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Short Communications

On the Monge–Kantorovich duality theorem

D. Ramachandran^a, L. Rüschendorf<sup>b

a</sup> California State University, Department of Mathematics and Statistics
^b Institut füur Mathematische Stochastik, Albert-Ludwigs-Universität, Germany

Abstract: The Monge–Kantorovitch duality theorem has a variety of applications in probability theory, statistics, and mathematical economics. There has been extensive work to establish the duality theorem under general conditions. In this paper, by imposing a natural stability requirement on the Monge–Kantorovitch functional, we characterize the probability spaces (called strong duality spaces) which ensure the validity of the duality theorem. We prove that strong duality is equivalent to each one of (i) extension property, (ii) projection property, (iii) the charge extension property, and (iv) perfectness. The resulting characterization enables us to derive many useful properties that such spaces inherit from being perfect.

Keywords: duality theorem, marginals, perfect measure, charge extension, Marczewski function.

Received: 01.04.1999

Language: English

DOI: 10.4213/tvp474

 English version: Theory of Probability and its Applications, 2001, 45:2, 350–356

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