This is question 5.D in Bartle's Elements of Integration.
If $f \in L(X,\mathcal X,\mu)$ and $\epsilon > 0$, then there exists a $\mathcal X$ − measurable simple function $\phi$ such that: $\int|f-\phi|d\mu<\epsilon$.
The answer seems trivial when $f$ is either (a.e.) positive or (a.e.) negative. However, when it is both positive and negative, I cannot define simple function that works for both cases.