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

Theodore P. Hill, Georgia Institute of Technology - Main Campus

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Copyright © 1987 Springer. The original publication is available at http://dx.doi.org/10.1007/BF00320087.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

Abstract

If μ₁, ... ,μ_n are non-atomic probability measures on the same measurable space (S, F), then there is an F-measurable partition {A_i }ⁿ_{i = 1} of S so that μ₁ (A_i )≥ (n – 1 + m)^–1 for all i=1, ..., n, where m = ∥ Δⁿ_i=1 μ_i∥ is the total mass of the largest measure dominated by each of the μ_i'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)^-1 of Elton et al. [5] 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 [10] and of Dubins and Spanier [2].