Fr\'echet barycenters in the Monge-Kantorovich spaces

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 $J$.