A note on duality theorems in mass transportation

The duality theory of the Monge-Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let $(\mathcal{X},\mathcal{F},\mu )$ and $(\mathcal{Y},\mathcal{G},\nu )$ be any probability spaces and $c:\mathcal{X}\times \mathcal{Y}\to \mathbb{R}$ a measurable cost function such that $f_1+g_1\le c\le f_2+g_2$ for some $f_1,f_2\in L_1(\mu )$ and $g_1,g_2\in L_1(\nu )$. Define $\alpha (c)={\inf }_P\int cdP$ and ${\alpha }^{\ast }(c)={\sup }_P\int cdP$, where $\inf$ and $\sup$ are over the probabilities $P$ on $\mathcal{F}\otimes \mathcal{G}$ with marginals $\mu$ and $\nu$. Some duality theorems for $\alpha (c)$ and ${\alpha }^{\ast }(c)$, not requiring $\mu$ or $\nu$ to be perfect, are proved. As an example, suppose $\mathcal{X}$ and $\mathcal{Y}$ are metric spaces and $\mu$ is separable. Then, duality holds for $\alpha (c)$ (for ${\alpha }^{\ast }(c)$) provided $c$ is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both $\alpha (c)$ and ${\alpha }^{\ast }(c)$ 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.