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 x Y -> [0, a] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification c (r) of the cost c, so that the Monge-Kantorovich duality holds true replacing c by c (r) . In particular, passing from c to c (r) only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function c (r) 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.