TITLE: How to prove simple function is measurable
QUESTION [7 upvotes]: 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.

REPLY [10 votes]: 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.