View source for Weakly-wandering set

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''for an invertible measurable transformation $T$ of a [[Measurable space|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|quasi-invariant measure]] $\mu$ (defined on $\mathfrak B$), then a necessary and sufficient condition for $T$ to have a finite [[Invariant measure|invariant measure]] equivalent to $\mu$ is that $\mu A=0$ for any weakly-wandering set $A$.

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

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