Characterization of Optimal Transport Plans for the Monge-kantorovich-problem

We prove that c-cyclically monotone transport plans π optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always satisfied for finitely valued, lower semi-continuous cost functions. In particular, this yields a positive answer to Problem 2.25 in C. Villani’s book. We emphasize that we do not need any regularity conditions as were imposed in the previous literature. We consider the Monge-Kantorovich optimal transport problem (μ, ν, c) for Borel measures μ, ν on polish spaces X,Y and a lower semi-continuous cost function c : X×Y → R≥0∪{+∞} (see C. Villani’s beautiful book [8] for all necessary details): we define the set of transport plans Π(μ, ν) as the set of probability measures π with marginal μ on X , and marginal ν on Y , respectively. Furthermore, we define Φ(μ, ν) as the set of pairs (φ, ψ) of Borel functions φ : X → R ∪ {−∞} and ψ : Y → R ∪ {−∞} with φ ∈ L(μ) and ψ ∈ L(ν), such that φ(x) + ψ(y) ≤ c(x, y) for all (x, y) ∈ X × Y . The Monge-Kantorovich problem is to minimize the cost