some nonnegative integer t and s ∈ [0 , 1]. Then the LHS in (A.4) is 2 . Moreover, since ϕ ( y$_{0}$ ) + ψ ( y$_{1}$ ) > 1, for ( y$_{0}$, y$_{1}$ ) ∈$_{R}$×$_{R}$ s.t. s + t ≤ ϕ ( y$_{0}$ ) < s + t + 1 and − ( s + t − 1) < ψ ( y$_{1}$ ) , by (A.1) and thus (A.4) is also satisfied from the following: Similarly, it can be proven that (A.4) is also satisfied for other nontrivial cases. Therefore it concludes that each ( ϕ, ψ ) ∈ Φ$_{c}$ can be written as a continuous convex combination of a continuum of pairs of the form Proof of Theorem 1 By Lemma A.1, ( ϕ, ψ ) ∈ Φ$_{c}$ can be represented as a continuous convex combi- nation of a continuum of pairs of the form Since Φ$_{c}$ is a convex set and J ( ϕ, ψ ) = ∫ ϕdF$_{0}$ + ∫ ψdF$_{1}$ is a linear functional, for all ( ϕ, ψ ) ∈ Φ$_{c}$, there exists s ∈ (0 , 1] such that 58 J ( ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s ) − 1$_{A}$ − k ( ϕ,s ) $^{)}$, ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ψ,s ) − 1$_{A}$ − k ( ψ,s ) ) ) ≥ J ( ϕ, ψ ) . (A.5) ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s $_{)}$( y$_{0}$ ) − 1$_{A}$ − k ( ϕ,s $_{)}$( y$_{0}$ ) $^{)}$+ ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ψ,s $_{)}$( y$_{1}$ ) − 1$_{A}$ − k ( ψ,s $_{)}$( y$_{1}$ ) ≤ 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) . ( ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s ) − 1$_{A}$ − k ( ϕ,s ) $^{)}$, ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ψ,s ) − 1$_{A}$ − k ( ψ,s ) ) ) , ( ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ϕ,s ) − 1$_{A}$ − k ( ϕ,s ) $^{)}$, ∞ ∑ k =0 $^{(}$1$_{A}$ + $_{k}$( ψ,s ) − 1$_{A}$ − k ( ψ,s ) ) ) . 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) = ∞ . 1 < ϕ ( y$_{0}$ ) + ψ ( y$_{1}$ ) ≤ 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) , ■ with