Published in The Annals of Probability, Volume 15, Issue 2, April 1, 1987, pages 804-813. Copyright © 1987 Institute of Mathematical Statistics. The definitive version is available at http://dx.doi.org/10.1214/aop/1176992173.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
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.