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Suppose ₁, , ₙ are probability measures on the same measurable space (, F). Then if all atoms of each ᵢ have mass or less, there is a measurable partition A₁, , Aₙ of so that ᵢ (Aᵢ) Vₙ () for all i = 1, , n, where Vₙ () is an explicitly given piecewise linear nonincreasing continuous function on 0, 1. Moreover, the bound Vₙ () is attained for all n and all. Applications are given to L₁ spaces, to statistical decision theory, and to the classical nonatomic case.
Theodore P. Hill (Wed,) studied this question.