the union of free orbits in ${\mathcal D}$. The restriction of the action to the set ${\mathcal C}\cup({\mathcal D}\setminus{\mathcal D}_{\mathsf{free}})$ is conservative, whereas the restriction to the set ${\mathcal D}_{\mathsf{free}}$ is completely dissipative, thus providing the so-called *Hopf decomposition* of the action space into the conservative and completely dissipative parts ([Theorem \[th:hopf\]]{}). \[Historically, such a decomposition was first established in the pioneering paper of Eberhard Hopf [@Hopf30] for one-parameter groups of measure preserving transformations.\] Although this fact is definitely known to the specialists, it rather belongs to the “folklore”, and the treatment of this issue in the literature is sometimes pretty confused, so that we felt it necessary to give a clear and concise proof.

The continual part ${\mathcal C}$ can be described as the set of points for which the orbitwise sum of the Radon–Nikodym derivatives of the action is infinite ([Theorem \[th:equiv\]]{}(iii)). Therefore, in the case of (mod 0) free actions this condition completely characterizes the conservative part of the action. Once again, the specialists in the theory of discrete equivalence relations will hardly be surprised by this result. However, to the best of our knowledge, in spite of its simplicity it has never been formulated explicitly. A 