26 1. The Kantorovich Duality (μk) in P one can extract a subsequence, still denoted (μk), and a probability measure μ∗ on X, such that for any ϕ Cb(X), lim k→∞ X ϕ dμk = X ϕ dμ∗. See [41] for a proof this statement can also be generalized outside the setting of Polish spaces. 4. If X is a metric space and F is a nonnegative lower semi-continuous function on X, then it can be written as the supremum of an increasing sequence of uniformly continuous nonnegative functions. To see this, just choose (1.11) Fn(x) = inf y∈X F (y) + nd(x, y) , where d is a metric on X, and check (excellent exercise) that the sequence (Fn) satisfies all the required properties. Note that each Fn is well-defined because F is bounded below. The property of uniform continuity will be useful to check that supΦc∩L1 J = supΦc∩Cb J. Proof of Theorem 1.3. We separate the proof into three steps, by in- creasing order of generality. The minimax principle will only be applied in the first step, which is the case when X and Y are compact and c is contin- uous. All the rest of the proof will consist in showing that this particular case implies the general statement, by approximation arguments. 1. Let us first assume that X, Y are compact and that c is continuous on X × Y . Let E = Cb(X × Y ) be the set of all (bounded) continuous functions on X × Y , equipped with its usual supremum norm ·∞. By Riesz’ theorem, its topological dual can be identified with the space of (regular) Radon measures, E∗ = M(X × Y ), normed by total variation. Moreover, a nonnegative linear form is defined by a regular nonnegative (i.e. Borel) measure. Then we introduce Θ : u Cb(X × Y ) −→ 0 if u(x, y) −c(x, y), +∞ else, Ξ : u Cb(X × Y ) −→ X ϕ + Y ψ if u(x, y) = ϕ(x) + ψ(y), +∞ else.
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