is measurable in view of Theorem C).

\[cr:equiv\] Under conditions of [Theorem \[th:equiv\]]{} the continual part ${\mathcal C}$ and the discontinual part ${\mathcal D}$ of the action coincide with the sets $$\label{eq:cons}
\left\{x \in X: \sum_{g\in G}\frac{dgm(x)}{dm(x)} = \infty \right\}$$ and $$\label{eq:diss}
\left\{x \in X: \sum_{g\in G}\frac{dgm(x)}{dm(x)} < \infty \right\} \;,$$ respectively.

[Theorem \[th:equiv\]]{} (except for the equivalence (i)$\;\Longleftrightarrow\;$(ii)) and [Corollary \[cr:equiv\]]{} (with the summation taken over the equivalence class of $x$) are also true for an arbitrary *countable non-singular equivalence relation* on a Lebesgue space $(X,m)$. In this case $\mu_x$ are the measures on the equivalence classes determined by the *Radon–Nikodym cocycle* of the equivalence relation. In particular, the equivalence of conditions (ii), (iii), (iv), (v) from [Theorem \[th:equiv\]]{} also holds for non-free actions. In what concerns [Corollary \[cr:equiv\]]{}, the only difference with the free case is that one has to replace the summation over $g$ in formulas and with the summation over the orbit of $x$. The proof is precisely the same; however, in order to spare the reader the trouble of going through the definitions from the ergodic theory of equivalence relations (see [@Feldman-Moore77]) we confine ourselves just to free actions. This generality is sufficient, on one 