Figure. A.2: A D k for A$_{k}$ = ( a$_{k}$, ∞ ) and A$_{k}$$_{+1}$ = ( a$_{k}$$_{+1}$, ∞ ) Since the function ϕ is nondecreasing in the support of the optimal joint distribution, A$_{k}$ reduces to ( a$_{k}$, ∞ ) with a$_{k}$ ≤ a$_{k}$$_{+1}$ and a$_{k}$ ∈ [ −∞ , ∞ ] where A$_{k}$ = φ for a$_{k}$ = ∞ . By Theorem 1, for each integer k and δ > 0 , Then, F$_{0}$ ( A$_{k}$ ) − F$_{1}$ $^{(}$A D k ) = 0 for a$_{k}$ = ∞ , while F$_{0}$ ( A$_{k}$ ) − F$_{1}$ $^{(}$A D k ) = min { F$_{1}$ ( a$_{k}$ + δ ) , F$_{1}$ ( a$_{k}$$_{+1}$ ) } − F$_{0}$ ( a$_{k}$ ) for a$_{k}$ < ∞ . Therefore, By Theorem 1, Now I show that it is innocuous to assume that a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ for each integer k. Suppose that there exists an integer l s.t. a$_{l}$$_{+1}$ > a$_{l}$ + δ . Consider { ˜ A$_{k}$ } ∞ k = −∞ with ˜ A$_{k}$ = ($_{˜}$ a$_{k}$, ∞ ) as follows: 64 ˜ a$_{k}$ = a$_{k}$ for k ≤ l, ˜ a$_{l}$$_{+1}$ = a$_{l}$ + δ, ˜ a$_{k}$$_{+1}$ = a$_{k}$ for k ≥ l + 1 . F L Δ ( δ ) = sup { A$_{k}$ } ∞ k = −∞ [ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$ ( A$_{k}$ ) − µ$_{1}$ $^{(}$A D k $^{)}$, 0 } ] = sup { a$_{k}$ } ∞ k = −∞ [ ∞ ∑ k = −∞ max { min { F$_{1}$ ( a$_{k}$ + δ ) , F$_{1}$ ( a$_{k}$$_{+1}$ ) } − F$_{0}$ ( a$_{k}$ ) , 0 }$^{]}$ . A D k = { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ > a$_{k}$ s.t. y$_{1}$ − y$_{0}$ ≥ δ } ∪ { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ > a$_{k}$$_{+1}$ s.t. 0 ≤ y$_{1}$ − y$_{0}$ < δ } = ( a$_{k}$ + δ, ∞ ) ∪ ( a$_{k}$$_{+1}$, ∞ ) = (min { a$_{k}$ + δ, a$_{k}$$_{+1}$ } , ∞ )