Let $X,Y$ be topological $G$-spaces, with (left) $G$-invariant probability measures $\mu_X,\mu_Y$ respectively, and let $f:X \to Y$ be a surjective $G$-equivariant map preserving the measures, i.e. $\mu_Y= \mu_X \circ f^{-1}$. The ergodic decomposition theorem (cf. theorem 4.2 in Varadarajan's paper referred to below) tells us that $X$ (resp. $Y$) can be decomposed in a measurable way into disjoint invariant Borel sets $X_e$ (resp. $Y_{e'}$) on each of which there is a unique ergodic probability measure $e$. My question is whether the map $f$ must preserve these decompositions, in the following sense: For every (or almost every) $X_e$, we have that $f(X_e)$ is one of the sets $Y_{e'}$ and $f: X_e\to Y_{e'}$ satisfies $e'= e\circ f^{-1}$.
Perhaps this is a standard fact but I couldn't find it stated in the literature. If there is a reference I'd be very grateful.
If it does not hold in the above generality, which conditions must be added? A basic case where the statement holds is the following: $X,Y$ are compact abelian groups with Haar measures, $f:X\to Y$ is a continuous surjective homomorphism, and $G\leq X$ is a closed subgroup acting on $X$ by translation and acting on $Y$ as expected: $g \cdot y = f(g)+y$. Then the decompositions are just that of $X$ into cosets of $G$ and that of $Y$ into cosets of $f(G)$, and the above fact holds with $e,e'$ being the (translates of) Haar measure on $G, f(G)$.
http://www.ams.org/journals/tran/1963-109-02/S0002-9947-1963-0159923-5/S0002-9947-1963-0159923-5.pdf