Space of Simple P-Integrable Functions is Everywhere Dense in Lebesgue Space

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space, and let $p \in \R$, $p \ge 1$.

Let $\mathcal{L}^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$.

Let $\mathcal E \left({\Sigma}\right) \cap \mathcal{L}^p \left({\mu}\right)$ be the space of $\Sigma$-simple, $p$-integrable functions.

Then $\mathcal E \left({\Sigma}\right) \cap \mathcal{L}^p \left({\mu}\right)$ is everywhere dense in $\mathcal{L}^p \left({\mu}\right)$.