of the action of $G$ on the space $(X,m)$, and the preimages $X_{{\overline}x}$ endowed with the conditional measures $m_{{\overline}x}$ are called the *ergodic components*. The ergodic components $X_{{\overline}x}$ are $G$-invariant, the conditional measures $m_{{\overline}x}$ are $G$-quasi-invariant, and the action of $G$ on the spaces $(X_{{\overline}x}, m_{{\overline}x})$ is ergodic (e.g., see [@Schmidt77]). [Lemma \[lem:RNsame\]]{} implies that the conditional measures $m_{{\overline}x}$ have the same Radon–Nikodym derivatives with respect to the action of $G$ as the original measure $m$.

Since the ergodic components are ergodic, each of them is either purely atomic (in which case it consists of a single $G$-orbit), or purely non-atomic.

\[def:hopf\] The *continual* ${\mathcal C}$ (resp., *discontinual* ${\mathcal D}$) *part* of the action $G{\circlearrowleft}(X,m)$ is the union of all the purely non-atomic (resp., purely atomic) components of the action. Denote by ${\overline}{\mathcal C}$ and ${\overline}{\mathcal D}$ the corresponding $G$-invariant subsets of the space of ergodic components ${\overline}X$ (their measurability follows from Theorem C). An orbit $Gx$ is *continual* (resp., *discontinual*) if it belongs to ${\mathcal C}$ (resp., to ${\mathcal D}$). The action is *discontinual* if $m({\mathcal D})>0$ and *continual* otherwise. If $m({\mathcal C})=0$ the action is called *completely discontinual*.

The quotient measure class on the space of ergodic components 