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