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'compactly supported continuous functions are dense in $L^p$'
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| Title of object: |
compactly supported continuous functions are dense in $L^p$ |
| Canonical Name: |
CompactlySupportedContinuousFunctionsAreDenseInLp |
| Type: |
Theorem |
| Created on: |
2008-12-26 22:28:53 |
| Modified on: |
2008-12-26 22:28:53 |
| Classification: |
msc:28C15, msc:46E30, msc:54C35 |
Preamble:
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Content:
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:
\begin{itemize}
\item $\mu(K) < \infty$ for all compact sets $K \subset X$.
\item $\mu$ is inner regular, meaning $\mu(A) = \sup\{ \mu(K) : K \subset A, \; K\,\text{is compact}\}$
\item $\mu$ is outer regular, meaning $\mu(A) = \inf\{ \mu(U) : A \subset U,\; U \in \mathcal{B} \text{and}\; U\,\text{is open}\}$
\end{itemize}
We denote by $C_c(X)$ the space of continuous functions $X \to \mathbb{C}$ with compact support.
{\bf Theroem -} For every $1 \leq p < \infty$, $C_c(X)$ is dense in \PMlinkname{$L^p(X)$}{LpSpace}.
{\bf \emph{\PMlinkescapetext{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 \PMlinkescapetext{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
\begin{align*}
\mu(U \setminus K) = \mu(U) - \mu(K) < \epsilon
\end{align*}
By the \PMlinkname{Urysohn's lemma for locally compact Hausdorff spaces}{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,
\begin{align*}
\int_X |\chi_A - f|^p \;d\mu = \int_{U \setminus K} |\chi_A - f|^p \;d\mu < \epsilon
\end{align*}
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 $L^p(X)$. $\square$ |
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