  Given two metric measure spaces $X$ and $Y$. Let $f:X\to Y$ be a measurable
mapping and $A\subset X$. The Banach indicatrix (multiplicity function) is
defined as $N(y,f,A) = \#\{x\in A \mid f(x) = y\}$. We prove measurability of
this multiplicity function. The question was also discussed on
http://mathoverflow.net/q/206780/15946.
