Theorem Let $(X,B,\mu)$ be a standard measure space and $g:(x,B,\mu) \rightarrow (x,B,\mu)$ be a measure preserving map. Let $C \subset B$ be a countably generated $g$-invariant $\sigma$-algebra. Then exists a compact metric space with $Y$ with Borel - $\sigma$ - algebra $\theta$ and probability measure $\nu$. And there exists $h: Y \rightarrow Y$ measure preserving and there exists a measure preserving factor map $F:X \rightarrow Y$ such that $F^{-1}(\theta) = C$ a.e..
The proof starts the following way:
First set $Y:= \{\nu | \nu \text{ a probability measure on } X\}$ with Borel $\sigma$ - Algebra $\theta$. Let $(\mu_y)_{y \in X}$ be the disintegration of $\mu$ with respect to $C$. We set $F: X \rightarrow Y$, $x \mapsto \mu_x$. By the properties of disintegration is $x \rightarrow \mu_x$ $C$-measurable and thus $F^{-1}(\theta) \subset C$. Clearly $\forall A \in C$ holds that $\mu_y(A) = \chi_A(x)$ for $\mu$-a.e. $x \in X$ and $D_A = \{p \in Y | p(A) = 1\} \in \theta$.
Why is $\forall A \in C$ $\mu_y(A) = \chi_A(x)$ for $\mu$-a.e. $x \in X$ true or clear? And why is $D_A \in \theta$?
For the second question, $D_A \in \theta$.
Open sets in $Y$ are generated by $\{\nu| \int_X \psi d_\nu \in (a,b)$ for $\psi \in C(X)$ and $a, b \in [0,1]$. Thus $\{\nu| \int_X \psi d_\nu = 1 \}$ is closed $\forall \psi \in C(X)$.
In our case, we have to approximate $\chi_A$ with continuous functions in the right way. Since characteristic functions can be approximated with increasing and decreasing sequences of continuous functions, there exists $(f_n)_n$ with $f_n \in C(X)$ such that $\chi_A \leq f_n$, $f_{n+1} \leq f_n$ and $f_n \rightarrow \chi_A$ pointwise.
Therefore $$\bigcap_n \{\nu | \int_X f_n d\nu = 1 \} = D_A.$$ Thus $D_A$ is closed and hence $D_A \in \theta$.