Let $G$ be a finitely generated group acting on a set $S$ (on the right). Define the heirarchy of "marginal sets" as follows:
So if we recall that $A$ is wandering if, for some $g$ in $G$, the sets $A \cdot g^i$ $(i < \infty)$ are pairwise disjoint, then 1-marginal just means that the set is a finite union of wandering sets.
The point is that marginal sets are assigned measure 0 by any invariant measure (or even by any invariant exhaustive submeasure --- see my other recent question).
Now for the question(s):