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For a family F (a collection of subsets of Z_+), the notion of F-independenceis defined both for topological dynamics (t.d.s.) and measurable dynamics(m.d.s.). It is shown that there is no non-trivial {syndetic}-independentm.d.s.; a m.d.s. is {positive-density}-independent if and only if it hascompletely positive entropy; and a m.d.s. is weakly mixing if and only if it is{IP}-independent. For a t.d.s. it is proved that there is no non-trivialminimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only ifit is {IP}-independent.  Moreover, a non-trivial proximal topological K system is constructed, and atopological proof of the fact that minimal topological K implies strong mixingis presented.

Family independence for topological and measurable dynamics