Approximation of Discrete Measures by Finite Point Sets

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