A Sharp Partitioning-Inequality for Non-Atomic Probability Measures Based on the Mass of the Infimum of the Measures

If Ill' ... , ~tn are non-atomic probability measures on the same measurable space (S, ff"), then there is an ff"-measurable partition {AJ7= 1 of S so that IlJAi)~(n-l+m)-l for all i=I, ... ,n, where m=l l i 01 lli l l is the total mass of the largest measure dominated by each of the Il/s; moreover, this bound is attained for all n ~ 1 and all m in [0,1]. This result is an analog of the bound (n + 1-M)-l of Elton et 0.1. [5J based on the mass M of the supremum of the measures; each gives a quantative generalization of a well-known cake-cutting inequality of Urbanik [10J and of Dubins and Spanier [2]. § 1. Introduction Suppose 111'112"'" Il n are non-atomic probability measures on the same measurable space (S, ff"), and let n denote the total mass of the sub-probability measure 1\ Ili, the largest measure i= 1 dominated by each of the measures Ili' The main purpose of this note is to prove the following result. Theorem 1. If Ill"'" Il n are non-atomic probability measures on the same measurable space (S, ff"), then there is an ff"-measurable partition {AJ7= 1 of S satisfying J1JAJ~~(n-l+m)-l (1) moreover, this bound is attained for all positive integers n and all mE [0, 1].