Optimal transportation of vector measures

The Monge and Kantorovich problems of optimal transportation of vector measures are studied. The existence of optimal Kantorovich transport plans is proved. We formulate the dual problem and prove the equality of optimal values in the direct and dual problems for lower semicontinuous cost functions. We prove the equality of the infima in the Monge and Kantorovich problems for vector measures under the conditions of Lyapunov's theorem, which guarantees the existence of Monge mappings for vector measures. We also give sufficient conditions under which the optimal Kantorovich plan is generated by a Monge mapping.