On the Monge–Kantorovich duality theorem

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.