  We consider probability measures on $\mathbb{R}^{\infty}$ and study optimal
transportation mappings for the case of infinite Kantorovich distance. Our
examples include 1) quasi-product measures, 2) measures with certain symmetric
properties, in particular, exchangeable and stationary measures. We show in the
latter case that existence problem for optimal transportation is closely
related to ergodicity of the target measure. In particular, we prove existence
of the symmetric optimal transportation for a certain class of stationary Gibbs
measures.
