Lemma 57 ([A common lower bound for two-scale readings]). For 1≤i≤k , let U^i_n,V^i_n take values in [-M,M] , with U^i_n measurable with respect to a sigma-field A^i_n . Suppose that |U^i_n-V^i_n|⟶0. Suppose there are probability measures …

Pass to a subsequence on which ν_n and the laws of all U^i_n converge. Conditional integration gives |U^i_n-V^i_n| ≥c\,[1_ G^i_n ∫|U^i_n-y|\,ν_n( y)]. The lost contribution outside G^i_n is at most 2M((G^i_n)^c) . If μ_i and ν are the limiting laws, bounded continuity yields ∫|x-y|\,μ_i( x)ν( y)=0 . A product of two probability measures supported on the diagonal must consist of the same point mass in each coordinate. The point is common to all i because ν is common. Agreement of the readings completes the proof.