1.4. Appendix: {0, 1}-valued costs and Strassen’s theorem 45 pointwise by a nondecreasing sequence (ck) of continuous functions, with 0 ≤ ck ≤ c. As we saw in the proof of Theorem 1.3, inf π∈Π(μ,ν) π[C] = lim k↑∞ inf π∈Π(μ,ν) ck dπ = lim k↑∞ sup X ϕ dμ + Y ψ dν (ϕ, ψ) ∈ Φck . In view of Remark 1.13, for each k we may restrict the supremum to those pairs (ϕ, ψ) in Φck such that 0 ≤ ϕ ≤ 1, −1 ≤ ψ ≤ 0, and ϕ, ψ are upper semi-continuous (as infima of continuous functions, see formula (1.18)) and satisfy the inequality ϕ(x) + ψ(y) ≤ ck(x, y) ≤ c(x, y) for all x, y. We deduce that (1.33) inf π∈Π(μ,ν) π[C] = sup ϕ dμ + ψ dν (ϕ, ψ) ∈ Φc , where Φc is the set of all pairs (ϕ, ψ) in L1(dμ) × L1(dν) such that (1.34) ⎧ ⎪ϕ(x) ⎨ ⎪ ⎩ + ψ(y) ≤ c(x, y) = 1C(x, y) for all (x, y), 0 ≤ ϕ ≤ 1, −1 ≤ ψ ≤ 0, ϕ is upper semi-continuous. Note that Φc is a convex set. 2. We claim that each (ϕ, ψ) ∈ Φc can be represented as a convex combination of pairs of the form (1A, −1B), where A is closed, belonging to Φc (i.e. 1A(x) − 1B(y) ≤ 1C(x, y) for all x, y). Let us postpone the proof of this claim for a while. Since the functional to maximize, J(ϕ, ψ) = ϕ dμ + ψ dν, is linear, we deduce that for all (ϕ, ψ) ∈ Φc, there exists such a pair (1A, −1B) with J(1A, −1B) ≥ J(ϕ, ψ). In particular, the value of the right-hand side in (1.33) is unchanged if one restricts the supremum to pairs of the form (1A, −1B). Once this is proven, Theorem 1.27 will follow: indeed, 1A − 1B ≤ 1C implies that for all y, 1B(y) ≥ sup[1A(x) x∈X − 1C(x, y)] = 1AC (y), which means AC ⊂ B, so μ[A] − ν[B] ≤ μ[A] − ν[AC]. 3. We prove the claim made above, by using the “layer cake rep- resentation”: any measurable mapping u : X → [0, 1] can be written as

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