X$ endowed with a transportation cost $J(\mu, \nu)$ generated by a
nonnegative continuous cost function. For a probability distribution on
$\mathcal{P}(X)$ we formulate a notion of average with respect to this
transportation cost, called here the Fr\'echet barycenter, prove a version of
the law of large numbers for Fr\'echet barycenters, and discuss the structure
of $\mathcal{P}(X)$ related to the transportation cost $J$.
