particular, $\B$ itself is corresponding to a partition into single points. An important property of Lebesgue spaces and measurable partitions is the following.

\[thm-2.1\] Let $\eta$ be a measurable partition of a Lebesgue space $(X, \B, m)$. Then, for every $x$ in a set of full $m$-measure, there is a probability measure $m^\eta_x$ defined on $\eta(x)$, the element of $\eta$ containing $x$. These measures are uniquely characterized (up to sets of $m$-measure $0$) by the following properties: if $A\subset X$ is a measurable set, then $x\mapsto m^\eta_x(A)$ is $\widehat{\eta}$-measurable and $m(A)=\int
m^\eta_x(A)d m(x)$. These properties imply that for any $f\in
L^1(X,\B, m)$, $m_x^\eta(f)=\E_m(f|\widehat{\eta})(x)$ for $m$-a.e. $x$, and $m(f)=\int \E_m(f|\widehat{\eta})dm$.

The family of measures $\{m^\eta_x\}$ in the above theorem is called the [*canonical system of conditional measures associated with $\eta$*]{}.

Throughout the remaining part of this subsection, we assume that $(X,\B,m)$ is a Lebesgue space. Suppose that $Y$ is a complete separable metric space and $\pi: X\to  Y$ is a $\B$-measurable map. Denote $\gamma:=\B(Y)$, the Borel-$\sigma$-algebra on $Y$. According to Rohlin’s theory (cf. [@Rohlin1949 Section 2.5], [@Parry1969 Chapter IV]), the mapping $\pi$ induces a measurable partition $$\label{e-xi1}
\xi=\{\pi^{-1}(y):\; y\in Y\}$$ of $X$ with $\widehat{\xi}=\pi^{-1}\gamma\mbox{ (mod 0)}$, and $(X_\xi, \B_\xi, m_\xi)$ is 