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Endow the space $\mathcal{P}(\mathbb{R})$ of probability measures on$\mathbb{R}$ with a transportation cost $J(\mu, \nu)$ generated by atranslation-invariant convex cost function.  For a probability distribution on $\mathcal{P}(\mathbb{R})$ we formulate anotion of average with respect to this transportation cost, called here theFr\'echet barycenter, prove a version of the law of large numbers for Fr\'echetbarycenters, and briefly discuss the structure of $\mathcal{P}(\mathbb{R})$related to the transportation cost $J$.

Fr\'echet Barycenters and a Law of Large Numbers for Measures on the Real Line