  It is well-known that for every $N \geq 1$ and $d \geq 1$ there exist point
sets $x_1, \dots, x_N \in [0,1]^d$ whose discrepancy with respect to the
Lebesgue measure is of order at most $(\log N)^{d-1} N^{-1}$. In a more general
setting, the first author proved together with Josef Dick that for any
normalized measure $\mu$ on $[0,1]^d$ there exist points $x_1, \dots, x_N$
whose discrepancy with respect to $\mu$ is of order at most $(\log
N)^{(3d+1)/2} N^{-1}$. The proof used methods from combinatorial mathematics,
and in particular a result of Banaszczyk on balancings of vectors. In the
present note we use a version of the so-called transference principle together
with recent results on the discrepancy of red-blue colorings to show that for
any $\mu$ there even exist points having discrepancy of order at most $(\log
N)^{d-\frac12} N^{-1}$, which is almost as good as the discrepancy bound in the
case of the Lebesgue measure.
