18 1. The Kantorovich Duality on the nonempty, convex set Π(μ, ν), defined as the set of all probability measures on X × Y with marginals μ on X and ν on Y . More explicitly, π Π(μ, ν) if and only if π is a nonnegative measure satisfying (1.1) π[A × Y ] = μ[A], π[X × B] = ν[B], for all measurable subsets A of X and B of Y . Note that this definition forces π to be a probability measure. Equivalently, π Π(μ, ν) if and only if it is a nonnegative measure on X ×Y such that, for all measurable functions (ϕ, ψ) L1(dμ) × L1(dν), or equivalently L∞(dμ) × L∞(dν), (1.2) X×Y [ϕ(x) + ψ(y)] dπ(x, y) = X ϕ + Y ψ dν. It is often convenient to use a narrower class of test functions in (1.2), but this can be done only under some topological assumptions on the mea- sure spaces (X, μ) and (Y, ν). Recall that a Borel probability measure is a probability measure defined on the Borel σ-algebra of some topological space, i.e. the σ-algebra generated by open sets. In the sequel, we shall only consider such probability measures, and P (X) will stand for the set of Borel probability measures on X. When X and Y are Polish spaces (i.e. complete separable metric spaces), and μ, ν are Borel probability measures, it is sufficient to impose (1.2) for (ϕ, ψ) Cb(X) × Cb(Y ) only. When in addition X and Y are locally compact, i.e. each point admits a com- pact neighborhood, then one can even be content with imposing (1.2) for (ϕ, ψ) C0(X) × C0(Y ). Recall that Cb(X) denotes the space of bounded continuous functions on X, and C0(X) the space of continuous functions going to 0 at infinity, i.e. those continuous functions ϕ such that for any ε 0 there is a compact set X satisfying supx/Kε |ϕ(x)| ε note that C0(X) Cb(X). This possibility to restrict the class of test functions to the narrower space C0 when X and Y are locally compact is due to Riesz’ theorem [217, p. 40], which identifies the space M(X) of Borel measures having finite total variation on X with the topological dual of C0(X). The basic examples one should keep in mind to appreciate the interest of these definitions are the following. Of course Rn is Polish and locally compact, and so is any finite-dimensional complete Riemannian manifold. On the other hand, Cb([0, 1] R) = C([0, 1] R), endowed with the supremum distance, and P (Rn), endowed with the weak topology of measures, are Polish spaces, but not locally compact. Both C([0, 1] R) and P (Rn) are typical examples of spaces in which one is led to use mass transportation when studying problems of statistical mechanics. Exercise 1.1. Let K be a compact of C([0, 1] R) show that K has empty interior (of course this a particular case of a well-known theorem: the unit ball of a normed vector space X is not compact if X has infinite dimension
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