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:

•
$$ for all compact sets $K\subset X$.

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

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$\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^{} $X\to \u2102$ with compact support.
Theroem  For every $$, ${C}_{c}(X)$ is dense in ${L}^{p}(X)$ (http://planetmath.org/LpSpace).
: It is clear that ${C}_{c}(X)$ is indeed contained in ${L}^{p}(X)$, where we identify each function in ${C}_{c}(X)$ with its class in ${L}^{p}(X)$.
We begin by proving that for each $A\in \mathcal{B}$ with finite measure, the characteristic function^{} ${\chi}_{A}$ can be approximated, in the ${L}^{p}$ norm, by functions in ${C}_{c}(X)$. Let $\u03f5>0$. By of $\mu $, we know there exist an open set $U$ and a compact set $K$ such that $K\subset A\subset U$ and
$$ 
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 ${L}^{p}(X)$. $\mathrm{\square}$
Title  compactly supported continuous functions are dense in ${L}^{p}$ 

Canonical name  CompactlySupportedContinuousFunctionsAreDenseInLp 
Date of creation  20130322 18:38:53 
Last modified on  20130322 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 ${L}^{p}(X)$ 