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

摘要

If μ1, ... ,μn are non-atomic probability measures on the same measurable space (S, F), then there is an F-measurable partition {Ai }ni = 1 of S so that μ1 (Ai )≥ (n – 1 + m)–1 for all i=1, ..., n, where m = ∥ Δni=1 μi∥ is the total mass of the largest measure dominated by each of the μi\u27s; 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].