(a) (b) Figure A.3: a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ at the optimum Part 2. The upper bound inf F ∈ Π( F$_{0}$,F$_{1}$ ) ∫ { 1 { y$_{1}$ − y$_{0}$ > δ } + λ ( 1 ( y$_{1}$ < y$_{0}$ )) } dπ = sup ( ϕ,ψ ) ∈ Φ$_{c}$ ∫ ϕ ( y$_{0}$ ) dµ$_{0}$ +$^{∫}$ ψ ( y$_{1}$ ) dµ$_{1}$, 66 the function ϕ is nonincreasing. Note that under Pr ( Y$_{1}$ = Y$_{0}$ ) = 0, Pr ( Y$_{1}$ ≥ Y$_{0}$ ) = Pr ( Y$_{1}$ > Y$_{0}$ ) = 1, and I prove that in the dual representation Proof of Lemma A.2 Since the proof of (ii) is very similar to the proof of (i), I provide only the proof for (i). Suppose not. There exist a and b in R with a < b s.t. f ( a ) > f ( b ). Define V = { x ∈ [ a, b ] ; f ( a ) > f ( x ) } . Since V is a nonempty set with b ∈ V and bounded below by a , V has an infimum x$_{0}$ ∈ [ a, b ] . Since f is continuous, f ( x$_{0}$ ) = f ( a ) . Note that a ≤ x$_{0}$ < b. Pick ε$_{x}$$_{0}$ > 0 satisfying f ( t$_{0}$ ) ≤ f ( t$_{1}$ ) whenever x$_{0}$ ≤ t$_{0}$ < t$_{1}$ < x$_{0}$ + ε$_{x}$$_{0}$. Since x$_{0}$ is an infimum of the set V , there exists t ∈ ( x$_{0}$$_{,}$x$_{0}$ + ε$_{x}$$_{0}$ ) s.t. f ( x$_{0}$ ) > f ( t ) . This is a contradiction. Thus, for any a < b, f ( a ) ≤ f ( b ) . ■ Lemma A.2 (i) Let f : R → R be a continuous function. Suppose that for any x ∈ $_{R}$, there exists ε$_{x}$ > 0 s.t. f ( t$_{0}$ ) ≤ f ( t$_{1}$ ) whenever x ≤ t$_{0}$ < t$_{1}$ < x + ε$_{x}$. Then f is a nondecreasing function in $_{R}$. (ii) If there exists ε$_{x}$ > 0 for any x ∈$_{R}$ s.t. f ( t$_{0}$ ) ≥ f ( t$_{1}$ ) whenever x − ε$_{x}$ ≤ t$_{0}$ < t$_{1}$ < x, then f is a nonincreasing function in$_{R}$. First, I introduce the following lemma, which is useful for deriving the upper bound under MTR.