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 infinite) 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 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.