A note on duality theorems in mass transportation

Abstract

The duality theory of the Monge-Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let $(\mathcal{X},\mathcal{F},μ)$ and $(\mathcal{Y},\mathcal{G},ν)$ be any probability spaces and $c:\mathcal{X}\times\mathcal{Y}\rightarrow\mathbb{R}$ a measurable cost function such that $f_1+g_1\le c\le f_2+g_2$ for some $f_1,\,f_2\in L_1(μ)$ and $g_1,\,g_2\in L_1(ν)$. Define $α(c)=\inf_P\int c\,dP$ and $α^*(c)=\sup_P\int c\,dP$, where $\inf$ and $\sup$ are over the probabilities $P$ on $\mathcal{F}\otimes\mathcal{G}$ with marginals $μ$ and $ν$. Some duality theorems for $α(c)$ and $α^*(c)$, not requiring $μ$ or $ν$ to be perfect, are proved. As an example, suppose $\mathcal{X}$ and $\mathcal{Y}$ are metric spaces and $μ$ is separable. Then, duality holds for $α(c)$ (for $α^*(c)$) provided $c$ is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both $α(c)$ and $α^*(c)$ if the maps $x\mapsto c(x,y)$ and $y\mapsto c(x,y)$ are continuous, or if $c$ is bounded and $x\mapsto c(x,y)$ is continuous. This improves the existing results in \cite{RR1995} if $c$ satisfies the quoted conditions and the cardinalities of $\mathcal{X}$ and $\mathcal{Y}$ do not exceed the continuum.