Duality for rectified Cost Functions

It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function c:X×Y→[0,∞] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification cr of the cost c, so that the Monge-Kantorovich duality holds true replacing c by cr. In particular, passing from c to cr only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function cr is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport