Let $f: [0,1] \rightarrow \mathbb{R}$ be a continuous function. Let also $g$ be a function, defined by the following condition:for each $a \in [0, + \infty]$ $$g(a)=\text{card} \{ x \in [0, 1] | f(x) = a \} \in [0, + \infty)$$ I would like to prove that $g$ is also a measurable function. Are there any hints that might help?
Probably, it is worth trying to prove it more or less directly, by checking that $\{ a \in [0, + \infty] | g(a) < c \}$ is measurable for any $c$, but this approach does not seem to be clear enough, since the preimage of $g$ would be some cardinal number.
By the intermediate value theorem, for any interval $I\subseteq [0,1]$, the set $A:=\{f(x),x\in I\}$ is an interval, and so is a Borel set.
Fix $n$ and let $I_i=[i/n,(i+1)/n)$ for $i=0,1,\dots, n-1$. Now define the bounded, Borel measurable function $$g_n(a)=\sum_{i=0}^{n-1}{\bf 1}_{A_i}(a)+{\bf 1}_{\{f(1)\}}(a),$$ where $A_i=\{f(x), x\in I_i\}$.
We have $g_n(a)\uparrow g(a)$ as $n\to\infty$, which shows that $g$ is measurable.