Let $X$ be a metric space, $\mu$ a $\sigma$-finite Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ be its semi-norm. Here we use the Bochner integral. Let $\mathcal C_c(X)$ be the space of all $E$-valued continuous functions on $X$ with compact supports.
To fix a logical mistake in this proof, I have come across this result, i.e.,
Theorem: If $X$ is locally compact separable, then $\mathcal C_c(X)$ is dense in $\big (\mathcal L_p, \|\cdot\|_{\mathcal L_p} \big)$ for all $p \in [1, \infty)$.
Could you have a check on my attempt?
Proof: We need the following results, i.e.,
Lemma 1: The space $\mathcal S := \mathcal S(X, \mu, E)$ of simple functions is dense in $\big (\mathcal L_p, \|\cdot\|_{\mathcal L_p} \big)$ for all $p \in [1, \infty)$. [Prop 4.8 in Amann's Analysis III]
Lemma 2 Let $X$ be a locally compact separable metric space. Then $X$ is a Radon space.
Lemma 3 Let $X$ be a locally compact Hausdorff topological space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ such that $K \subset U$. Then there is an open subset $V$ of $X$ such that $K \subset V \subset \overline V \subset U$ and $\overline V$ is compact.
Let $\mathcal X$ be the Borel $\sigma$-algebra of $X$. By Lemma 1 and Minkowski's inequality, it suffices to approximate the map $e1_B$ with $e\in E \setminus \{0\}$ and $B \in \mathcal X$ such that $\mu(B) < \infty$ by a map $f \in \mathcal C_c(X)$.
Fix $\varepsilon>0$. By Lemma 2, there is a compact subset $K$ and an open subset $O$ of $X$ such that $K \subset B \subset O$ and $\mu(O \setminus K) < \varepsilon$.
By Lemma 3, then there is an open subset $V$ of $X$ such that $K \subset V \subset \overline V \subset O$ and $\overline V$ is compact. By Urysohn's lemma, there is a continuous $h:X \to [0, 1]$ such that $h(K)=1$ and $h(X \setminus V) = 0$. Let $f (x) := e h (x)$ for all $x\in X$. Then $$ \operatorname{supp} f = \operatorname{supp} h \subset \overline V. $$
Hence $f$ is compactly supported. We have $$ \| eh-e1_B\|_{\mathcal L_p} \le |e|^p \mu (O \setminus K) < |e|^p \varepsilon. $$
The proof then completes by taking the limit $\varepsilon \to 0^+$.
