# 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 $A$.