Duality for Borel measurable cost functions

TL;DR

This paper proves duality in the Monge-Kantorovich transport problem for any Borel measurable cost function on Polish spaces, extending the theoretical understanding of optimal transport.

Contribution

It establishes duality for Borel measurable costs in an abstract measure-theoretic setting, including methods to relate non-optimal plans and identify dual optimizers.

Findings

Duality holds for arbitrary Borel measurable costs on Polish spaces.

Introduces a subsidy function to relate non-optimal plans to optimal costs.

Provides examples illustrating limitations of duality relations.

Abstract

We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if $c:X\times Y\to [0,\infty)$ is an arbitrary Borel measurable cost function on the product of Polish spaces $X,Y$. 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.