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.

### 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},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.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11571/1272266`
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