If $|E|=0$ then $f$ is measurable

Question

Let $f:E\subset\mathbb{R}^n \rightarrow \mathbb{R} \:\cup \: \{-\infty,\infty\}$ a function defined in a measurable set $E$. Show that, if $|E|=0$ then $f$ is measurable.

I know I have to show that set $\{f>a\} $ is mesurable, for all $a\in\mathbb{R}$. However, I couldn't relate that to $| E | = 0.$

Answer 1

$\{x\in E: f(x)>a\}$ is a subset of the set $E$ which is of measure zero. We know that Lebesgue measure space is complete, so $\{x\in E: f(x)>a\}$ is Lebesgue measurable.

Answer 2

Hint: You should use the fact that all sets of Lebesgue measure 0 are mesurable.