Mesurable Functions by definition(from Stein and Shakarchi):
A function $f$ defined on a measurable subset $E$ of $\mathbb{R}^d$ is measurable, if for all $a\in \mathbb{R}$, the set $$f^{-1}([-\infty,a))=\{x\in E: f(x) is measurable.
Now a set $E$ is called measurable if $m_*(E)=0.$
Intuitively, the definition doesn't make much sense to me and would appreciate it if someone can explain it to me. A bonus would be if you can give me some simple examples of measurable and non measurable functions? Thanks.