32 1. The Kantorovich Duality Since cn c by construction, it follows that Φcn is a subset of Φc, on which Jn coincides with J, so (1.16) is trivial. Moreover, here it does not matter whether we define Φc and Φcn as subsets of Cb(X) × Cb(Y ), or as subsets of L1(dμ) × L1(dν). Since In is a nondecreasing sequence of functionals, it is also clear that inf In is a nondecreasing sequence, bounded above by inf I. Therefore, we only have to prove that (1.17) lim n→∞ inf π∈Π(μ,ν) In[π] inf π∈Π(μ,ν) I[π]. Next, we note that the set Π(μ, ν) is tight. Indeed, since μ and ν are tight, for any ε 0 there exist compact sets X and Y such that μ[Kε] c ε/2, ν[Lε] c ε/2 then for any π Π(μ, ν), π[(Kε × Lε)c] π[Kε × Y ] + π[X × Lε] = μ[Kε] + ν[Lε] ε. By Prokhorov’s theorem, this implies that Π(μ, ν) is relatively compact for the weak topology. In particular, if (πn)k∈Æ k is any minimizing sequence for the problem inf In[π], then we know that, up to extraction of a subse- quence, πn k converges weakly to some probability measure πn P (X ×Y ) as k ∞, in the sense that for any bounded continuous function θ on X × Y , θ(x, y) dπn(x, k y) −− k→∞ θ(x, y) dπn(x, y). From this we immediately see that πn belongs to Π(μ, ν) and that inf In = lim k→∞ cn dπn k = cn dπn, which shows the existence of a minimizing probability measure πn. Similarly, the sequence (πn)n∈Æ admits a cluster point π∗ by compactness of Π(μ, ν). Whenever n m, one has In[πn] Im[πn]. So, by continuity of Im, lim n→∞ In[πn] lim sup n→∞ Im[πn] Im[π∗]. By monotone convergence, Im[π∗] −→ I[π∗] as m ∞, so lim n→∞ In[πn] lim m→∞ Im[π∗] = I[π∗] inf π∈Π(μ,ν) I[π], which proves (1.17), and concludes the proof of (1.4). 4. To complete the proof of Theorem 1.3, it only remains to check that the infimum is attained. This is again a consequence of the compactness
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