A generalized dual maximizer for the Monge--Kantorovich transport problem

TL;DR

This paper proves the existence of dual optimizers in the Monge--Kantorovich transport problem under general conditions, using functional analysis and finitely additive measures.

Contribution

It introduces a novel approach to dual attainment by interpreting dual optimizers as projective limits of finitely additive measures, broadening the theoretical understanding.

Findings

Dual optimizers always exist under general conditions.

The approach uses Fenchel's perturbation technique.

Dual solutions are characterized as projective limits of finitely additive measures.

Abstract

The dual attainment of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $\mu$ and $\nu$. The transport cost function $c:\XY \to [0,\infty]$ is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel's perturbation technique.