Let $f:X\to Y$ is a measurable function. Banach indicatrix
$$
N(y,f) = \#\{x\in X \mid f(x) = y\}
$$
is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then $N(y,f) = \infty$. 

*I am interesting if $N(y,f)$ is measurable function?* 

 - If $X$ is an interval (say $X=[a,b]$) and $f$ be of bounded variation, the answer is some how positive (http://math.stackexchange.com/q/68635/23566).
 - In Federer's Geometric measure theory we find following theorem 

> Let $X$ be a separable metric space and let $f(A)$ be $\mu$-measurable for all Borel subsets $A$ of $X$. 
Let $\zeta(S) = \mu(f(S))$ for $S\subset X$ and let $\psi$ be the measure on $X$ defined by the Carathéodory construction from $\zeta$. Then
$$
\psi(A) = \int\limits_{A}N(y,f)\, d\mu_{Y}
$$
for every Borel set $A\subset X$.

*Does it say something about measurability of $N(y,f)$ ?*