and the measure classes of the conditional measures on the ergodic components do not change when the measure $m$ is replaced with an equivalent one, so that the above Definition (as well as various definitions below related to conservativity and dissipativity) depends only on the measure class of $m$.

\[lem:choice\] Let $A\subset X$ be a measurable $G$-invariant subset. It is contained in the discontinual part ${\mathcal D}$ if and only if one can select, in a measurable way, a representative from each $G$-orbit contained in $A$, i.e., if and only if there exists a measurable map $\pi:A\to A$ which is constant along the orbits of the action.

If $\pi$ is such a map, then it identifies the space of ergodic components of $A$ with $\pi(A)$, so that in particular $A\subset{\mathcal D}$. Conversely, by Theorem C the discontinual part ${\mathcal D}$ can be identified, in a measurable way, with the product ${\overline}{\mathcal D}\times\{1,2,\dots\}$. Then one can define the map $\pi:{\mathcal D}\to{\mathcal D}$ with the required properties as $\pi({\overline}x,n)=({\overline}x,1)$.

\[rem:choice\] Below we shall also encounter the situation when instead of a map with the properties from the above Lemma one has an orbit constant measurable map $x\mapsto M_x$, where $M_x$ is a 