“fundamental domain”), i.e., a measurable set $A\subset X$ with pairwise disjoint translates $gA,\,g\in G$.

We shall now explain the connection between these notions and our [Definition \[def:hopf\]]{}.

\[prp:rec\] Any measurable subset of the continual part ${\mathcal C}$ is infinitely recurrent.

For a measurable subset $A\subset{\mathcal C}$ put $$A_0 = \{x\in A: gx\in A \;\text{for finitely many}\; g\in G \} \subset F \;,$$ where $$F = \{x\in X: Gx\cap A \; \text{is non-empty and finite} \} \;.$$ The set $F$ is $G$-invariant and measurable, and by [Lemma \[lem:choice\]]{} and [Remark \[rem:choice\]]{} $F\subset{\mathcal D}$. Therefore $A_0$ must be a null set, whence the claim.

Denote by ${\mathcal D}_{\mathsf{free}}$ (resp., ${\mathcal D}_{\mathsf{cofinite}}$) the union of all the *free* (resp., *cofinite*) discontinual orbits, i.e., such that the stabilizers of their points are trivial (resp., finite). Obviously, ${\mathcal D}_{\mathsf{free}}\subset{\mathcal D}_{\mathsf{cofinite}}$, and both these sets are measurable. Let ${\overline}{\mathcal D}_{\mathsf{free}}$ and ${\overline}{\mathcal D}_{\mathsf{cofinite}}$ be the corresponding subsets of the space of ergodic components ${\overline}X$. The definition immediately implies

\[pr:disc\] Any measurable subset of ${\mathcal D}\setminus{\mathcal D}_{\mathsf{free}}$ (resp., of ${\mathcal D}\setminus{\mathcal D}_{\mathsf{cofinite}}$) is recurrent (resp., infinitely recurrent).

Let us now look at the wandering sets.

\[pr:wand\] Any wandering set is contained in ${\mathcal D}_{\mathsf{free}}$, and there is 