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A note on duality theorems in mass transportation. (English) Zbl 07268533
Summary: 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 \(c:\mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}\) a measurable cost function such that \(f_1+g_1\le c\le f_2+g_2\) for some \(f_1,\,f_2\in L_1(\mu )\) and \(g_1,\,g_2\in L_1(\nu )\). Define \(\alpha (c)=\inf_P\int c\,dP\) and \(\alpha^*(c)=\sup_P\int c\,dP\), 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 \(\alpha^*(c)\), 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 \(\alpha^*(c))\) provided \(c\) is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both \(\alpha (c)\) and \(\alpha^*(c)\) 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 Ramachandran and Ruschendorf (Probab Theory Relat Fields 101:311-319, 1995) if \(c\) satisfies the quoted conditions and the cardinalities of \(\mathcal{X}\) and \(\mathcal{Y}\) do not exceed the continuum.
MSC:
60A10 Probabilistic measure theory
60E05 Probability distributions: general theory
28A35 Measures and integrals in product spaces
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