Partitioning General Probability Measures

Abstract

Suppose $\mu_1,\ldots,\mu_n$ are probability measures on the same measurable space $(\Omega, \mathscr{F})$. Then if all atoms of each $\mu_i$ have mass $\alpha$ or less, there is a measurable partition $A_1,\ldots, A_n$ of $\Omega$ so that $\mu_i(A_i) \geq V_n(\alpha)$ for all $i = 1,\ldots, n$, where $V_n(\cdot)$ is an explicitly given piecewise linear nonincreasing continuous function on [0, 1]. Moreover, the bound $V_n(\alpha)$ is attained for all $n$ and all $\alpha$. Applications are given to $L_1$ spaces, to statistical decision theory, and to the classical nonatomic case.