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# Optimal-Partitioning Inequalities for Nonatomic Probability Measures

John Elton, Theodore P. Hill and Robert P. Kertz
Transactions of the American Mathematical Society
Vol. 296, No. 2 (Aug., 1986), pp. 703-725
DOI: 10.2307/2000385
Stable URL: http://www.jstor.org/stable/2000385
Page Count: 23
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## Abstract

Suppose μ1,...,μn are nonatomic probability measures on the same measurable space (S, B). Then there exists a measurable partition {Si}n i=1 of S such that μi(Si) ≥ (n + 1 - M)-1 for all i = 1,...,n, where M is the total mass of $\bigvee^n_{i=1} \mu_i$ (the smallest measure majorizing each μi). This inequality is the best possible for the functional M, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to L1-functions, discrete allocation problems, statistical decision theory, and a dual problem.

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