Proposition 14.6.6. Let $X$ be a LCH space, $\mu: \cb_{X} \to [0, \infty]$ be a Radon measure, $E$ be a normed space, and $p \in [1, \infty)$, then $C_{c}(X; E)$ is dense in $L^{p}(X; E)$.
Proof. By Proposition 12.1.8, $\Sigma(X, \cm; E) \cap L^{p}(X; E)$ is dense in $L^{p}(X; E)$. Using linearity, it is sufficient to approximate indicator functions of Borel sets with finite measure.
Let $A \in \cb_{X}$ and $\eps > 0$. By Proposition 14.6.3, there exists $U \in \cn^{o}(A)$ and $K \subset A$ compact such that $\mu(U \setminus A), \mu(A \setminus K) < \eps/2$. By Urysohn’s lemma, there exists $f \in C_{c}(X; [0, 1])$ such that $f|_{K} = 1$ and $\supp{f}\subset U$. In which case, for any $y \in E$,
\[\norm{x \cdot \one_A - x \cdot f}_{L^p(X; E)}\le \norm{x}_{E} \mu(\bracs{f \ne \one_A})^{1/p}\le \norm{x}_{E}\mu(U \setminus K)^{1/p}< \eps^{1/p}\norm{x}_{E}\]
$\square$