  We provide an algorithm to approximate a finitely supported discrete measure
$\mu$ by a measure $\nu_{N}$ corresponding to a set of $N$ points so that the
total variation between $\mu$ and $\nu_N$ has an upper bound. As a consequence
if $\mu$ is a (finite or infinitely supported) discrete probability measure on
$[0,1]^{d}$ with a sufficient decay rate on the weights of each point, then
$\mu$ can be approximated by $\nu_N$ with total variation, and hence
star-discrepancy, bounded above by $(\log N) N^{-1}$. Our result improves, in
the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that
for any normalized Borel measure $\mu$, there exist finite sets whose
star-discrepancy with respect to $\mu$ is at most $(\log N)^{d-\frac{1}{2}}
N^{-1}$. Moreover we close a gap in the literature for discrepancy in the case
$d=1$ showing both that Lebesgue is indeed the hardest measure to approximate
by finite sets and also that all measures without discrete components have the
same order of discrepancy as the Lebesgue measure.
