Proof of Lemma A.1 By Lemma 2, where Φ$_{c}$ is the set of all pairs ( ϕ, ψ ) in L $^{1}$( dF$_{0}$ ) × L $^{1}$( dF$_{1}$ ) such that Note that Φ$_{c}$ is a convex set. From the definition of A + k ( f, s ) and A − k ( f, s ) , for any function f :$_{R}$ →$_{R}$ and s ∈ (0 , 1] , as illustrated in Figure A.1. Let ϕ$_{+}$ ( x ) = max { ϕ ( x ) , 0 } ≥ 0 , ϕ$_{−}$ ( x ) = min { ϕ ( x ) , 0 } ≤ 0 . 55 Figure A.1: Monotonicity of $^{{}$A + k ( f, s ) } ∞ k =0 and $^{{}$A − k ( f, s ) } ∞ k =0 inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) I [ π ] = sup ( ϕ,ψ ) ∈ Φ$_{c}$ J ( ϕ, ψ ) , ϕ ( y$_{0}$ ) + ψ ( y$_{1}$ ) ≤ 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) for all ( y$_{0}$, y$_{1}$ ) . (A.1) . . . ⊆ A + 1 ( f, s ) ⊆ A + 0 ( f, s ) ⊆ $^{(}$A − 0 ( f, s ) c ⊆ $^{(}$A − 1 ( f, s ) ) c ⊆ . . . ., (A.2)