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44 1. The Kantorovich Duality 1.4. Appendix: {0, 1}-valued costs and Strassen’s theorem The Kantorovich duality takes a particular form when the cost function c only takes values 0 and 1, i.e. when it is of the form 1C(x, y): Theorem 1.27 (Kantorovich duality for {0, 1}-valued costs). Let X and Y be Polish spaces, μ ∈ P (X), ν ∈ P (Y ), and let C be a nonempty open set in X × Y . Then, inf π∈Π(μ,ν) π[C] = sup μ[A] − ν[AC] A ⊂ X, A closed , where AC := {y ∈ Y ∃x ∈ A, (x, y) / ∈ C}. The following corollary is a famous and useful theorem due to Strassen. To state it, we use the standard notation d(y, A) = inf {d(y, z) z ∈ A}. Corollary 1.28 (Strassen’s theorem). Let X be a Polish space, μ, ν ∈ P (X), and ε ≥ 0. Then inf π∈Π(μ,ν) π {d(x, y) ε} = sup A closed μ[A] − ν[Aε] , where Aε = {y ∈ X d(y, A) ≤ ε}. Note that the case ε = 0 reduces to (14). Dudley [116, Lecture 18] [119, Section 11.6] proves a very slight variation of Strassen’s theorem without the assumption of X being complete his argument is based on the so-called “pairing theorem” (or marriage lemma). Another proof is given in Rachev and R¨ uschendorf [211]. It is in fact possible to prove Strassen’s theorem by direct use of the Hahn-Banach theorem so our goal here is not to provide the simplest proof, but only to show that this theorem can be seen as a particular case of the Kantorovich duality. Remark 1.29. From Strassen’s theorem follows the equality inf {ε 0 for any closed A, μ[A] − ν[Aε] ≤ ε} (1.32) = inf {ε 0 inf P[d(U, V ) ε] ≤ ε}, where the last infimum runs over all random variables U and V with respec- tive laws μ and ν. The expression in (1.32) is a distance metrizing weak topology, called the Prokhorov distance (or sometimes L´ evy distance). Proof of Theorem 1.27. 1. Since C is open, the cost function c(x, y) = 1C(x, y) is lower semi-continuous on X × Y , and can be approximated