Approximation of Discrete Measures by Finite Point Sets

Abstract For a probability measure μ on [0, 1] without discrete component, the best possible order of approximation by finite point set in terms star-discrepancy is &inline as has been proven relatively recently. However, if contains component no non-trivial lower bound holds general because it straightforward to construct examples any error this case. This might explain, why measures sets so far not completely covered existing literature. In note, we close gap giving complete description for measures. Most importantly, prove that (not supported one only) infinitely many N bounded from below some constant 6 ≥ c> 2 which depends measure. implies, finitely d known indeed optimal one.