Prove a function is Lebesgue measurable

Question

Problem

$f$ is defined on a measurable set $E$, $D$ is a dense subset of $\mathbb{R}$ ($\overline{D}=\mathbb{R}$), prove $\forall\ r\in D$, if the set $\{x\in E:f(x)>r\}$ is measurable then $f$ is measurable on $E$

Since $D$ is dense in $\mathbb{R}$, $\forall\ a\in\mathbb{R}$, there exist $\{r_n\}_{n=1}^\infty\subset D$ such that $\lim_{n\to\infty}r_n=a$. We know $\{x\in E:f(x)>r_n\}$ is measurable for any $n$, so we only need to relate $\{x\in E:f(x)>r_n\}$ with $\{x\in E:f(x)>a\}$.

Answer 1

You have to take sequence $(r_n)$ from $D$ decreasing to $a$. In that case $\{x \in E: f(x) >a\}=\bigcup_n\{x\in E: f(x) >r_n\}$.