to the *preimage sub-${\sigma}$-algebra* in $X$ which consists of the preimages of all the measurable sets in ${\overline}X$. Another important feature of the Lebesgue spaces is that, in fact, an arbitrary sub-${\sigma}$-algebra in $X$ can be obtained in this way for a certain uniquely defined quotient map.

Below we shall use the following elementary fact which follows at once from the uniqueness of the system of conditional measures.

\[lem:RNsame\] Let $T$ be an invertible measure class preserving transformation of a Lebesgue space $(X,m)$, and let $p:(X,m)\to ({\overline}X,{\overline}m)$ be its $T$-invariant projection, i.e., $p(Tx)=p(x)$ for a.e. $x\in X$. Then the conditional measures $m_{{\overline}x}, \,{\overline}x\in{\overline}X,$ of the projection $p$ are quasi-invariant with respect to $T$ and have the same Radon–Nikodym derivatives as the measure $m$: $$\frac{d\,Tm}{dm}(x) = \frac{d\,Tm_{{\overline}x}}{dm_{{\overline}x}}(x) \;, \qquad
\text{where} \quad {\overline}x = p(x) \;,$$ for a.e. $x\in X$.

Ergodic components, continuality and discontinuality {#sec:b}
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*Let now $G{\circlearrowleft}(X,m)$ be an action of an infinite countable group $G$ by measure class preserving transformations on a Lebesgue space $(X,m)$ — which will be our standing assumption through the rest of this Section.*

The quotient space $({\overline}X,{\overline}m)$ of $(X,m)$ determined by the ${\sigma}$-algebra of $G$-invariant sets is called the *space of ergodic components* 