  For different classes of measure preserving transformations, we investigate
collections of sets that exhibit the property of lightly mixing. Lightly mixing
is a stronger property than topological mixing, and requires that a lim inf is
positive. In particular, we give a straightforward proof that any mildly mixing
transformation T has a dense algebra C such that T is lightly mixing on C.
Also, we provide a hierarchy for the properties of lightly mixing, sweeping out
and uniform sweeping out for dense collections, and dense algebras of sets. As
a result, stronger mixing realizations are given for several types of
transformations than those given by previous extensions of the Jewett-Krieger
Theorem.
