Duality for Borel measurable cost functions

We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if c : X × Y → [ 0 , ∞ ) c:X× Y→ [0,∞ ) is an arbitrary Borel measurable cost function on the product of Polish spaces X , Y X,Y . In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a “subsidy” function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.