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

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $\map {\LL^p} \mu$ be Lebesgue $p$-space for $\mu$.

Let $\map \EE \Sigma \cap \map {\LL^p} \mu$ be the space of $\Sigma$-simple, $p$-integrable functions.

Then $\map \EE \Sigma \cap \map {\LL^p} \mu$ is everywhere dense in $\map {\LL^p} \mu$.