Let $f$ be an integrable function on $\mathbb{R}^n$. Then there exists a sequence of simple functions $\{f_n\}$ such that $\mid f_n \mid \leq \mid f \mid$ for all $n$ and $f_n$ converges to $f$ almost everywhere.
Now, my question is that is it possible to have each $f_n$ as a (nonzero) linear combination of characteristic functions on compact subsets? The Lebesgue measure allows each measurable set of finite measure to be approximated by compact subsets.
Then, is it possible for me to utilize this fact and setting 'the underlying sets' of each $f_n$ to be compact? I am very curious...