  Endow the space $\mathcal{P}(\mathbb{R})$ of probability measures on
$\mathbb{R}$ with a transportation cost $J(\mu, \nu)$ generated by a
translation-invariant convex cost function.
  For a probability distribution on $\mathcal{P}(\mathbb{R})$ 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 briefly discuss the structure of $\mathcal{P}(\mathbb{R})$
related to the transportation cost $J$.
