Let $(X, \mathcal{B}, \mu)$ be a measure space, where $X$ is a locally compact Hausdorff space, $\mathcal{B}$ a $\sigma$ -algebra that contains all compact subsets of $X$ and $\mu$ a measure such that:

• $\mu(K) < \infty$ for all compact sets $K \subset X$ .
• $\mu$ is inner regular, meaning $\mu(A) = \sup\{ \mu(K) : K \subset A, \; K\,{is compact}\}$
• $\mu$ is outer regular, meaning $\mu(A) = \inf\{ \mu(U) : A \subset U,\; U \in \mathcal{B} {and}\; U\,{is open}\}$

We denote by $C_c(X)$ the space of continuous functions $X \to \mathbb{C}$ with compact support.

Theroem - For every $1 \leq p < \infty$ , $C_c(X)$ is dense in $L^p(X)$ .

Proof: It is clear that $C_c(X)$ is indeed contained in $L^p(X)$ , where we identify each function in $C_c(X)$ with its class in $L^p(X)$ .

We begin by proving that for each $A \in \mathcal{B}$ with finite measure, the characteristic function $\chi_A$ can be approximated, in the $L^p$ norm, by functions in $C_c(X)$ . Let $\epsilon > 0$ . By inner and outer regularity of $\mu$ , we know there exist an open set $U$ and a compact set $K$ such that $K \subset A \subset U$ and

Now, it follows easily that any simple function $\sum_{i=1}^n c_i \chi_{A_i}$ , where each $A_i$ has finite measure, can also be approximated by a compactly supported continuous function. Since this kind of simple functions are dense in $L^p(X)$ we see that $C_c(X)$ is also dense in $L^p(X)$ . $\square$