A simple proof in Monge–Kantorovich duality theory

D. A. Edwards¹

¹ Mathematical Institute 24–29 St Giles' Oxford OX1 3LB, United Kingdom

Studia Mathematica, Tome 200 (2010) no. 1, pp. 67-77

Résumé

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.

Détail

DOI : 10.4064/sm200-1-4

Keywords: simple proof given monge kantorovich duality theorem lower bounded lower semicontinuous cost function product completely regular spaces proof uses only hahn banach theorem properties radon measures allows bounded continuous cost function product completely regular spaces treated directly without consider intermediate cases duality semicontinuous cost function deduced via approximating net duality result completely regular spaces allows extend arbitrary metric spaces known duality theorem polish spaces time simplifying proof deep investigation kellerer warsch verw gebiete yielded wide range conditions sufficient duality limited aims present paper make possible simpler direct proofs which offer alternative recent accounts duality