part ${\mathcal C}$. Again, as with the infinite conservativity, the action on the set ${\mathcal D}\setminus{\mathcal D}_{\mathsf{free}}$ will be neither strictly conservative nor dissipative.

A continuality criterion
------------------------

\[th:equiv\] Let $G{\circlearrowleft}(X,m)$ be a free measure class preserving action of a countable group $G$ on a Lebesgue space. Denote by $\mu_x,\,x\in X$, the measure on the orbit $Gx$ defined as $$\mu_x(gx)=\frac{dg^{-1} m}{dm}(x)=\frac{dm(gx)}{dm(x)}$$ (obviously, the measures $\mu_x$ corresponding to different points $x$ from the same $G$-orbit are proportional). Then for a.e. point $x\in X$ the following conditions are equivalent:

-   The orbit $Gx$ is dissipative;

-   The orbit $Gx$ is discontinual;

-   The measure $\mu_x$ is finite;

-   For any $t>0$ the set $\{y\in Gx: \mu_x(y)\ge t\}$ is finite;

-   The set $M_x$ of maximal weight atoms of the measure $\mu_x$ is non-empty and finite.

(i)$\;\Longleftrightarrow\;$(ii). This is [Corollary \[cor:DD\]]{}.

(ii)$\implies$(iii). By [Definition \[def:hopf\]]{}, the orbit $Gx$ is discontinual if and only if it is an ergodic component of the $G$-action on $X$. By [Lemma \[lem:RNsame\]]{} in this case the measure $\pi_x$ is proportional to the conditional measure on this ergodic component, and therefore it is finite.

(iii)$\implies$(iv)$\implies$(v). Obvious.

(v)$\implies$(i). Follows from [Lemma \[lem:choice\]]{} and [Remark \[rem:choice\]]{} (because the map $x\mapsto M_x$ 