1.1. General duality 27 Note that Ξ is well-defined: if ϕ(x) + ψ(y) = ϕ(x) + ψ(y) for all x, y, then ϕ = ϕ + s, ψ = ψ − s, for some s ∈ R, and therefore ϕ dμ + ψ dν = ϕ dμ + ψ dν. The assumptions of Theorem 1.9 are obviously satisfied with z0 ≡ 1, so formula (1.9) holds true. Now, let us compute both sides of (1.9). Obviously, the left-hand side has to be inf X ϕ dμ + Y ψ dν ϕ(x) + ψ(y) ≥ −c(x, y) = − sup J(ϕ, ψ) (ϕ, ψ) ∈ Φc . Next, we compute the Legendre-Fenchel transforms of Θ, Ξ. First, for any π ∈ M(X × Y ), Θ∗(−π) = sup u∈Cb(X×Y ) − u(x, y) dπ(x, y) u(x, y) ≥ −c(x, y) = sup u∈Cb(X×Y ) u(x, y) dπ(x, y) u(x, y) ≤ c(x, y) . • If π is not a nonnegative measure, then there exists a nonpositive function v ∈ Cb(X × Y ) such that v dπ 0. Then, the choice u = λv, with λ → +∞, shows that the supremum is +∞. • On the other hand, if π is nonnegative, then the supremum is clearly c dπ. Thus Θ∗(−π) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c(x, y) dπ(x, y) if π ∈ M+(X × Y ), +∞ else. A similar argument shows that Ξ∗(π) = ⎧ ⎪0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ if ∀(ϕ, ψ) ∈ Cb(X) × Cb(Y ), [ϕ(x) + ψ(y)] dπ(x, y) = ϕ dμ + ψ dν , +∞ else. In other words, Θ∗ and Ξ∗ are the indicator functions of M+(X × Y ) and Π(μ, ν), respectively. Putting everything together and changing signs, we recover inf Π(μ,ν) I[π] = sup Φc∩Cb J(ϕ, ψ). Combining this with (1.5) finishes the proof.

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