Proposition 14.1.9.label Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $p \in [1, \infty)$, then $\Sigma(X, \cm; E) \cap L^{p}(X; E)$ is dense in $L^{p}(X; E)$.
Proof. Let $f \in L^{p}(X; E)$. By Definition 23.1.1, there exists $\seq{f_n}\subset \Sigma(X, \cm; E)$ such that $\norm{f_n}_{E} \le \norm{f}_{E}$ and $\norm{f_n - f}_{E} \to 0$ strongly pointwise as $n \to \infty$. By the Dominated Convergence Theorem, $f_{n} \to f$ in $L^{p}(X; E)$.$\square$