Let $(X, \mathcal{A}, \mu)$ be a measure space and let $1 \le p < \infty$. Then the set of simple functions in $L^p(X, \mu)$ is dense in $L^p(X, \mu)$. That is, for every $f \in L^p(X, \mu)$ and every $\varepsilon > 0$, there exists a simple function $s: X \to \mathbb{R}$ such that