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 $N(y,f) = \infty$.

Let $X\subset\mathbb R^n$, $Y\subset\mathbb R^m$ with Lebesgue measure.

I am interested to know if $N(y,f)$ is a measurable function (?)

• If $X$ is an interval (say $X=[a,b]$) and $f$ is a continuous function, 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 anything about measurability of $N(y,f)$ ?