Random sets are close to low-discrepancy sets

We show that a random sample from an arbitrary probability measure on ${\mathbb{R}}^d$ is close to a low-discrepancy point set. Namely, after moving only a small fraction of the sample points in expectation, one obtains an $n$-point set with star discrepancy $\mathrm{polylog}(n)\mathrm{/}n$ with respect to the original measure.