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

Tusn\'ady's problem, the transference principle, and non-uniform QMC sampling