We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if c : X × Y → [0,∞) is an arbitrary Borel measurable cost function on the product of Polish spaces X,Y . In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a “subsidy” function and how to identify t...

In the paper [29], N. Weaver introduced the notion of measurable metric and that of Lipschitz function with respect to such a metric. The study of these notions was pursued by the same author in subsequent papers ([30, 32, · · ·]) and in the book [31]. In our paper [16], we treated various important examples and, in particular, we studied the intrinsic measurable metric associated with a local ...

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In this article we introduce some definitions concerning measurable functions and prove related properties. In this paper k is a natural number, r is a real number, i is an integer, and q is a rational number. The subset Z − of R is defined by: (Def. 1) r ∈ Z − iff there exists k such that r = −k. Let us note that Z − is non empty. The following three propositions are true: (1) N ≈ Z −. Z is a ...