Suppose μ1, ..., μn are probability measures on the same measurable space (Ω, F). Then if all atoms of each μi have mass α or less, there is a measurable partition A1,..., An of Ω so that μi(Ai) ≥ Vn(α) for all i = 1, ... , n, where Vn(•) is an explicitly given piecewise linear nonincreasing continuous function on [0, 1]. Moreover, the bound Vn(α) is attained for all n and all α. Applications are given to L1 spaces, to statistical decision theory, and to the classical nonatomic case.



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