How to prove simple function is measurable

Question

I'm aware of the definition of the measurable function. But I was wondering how to prove simple function is measurable? It would be better have some detailed proof.

Answer 1

Consider a simple function $$ f = \sum_{i=1}^na_i1_{E_i}, $$ where $E_1,\ldots,E_n$ are disjoint measurable sets. To see that $f$ is measurable, it suffices to show that $$ f^{-1}(-\infty,a] $$ is measurable for every $a\in\mathbb{R}$. Given $a\in\mathbb{R}$, note that $$ f^{-1}(-\infty,a] = \bigcup\{E_i \mid i\in\{1,\ldots,n\}\ \text{such that}\ a_i\le a\} $$ is a union of measureable sets. Therefore $f$ is measurable.