non-empty finite *subset* of the orbit $Gx$. This situation can be easily reduced to [Lemma \[lem:choice\]]{} by choosing (in a measurable way!) just a single point from each of the sets $M_x$. This can be done, for instance, by identifying the space $(X,m)$ with the unit interval (with possible collapsing corresponding to the atoms of the measure $m$) and taking then the minimal of the points of $M_x$.

Recurrent and wandering sets {#sec:rw}
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Let us first remind the definitions (e.g., see [@Krengel85] for the case when $G$ is the group of integers ${\mathbb Z}$). Actually, we have to slightly modify them (and to distinguish recurrence from infinite recurrence) in order to take into account certain effects which do not arise for the group ${\mathbb Z}$ (see below the remarks after [Theorem \[th:hopf\]]{}).

\[dfn:rec\] A measurable set $A\subset X$ is called *recurrent* (resp., *infinitely recurrent*) if for a.e. point $x\in A$ the trajectory $Gx$ eventually returns to $A$, i.e., $gx\in A$ for a certain element $g\in G$ other than the group identity $e$ (resp., returns to $A$ infinitely often, i.e., $gx\in A$ for infinitely many elements $g\in G$). The opposite notion is that of a *wandering set* (kind of a 