Theorem: If $f$ is a measurable function on a set $E$, then it is a limit of a uniform convergent sequence of simple measurable functions $\{g_n\}_{n=1}^\infty$, that is
$f(x) = \lim_{n \rightarrow\infty}g_n(x)$.
The proof is rather simple. We just take $E_{l,m} = \{x\in E | f(x) \in [\frac{l}{2^m}; \frac{l+1}{2^m})\} $ and $g_n(x) = \frac{l}{2^m}$. But what is special about measurability? What if I miss italic words? I know that with this theorem I may consider a measurable function as "nearly" continous, otherwise I just don't the construction of Lebesgue sets (it may be continous not "almost everywhere"). So, is the reason we don't use this theorem without measurability just because it would be useless?