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 $(mathcal{Y},mathcal{G}, u)$ be any probability spaces and $c:mathcal{X} imesmathcal{Y} ightarrowmathbb{R}$ a measurable cost function such that $f_1+g_1le cle f_2+g_2$ for some $f_1,,f_2in L_1(mu)$ and $g_1,,g_2in L_1( u)$. Define $alpha(c)=inf_Pint c,dP$ and $alpha^*(c)=sup_Pint c,dP$, where $inf$ and $sup$ are over the probabilities $P$ on $mathcal{F}otimesmathcal{G}$ with marginals $mu$ and $ u$. Some duality theorems for $alpha(c)$ and $alpha^*(c)$, not requiring $mu$ or $ u$ 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 $alpha^*(c)$) provided $c$ is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both $alpha(c)$ and $alpha^*(c)$ if the maps $xmapsto c(x,y)$ and $ymapsto c(x,y)$ are continuous, or if $c$ is bounded and $xmapsto c(x,y)$ 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.