a maximal wandering set, i.e., such that ${\mathcal D}_{\mathsf{free}}=\bigcup_{g\in G} gA$.

If $A$ is a wandering set, then the map $\pi$ from the $G$-invariant union ${\widetilde}A=\bigcup_{g\in G} gA$ to $A$ defined as $\pi(x)=Gx\cap A$ is measurable and $G$-invariant, so that $A\subset{\mathcal D}$ by [Lemma \[lem:choice\]]{}. Moreover, all the orbits intersecting $A$ are obviously free, whence $A\subset{\mathcal D}_{\mathsf{free}}$.

As for the maximality, one can take for such a wandering set any measurable section of the projection ${\mathcal D}_{\mathsf{free}}\to{\overline}{\mathcal D}_{\mathsf{free}}$ (which exists by [Lemma \[lem:choice\]]{}).

Hopf decomposition
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An action $G{\circlearrowleft}(X,m)$ is called *conservative* (resp., *infinitely conservative*) if any measurable subset $A\subset X$ is recurrent (resp., infinitely recurrent). It is called *dissipative* if there is a non-trivial wandering set, and *completely dissipative* if the whole action space $X$ is the union of translates of a certain wandering set.

The action on the disjoint union of two $G$-invariant sets is conservative (resp., infinitely conservative) if and only if the action on each of these sets has the same property. Taking stock of the Propositions from [Section \[sec:rw\]]{} we now obtain

\[th:hopf\] The action space $X$ can be decomposed into the disjoint union of two $G$-invariant measurable sets (called its *conservative* and *dissipative parts*, 