I’ve found this result in my Measure Theory teacher’s notes.
If $f: \Omega \longrightarrow [-\infty, \infty]$ is measurable, then there exists an increasing sequence of step functions $f_n: \Omega \longrightarrow [- \infty , \infty)$ which has $f$ as pointwise limit.
I just can’t manage to find it on the web, and I’m using Zygmund, Measure and Integral as a guide as well, and it says that the sequence might not be increasing. I'm trying to see if this result is right, it just looks very important, but if I can't find it anywhere then there must be some mistake.
EDIT: I'm going to add my teacher's proof, so that you can check what he's saying.
For every $m \in \mathbb{N}$, let's consider the partition of [-m,m) given by clopen intervals
$$ A_i=\bigg[-m + \frac{i-1}{2^m}, -m + \frac{i}{2^m} \bigg)$$
for $i=1,...,2m2^m$. We define the increasing sequence of step functions as follows:
$$ f_m(x) = \begin{cases} -\infty & f(x)< -m \\ -m + \frac{i-1}{2^m} & f(x) \in A_i (i=1,...,2m2^m)\\ m & f(x) \geq m \end{cases} $$
It's clear that this sequence's point wise limit is $f$, and if $|f(x)| < M \ \ \ \forall x$, then convergence is uniform.