Let $A$ be a bounded measurable subset of $\mathbb{R}$. Show that if $f:A\rightarrow \mathbb{R}$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$.
Choose real $c$. Since $f$ is measurable we know that each set $S_n=\{x\in A : c-\frac{1}{n} \le f(x)\le c + \frac{1}{n}\}$ is measurable. Since
$$\{x\in A:c=f(x)\}=\bigcap_n S_n$$
is a countable intersection of measurable sets, it is measurable itself.
Is this correct? I'm asking because up until now I have not had much experience with measure theory.