I am aware that this is a duplicate questions, but none of the answers on here were able to help me. Here is the problem.
Suppose $f$ is non-negative and measurable and $\mu$ is $\sigma$-finite. Show there exist simple functions $s_n$ increasing to $f$ at each point such that $\mu(\left\lbrace x:s_n(x)\neq 0\right\rbrace)<\infty$ for each $n$.
I basically know two things from previous work in this class.
1) There exists a sequence of non-negative simple functions $s_n$ increasing to $f$, and
2) There exist $E_i\in\mathcal{A}$ for $i=1,2,\dots$, such that $\mu(E_i)<\infty$ for each $i$ and $X=\cup_{i=1}^\infty E_i$.
It seems to me that these two facts should be enough to get me the proof, but I am stuck and don't know how to proceed. I wanted to define the simple functions somehow in terms of the sets $E_i$ and see where I could go with that. But I really need a strong hint (easy hints aren't doing it for me this semester. This measure theory stuff is ridiculous).