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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