  We formulate and study an optimal transportation problem with infinitely many
marginals; this is a natural extension of the multi-marginal problem studied by
Gangbo and Swiech (1998). We prove results on the existence, uniqueness and
characterization of the optimizer, which are natural extensions of the results
of Gangbo and Swiech. The proof relies on a relationship between this problem
and the problem of finding barycenters in the Wasserstein space, a connection
first observed for finitely many marginals by Agueh and Carlier (2011).
