alent to the that of ∞ ∑ k = −∞ $^{{}$F$_{0}$ ( A$_{k}$ ) − F$_{1}$ $^{(}$A D k )} over { A$_{k}$ } ∞ k = −∞ with the following monotonicity condition: Therefore, it follows that where Note that the expression (A.9) can be equivalently written as follows: That is, F$_{0}$ ( A$_{k}$ ) − F$_{1}$ $^{(}$A D k ) ≥ 0 for each integer k at the optimum in the expression (A.9) . This is easily shown by proof by contradiction. Suppose that there exists an integer p s.t. F$_{0}$ ( A$_{p}$ ) − F$_{1}$ $^{(}$A D p ) < 0 at the optimum. If there exists an integer q > p s.t. F$_{0}$ ( A$_{q}$ ) − F$_{1}$ $^{(}$A D q ) > 0 , then there exists another monotonically decreasing sequence of open sets { ˜ A$_{k}$ } ∞ k = −∞ s.t. where ˜ A$_{k}$ = A$_{k}$ for k < p and ˜ A$_{k}$ = A$_{k}$$_{+1}$ for k ≥ p. If there is no integer q > p s.t. F$_{0}$ ( A$_{q}$ ) − F$_{1}$ $^{(}$A D q ) > 0 , then also there exists a monotonically decreasing sequence of open sets { ̂ A$_{k}$ } ∞ k = −∞ s.t. where ̂ A$_{k}$ = A$_{k}$ for k < p and ̂ A$_{k}$ = φ for k ≥ p. This contradicts the optimality of { A$_{k}$ } ∞ k = $_{−∞}$. ■ 62 ∞ ∑ k = −∞ $^{{}$µ$_{0}$ ( ̂ A$_{k}$ $^{)}$− µ$_{1}$ ( ̂ A D k )} > $^{{}$µ$_{0}$ ( A$_{k}$ ) − µ$_{1}$ $^{(}$A D k $^{)}}$, ∞ ∑ k = −∞ $^{{}$µ$_{0}$ ( ˜ A$_{k}$ $^{)}$− µ$_{1}$ ( ˜ A D k )} > ∞ ∑ k = −∞ $^{{}$µ$_{0}$ ( A$_{k}$ ) − µ$_{1}$ $^{(}$A D k $^{)}}$, inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) I [ F ] = sup { A$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$ ( A$_{k}$ ) − µ$_{1}$ $^{(}$A D k $^{)}$, 0 $^{}}$. { A$_{k}$ } ∞ k = −∞ is a monotonically decreasing sequence of open sets, A D k = { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ ∈ A$_{k}$ s.t. y$_{1}$ − y$_{0}$ ≥ δ and ( y$_{0}$, y$_{1}$ ) ∈ C } ∪ { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ ∈ A$_{k}$$_{+1}$ s.t. y$_{1}$ − y$_{0}$ < δ and ( y$_{0}$, y$_{1}$ ) ∈ C } for any integer k. inf F ∈ Π( µ$_{0}$,µ$_{1}$ ) I [ F ] = sup { A$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ $^{(}$µ$_{0}$ ( A$_{k}$ ) − µ$_{1}$ $^{(}$A D k $^{))}$, (A.9) . . . ⊆ A$_{k}$$_{+1}$ ⊆ A$_{k}$ ⊆ A$_{k}$$_{−}$$_{1}$ ⊆ . . . .