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 $\operatorname{polylog}(n)/n$ with respect to the original measure.