  For a probability measure $\mu$ on $[0,1]$ without discrete component, the
best possible order of approximation by a finite point set in terms of the
star-discrepancy is $\frac{1}{2N}$ as has been proven relatively recently.
However, if $\mu$ 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 this gap
by giving a complete description of the discrete case. Most importantly, we
prove that for any discrete measure the best possible order of approximation is
for infinitely many $N$ bounded from below by $\frac{1}{cN}$ for some constant
$c \geq 2$ which depends on the measure. This implies, that for a finitely
supported discrete measure on $[0,1]^d$ the known possible order of
approximation $\frac{1}{N}$ is indeed the optimal one.
