  The main subject of the paper, motivated by a question raised by
Boshernitzan, is to give criteria for a bounded complex-valued sequence to be
uncorrelated to any strictly ergodic sequence. As a tool developed to study
this problem we introduce the notion of correlation between two shift-invariant
measures supported by the symbolic space with complex symbols. We also prove a
"lifting lemma" for generic points: given a joining $\xi$ of two
shift-invariant measures $\mu$ and $\nu$, every point $x$ generic for $\mu$
lifts to a pair $(x,y)$ generic for $\xi$ (such $y$ exists in the full symbolic
space). This lemma allows us to translate correlation between bounded sequences
to the language of correlation of measures. Finally, to establish that the
property of an invariant measure being uncorrelated to any ergodic measure is
essentially weaker than the property of being disjoint from any ergodic
measure, we develop and apply criteria for ergodicity of four-jump cocycles
over irrational rotations. We believe that apart from the applications to
studying the notion of correlation, the two developed tools: the lifting lemma
and the criteria for ergodicity of four-jump cocycles, are of independent
interest. This is why we announce them also in the title. In the Appendix we
also introduce the notion of conditional disjointness.
