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

Let $(X,\mathcal{B},\mu)$ be a measure space, where $X$ is a locally compact Hausdorff space, $\mathcal{B}$ a $\sigma$ -algebra (http://planetmath.org/SigmaAlgebra) 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 $\mu(A)=\sup\{\mu(K):K\subset A,\;K\,\text{is compact}\}$
•

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

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 $L^{p}(X)$ (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 $A\in\mathcal{B}$ 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 of $\mu$ , we know there exist an open set $U$ and a compact set $K$ such that $K\subset A\subset U$ and

\displaystyle\mu(U\setminus K)=\mu(U)-\mu(K)<\epsilon

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 $0\leq f\leq 1$ , $f|_{K}=1$ and $\mathrm{supp}\,f\subset U$ . Hence,

\displaystyle\int_{X}|\chi_{A}-f|^{p}\;d\mu=\int_{U\setminus K}|\chi_{A}-f|^{p% }\;d\mu<\epsilon

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

Title	compactly supported continuous functions are dense in $L^{p}$
Canonical name	CompactlySupportedContinuousFunctionsAreDenseInLp
Date of creation	2013-03-22 18:38:53
Last modified on	2013-03-22 18:38:53
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	6
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 54C35
Classification	msc 46E30
Classification	msc 28C15
Synonym	$C_{c}(X)$ is dense in $L^{p}(X)$

compactly supported continuous functions are dense in Lp

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