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