It is well-known that for every N ≥ 1 and d ≥ 1 there exist point sets x1, . . . , xN ∈ [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 µ on [0, 1] d there exist points x1, . . . , xN whose discrepancy with respect to µ 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 µ there even exist points having discrepancy of order at most (log N ) d− 1 2 N −1 , which is almost as good as the discrepancy bound in the case of the Lebesgue measure.