compactly supported continuous functions are dense in $L^p$

Let $(X,ℬ,\mu )$ be a measure space, where $X$ is a locally compact Hausdorff space, $ℬ$ a $\sigma$-algebra (http://planetmath.org/SigmaAlgebra) that contains all compact subsets of $X$ and $\mu$ a measure such that:

• for all compact sets $K\subset X$.

• $\mu$ is inner regular, meaning $\mu (A)=sup\{\mu (K):K\subset A,K\text{is compact}\}$

• $\mu$ is outer regular, meaning $\mu (A)=inf\{\mu (U):A\subset U,U\in ℬ\text{and}U\text{is open}\}$

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

Theroem - For every , $C_c(X)$ is dense in $L^p(X)$ (http://planetmath.org/LpSpace).

: 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 $A\in ℬ$ with finite measure, the characteristic function ${\chi }_A$ can be approximated, in the $L^p$ norm, by functions in $C_c(X)$. Let $ϵ>0$. By of $\mu$, we know there exist an open set $U$ and a compact set $K$ such that $K\subset A\subset U$ and

By the Urysohn’s lemma for locally compact Hausdorff spaces (http://planetmath.org/ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces), we know there is a function $f\in C_c(X)$ such that $0\le f\le 1$, ${f|}_K=1$ and $\mathrm{supp}f\subset U$. Hence,

Thus, ${\chi }_A$ can be approximated in $L^p$ by functions in $C_c(X)$.

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 $L^p(X)$. $\mathrm{\square }$