1.1. General duality 25 In other words, there exist w∗ ∈ E∗ and α ∈ R, (w∗,α) = (0, 0), such that w∗,x + αλ ≥ w∗,y + αμ, as soon as λ Θ(x) and μ ≤ m − Ξ(y). As one can easily check, this is possible only if α 0 thus, with z∗ = w∗/α we have z∗,x + λ ≥ z∗,y + μ, in particular z∗,x + Θ(x) ≥ z∗,y + m − Ξ(y). Since this holds true for all x and y in E, (1.10) is proven. 1.1.7. Proof of the Kantorovich duality. First of all, let us list the basic properties of Polish spaces which will underlie the proof. This is just for the convenience of readers who would try to adapt the argument to a more general setting, or for those who would need some reminders about Polish spaces. A reader who is not in one of these situations may safely skip this bit. 1. First, a Borel probability measure μ on a Polish space X is automat- ically regular [41], which means that for any Borel set A, one has μ[A] = sup μ[K] K compact, K ⊂ A = inf μ[O] O open, A ⊂ O . Regularity appears naturally when one invokes the Riesz theorem [217, p. 40], which, roughly speaking, asserts the equivalence between nonneg- ative linear functionals on C0(X) and regular nonnegative measures with bounded mass when X is a locally compact Hausdorff topological space. 2. A probability measure μ on a Polish space is automatically concen- trated on a σ-compact set [41]: there exists a measurable set S, which can be written as the union of countably many compact sets, such that μ[S] = 1. Equivalently (exercise), μ is tight, which means that for any ε 0 there exists a compact Kε such that μ[Kε] c ≤ ε. This result is known as Ulam’s lemma. 3. A family P of probability measures on a topological space X is said to be tight if for any ε 0 there exists a compact set Kε ⊂ X for which sup μ∈P μ[Kε] c ≤ ε. Let X be a Polish space Prokhorov’s theorem asserts that any tight fam- ily in P (X) is relatively sequentially compact in P (X): from any sequence

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