A simple proof in Monge–Kantorovich duality theory
A simple proof is given of a Monge–Kantorovich duality theorem for a lower bounded lower semicontinuous cost function on the product of two completely regular spaces. The proof uses only the Hahn–Banach theorem and some properties of Radon measures, and allows the case of a bounded continuous cost function on a product of completely regular spaces to be treated directly, without the need to consider intermediate cases. Duality for a semicontinuous cost function is then deduced via the use of an approximating net. The duality result on completely regular spaces also allows us to extend to arbitrary metric spaces a well known duality theorem on Polish spaces, at the same time simplifying the proof. A deep investigation by Kellerer [Z. Warsch. Verw. Gebiete 67 (1984)] yielded a wide range of conditions sufficient for duality to hold. The more limited aims of the present paper make possible simpler, very direct, proofs which also offer an alternative to some recent accounts of duality.