Weakly-wandering set

for an invertible measurable transformation $T$ of a measurable space $(X,\mathfrak B)$

A measurable subset $A\subset X$ for which there is an infinite sequence of integers $n_i$ such that the sets $T^{n_i}$ are mutually disjoint (here, invertibility of $T$ is understood to mean existence and measurability of $T^{-1}$). If $T$ has a $\sigma$-finite quasi-invariant measure $\mu$ (defined on $\mathfrak B$), then a necessary and sufficient condition for $T$ to have a finite invariant measure equivalent to $\mu$ is that $\mu A=0$ for any weakly-wandering set $A$.

A word of warning. In topological dynamics there is a notion of weakly non-wandering point (see [3]) which has nothing to do with the notion defined above.

References

[1]	A.B. Hajian, S. Kakutani, "Weakly wandering sets and invariant measures" Trans. Amer. Math. Soc. , 110 : 1 (1964) pp. 136–151
[2]	A. Hajian, Y. Itô, "Weakly wandering sets and invariant measures for a group of transformations" J. Math. Mech. , 18 : 12 (1969) pp. 1203–1216
[3]	D.V. Anosov, I.V. Bronshtein, "Topological dynamics" , Dynamical Systems I , Encycl. Math. Sci. , I , Springer (1988) pp. 197–219 (Translated from Russian)