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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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