A general duality theorem for the Monge-Kantorovich transport problem

The duality theory of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y → [0,∞] is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses 1− ε from (X,μ) to (Y, ν), as ε > 0 tends to zero. The classical duality theorems of H. Kellerer, where c is lower semi-continuous or uniformly bounded, quickly follow from these general results.