1.1. General duality 33 of Π(μ, ν). Indeed, let (πk) be a minimizing sequence for I, and let π∗ be any weak cluster point of (πk) then, by invoking the monotone convergence theorem for the increasing sequence (cn), we have I[π∗] = lim n→∞ In[π∗] ≤ lim n→∞ lim sup k→∞ In[πk] ≤ lim sup k→∞ I[πk] = inf I. Remark 1.12 (c-concave functions). It follows from the proof that, when c is bounded, one can restrict the supremum in the right-hand side of (1.4) to those pairs (ϕcc,ϕc), where ϕ is bounded and (1.18) ϕc(y) = inf x∈X [c(x, y) − ϕ(x)], ϕcc(x) = inf y∈Y [c(x, y) − ϕc(y)]. An easy argument shows that (ϕcc)c = ϕc (see Exercise 2.35). The pair (ϕcc,ϕc) is called a pair of conjugate c-concave functions. Note that ϕc is measurable, since it can be written (exercise) as lim →∞ ψ , where ψ (y) = inf x∈X [c (x, y) − ϕ(x)], and c is an increasing family of bounded uniformly continuous functions converging pointwise to c. Indeed, each ψ is uniformly continuous, and therefore ϕc is measurable. Similarly, ϕcc is measurable. One can give an alternative derivation of the Kantorovich duality via the study of c-concave functions, see Remark 2.40. Although this notion is elementary, we prefer to develop it only in Chapter 2, after some reminders about the classical duality of convex functions. Remark 1.13 (Estimates for bounded cost functions). In the case when c is bounded, it is sometimes useful to know that the supremum can be further restricted: sup J(ϕ, ψ) (ϕ, ψ) ∈ Φc = sup J(ϕ, ψ) (ϕ, ψ) ∈ Φc, 0 ≤ ϕ ≤ c ∞, −c ∞ ≤ ψ ≤ 0 . This is a consequence of Remark 1.12 and the following estimates on pairs of conjugate c-concave functions: (1.19) − sup ϕ ≤ ϕc ≤ c ∞ − sup ϕ, − sup ϕc ≤ ϕ = ϕcc ≤ c ∞ − sup ϕc. Since J(ϕ + s, ψ − s) = J(ϕ, ψ) for all s ∈ R, and (ϕ + s)c = ϕc − s, we can assume without loss of generality that sup ϕ = c ∞. Then it follows from (1.19) that −c ∞ ≤ ϕc ≤ 0, and this in turns implies inf ϕ ≥ 0.

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