A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures

Summary

If μ ₁, ... , μ _η 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 μ _i (A _i )≧(n − 1 + m)⁻¹ for all i=1, ..., n, where $m = \left\| {\mathop \Lambda \limits_{i = 1}^n \mu _i } \right\|$ 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) ^-1of 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].