This research is partly motivated by a question raised by Boshernitzan whether the Möbius function is correlated with any strictly ergodic sequence, partly by the desire to better understand disjointness of measures. We (Jacek Serafin and myself) are studying abstractly, when a sequence generic for an invariant measure is correlated to a strictly ergodic sequence. It turns out, this depends exclusively on the measure, for which the point is generic. We show that the responsible property resembles, but is essentially weaker than, disjointness from all ergodic measures. We give appropriate examples.