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I have the following questions: assume that $X$ be a standard Borel space (i.e. a Polish space equipped with the $\sigma$-algebra generated by open sets) with a (possibly Borel, invariant, ergodic) probability measure $\mu$ , $Y$ a Borel space. Suppose that $E$ be an equivalence relation on $X$ (I,'m interested in the case there it is induced by the Borel action of a Polish locslly compact group like, for instance, $SL_m(\mathbb{Z})$), that $X_1\subseteq X$ has measure $1$ and that there exists a Borel function $f\colon X_1\to Y$.

$(1)$ How can I define (if it exists) a Borel function $c\colon X\to X$ in such a way that $c(X)\subseteq X_1$, so that $f\circ c$ is a Borel function from $X$ to $Y$ and $c(x)Ex$?

$(2)$ If the previous question has negative answer, under which assumptions this holds?

My intuition is that $(a)$ $X_1$ meets every equivalence class and then $(b)$ I could define $c$ on each such equivalence class to be one fixed element of this nonempty intersection, but I don't know how to prove it (it seems I need the Axiom of Choice to do this; can I avoid it?)

Here is my attempt. About $(a)$, I can prove that measure $1$ implies dense as follows: assume $A\subseteq X$ is a non-empty open set (hence $\mu (A)>0$) such that $A\cap X_1=\emptyset$. Then $1=\mu(X_1)<\mu(X_1)+\mu(A)=\mu(X_1\cup A)=1$, contraddiction. But does it suffice to prove that $X_1$ has non-empty intersection with each equivalence class?

What about $(b)$?

Thank you in advance for your help.

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