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 $\mu$ and $\nu$. The transport cost function $c:X\times Y \to [0,\infty]$ 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 $1-\varepsilon$ from $(X,\mu)$ to $(Y, \nu)$, as $\varepsilon >0$ 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.