A function arising in the estimation of unobserved probability

We study a function that arises naturally in the estimation of unobserved probability in random sampling. To define it, let μ=(μ1,μ2,…)μ=(μ1,μ2,…) be an infinite discrete probability measure, with μ(i)=μiμ(i)=μi. Let μi>0μi>0 and μi⩾μi+1∀iμi⩾μi+1∀i. Given s∈[0,1]s∈[0,1], let σ(s)=inf{μ(A):μ(A)⩾s}σ(s)=inf{μ(A):μ(A)⩾s}. Several questions arise: what are the measures μμ for which σσ is continuous? What are the measures for which σ(s)=sσ(s)=s for all s  ? When σσ is not continuous, can one say more about its discontinuities? Work by Rényi yields complete answers to the first two questions, and surprisingly the two classes of measures turn out to be the same. The last question is more complex, and we answer it by considering different cases. In that process, we discover a class of measures which have the interesting property that, for them, events are uniquely determined by their probabilities. We call such measures sharp.Analogous results hold for the function τ(s)=sup{μ(A):μ(A)⩽s}τ(s)=sup{μ(A):μ(A)⩽s}.