Villani, C. (2003). “Topics in Optimal Transportation,” Graduate Studies in Mathematics 58, American Mathematical Society. Williamson, R. C. and T. Downs. (1990). “Probabilistic Arithmetic I: Numerical Methods for Calculating Convolutions and Dependency Bounds,” International Journal of Approximate Reasoning , 4, 89–158. In Appendix A. I provide technical proofs for Theorem 1, Corollary 1 and Corollary 2. Throughout Appendix A, the function ϕ is assumed to be bounded and continuous without loss of generality by Lemma 2. Since the proofs of characterization of F U Δ and F L Δ are very similar, I provide a proof for characterization of F L Δ only. Let for λ = ∞ . To prove Theorem 1, I introduce Lemma A.1: Lemma A.1 For any function f :$_{R}$ →$_{R}$ , s ∈ [0 , 1] , and nonnegative integer k, define A + k and A − k to be level sets of a function f as follows: Then for the following dual problems each ( ϕ, ψ ) ∈ Φ$_{c}$ can be represented as a continuous convex combination of a continuum of pairs of the form 54 ( ∞ ∑ k =0 1$_{A}$ + $_{k}$( ϕ,s ) − ∞ ∑ k =0 1$_{A}$ − k ( ϕ,s $_{)}$, ∞ ∑ k =0 1$_{A}$ + $_{k}$( ψ,s ) − ∞ ∑ k =0 1$_{A}$ − k ( ψ,s ) ) ∈ Φ$_{c}$ inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) I [ π ] = sup ( ϕ,ψ ) ∈ Φ$_{c}$ J ( ϕ, ψ ) , A + k ( f, s ) = { y ∈$_{R}$ ; f ( y ) > s + k } , A − k ( f, s ) = { y ∈$_{R}$ ; f ( y ) ≤ − ( s + k ) } . I [ π ] = ∫ { 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) } dπ, J ( ϕ, ψ ) = ∫ ϕdµ$_{0}$ +$^{∫}$ ψdµ$_{1}$, Proof of Theorem 1 Appendix A