A general duality theorem for the Monge–Kantorovich transport problem
The duality theory for 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 semicontinuous or uniformly bounded, quickly follow from these general results.