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

Summary If ₁, ... , are non-atomic probability measures on the same measurable space (S, ), then there is an -measurable partition {A _i } _{i = 1} ⁿ of S so that _i (A _i )(n – 1 + m)^–1 for all i=1, ..., n, where is the total mass of the largest measure dominated by each of the _i S; moreover, this bound is attained for all n1 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].Research partly supported by NSF Grants DMS-84-01604 and DMS-86-01608