40 1. The Kantorovich Duality Thus we can write = π + R, where R is a continuous linear functional supported at infinity, in the sense that u C0(X × Y ) =⇒ R, u = 0. As the following exercise shows, some of these R’s may be weird, and we should be careful when handling them. Exercise 1.23 (Be cautious when working in (Cb)∗). We use the same notation as above. A function u Cb(X) is said to admit a limit u(∞) at infinity, if for any ε 0 there exists a compact X such that x / =⇒ |u(x) u(∞)| ε. (i) Use the Hahn-Banach extension theorem to construct a continuous ex- tension of the linear functional “limit at infinity”. Show that this extension is supported at infinity. Note that it attributes a “limit” to any bounded continuous function. (ii) Let μ P (X), ν P (Y ) be Borel probability measures. Construct a continuous linear functional on Cb(X ×Y ), supported at infinity, such that (1.24) ∀(ϕ, ψ) C0(X) × C0(Y ), , ϕ + ψ = X ϕ + Y ψ dν. Here we used the notation ·, · to denote the duality bracket between E and E∗, and the shorthand + ψ)(x, y) = ϕ(x) + ψ(y). Note that, as a consequence of this exercise, equation (1.24) does not at all guarantee that Π(μ, ν)! Hint: Think of something like X lim y→∞ u(x, y) dμ(x) + Y lim x→∞ u(x, y) dν(y). There are representation theorems for Cb ∗, in the form of finitely additive (as opposed to σ-additive) measures, but we prefer to avoid using them. The following two lemmas will be enough to keep us on safe ground. The notation is the same as above we recall that a σ-compact set is a countable union of compact sets. Lemma 1.24 (Decomposition of nonnegative elements in (Cb)∗). Let X and Y be locally compact, σ-compact Polish spaces. Let be a nonnegative linear form on Cb(X × Y ). Then it can be written as π + R, where π is a nonnegative measure and R a nonnegative continuous linear functional supported at infinity. Lemma 1.25 (Marginal condition in (Cb)∗). Let X and Y be locally compact, σ-compact Polish spaces, and let μ P (X), ν P (Y ) be Borel
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