On the Kantorovich–Rubinstein theorem

Abstract

The Kantorovich–Rubinstein theorem provides a formula for the Wasserstein metric W 1 on the space of regular probability Borel measures on a compact metric space. Dudley and de Acosta generalized the theorem to measures on separable metric spaces. Kellerer, using his own work on Monge–Kantorovich duality, obtained a rapid proof for Radon measures on an arbitrary metric space. The object of the present expository article is to give an account of Kellerer’s generalization of the Kantorovich–Rubinstein theorem, together with related matters. It transpires that a more elementary version of Monge–Kantorovich duality than that used by Kellerer suffices for present purposes. The fundamental relations that provide two characterizations of the Wasserstein metric are obtained directly, without the need for prior demonstration of density or duality theorems. The latter are proved, however, and used in the characterization of optimal measures and functions for the Kantorovich–Rubinstein linear programme. A formula of Dobrushin is proved.