The function spaces $L^p\left(\mu \right),L^{\infty }\left(\mu \right),C_c\left(X\right)$, and $C_0\left(X\right)$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 19 March 2011

The Banach spaces $L^p\left(\mu \right)$, $L^{\infty }\left(\mu \right)$, and $C_0\left(X\right)$

Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ be a positive measure on $ℳ$.
Let $p\in {ℝ}_{>0}$ and let $f:X\to ℂ$ be a measurable function. The $L_p$-norm of $f$ is

${\Vert f\Vert }_p={\left({\int }_X{\left|f\right|}^{p}d\mu \right)}^{1/p}$.

Define

${\displaystyle L^p\left(\mu \right)=\{f:X\to ℂ|f\text{is measurable and}{\Vert f\Vert }_p<\infty }\}$

Let $f:X\to ℂ$ be a measurable function. The essential supremum of $f$ is

$\mathrm{essup}f=\inf \{\alpha \in ℝ|\mu \left(f^{-1}\right((\alpha ,\infty ]\left)\right)=0\}$

with $\mathrm{essup}f=\infty $ if$ \{\alpha \in ℝ|\mu \left({\left|f\right|}^{-1}\right((\alpha ,\infty ])=0\}=\varnothing $.$$

Let $f:X\to ℂ$ be a measurable function. The $L_{\infty }$-norm of $f$ is

${\Vert f\Vert }_{\infty }=\mathrm{essup}\left|f\right|$.

The essentially bounded measurable functions are the elements of

${\displaystyle L^{\infty }\left(\mu \right)=\{f:X\to ℂ|f\text{is measurable and}{\Vert f\Vert }_{\infty }<\infty }\}$

Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ be a positive measure on $ℳ$.

(a)   If $p\in {ℝ}_{\ge 1}$ then $L^p\left(\mu \right)$ is a complete metric space with respect to ${\Vert \Vert }_p$.

(b)   $L^{\infty }\left(\mu \right)$ is a complete metric space with respect to ${\Vert \Vert }_{\infty }$.

Let $X$ be a locally compact Hausdorff topological space. Let $f:X\to ℂ$ be a continuous function. The support of $f$ is

$\mathrm{supp}\left(f\right)=\overline{\{x\in X|f\left(x\right)\ne 0\}}$,

the closure of $\{x\in X|f\left(x\right)\ne 0\}$. Define

${\displaystyle C_c\left(X\right)=\{f:X\to ℂ\left|f\text{is continuous and}\mathrm{supp}\right(f\left)\text{is compact}\right\}}$

A function $f:X\to ℂ$ vanishes at infinity if $f$ satisfies

if $\epsilon \in {ℝ}_{>0}$ then there exists a compact set $K\subseteq X$ such that if $x\notin K$ then $\left|f\right(x\left)\right|<\epsilon$.

Define

${\displaystyle C_0\left(X\right)=\{f:X\to ℂ\left|f\text{is continuous and}f\text{vanishes at infinity}\right\}}$

Define $\Vert \Vert :C_0\left(X\right)\to {ℝ}_{>0}$ by

$\Vert f\Vert =\sup \left\{\right|f\left(x\right)\left|\right|x\in X\}$.

Let $X$ be a locally compact Hausdorff topological space.

(c)   $C_0\left(X\right)$ is a complete metric space with respect to with $\Vert \Vert$.

HW: Give an example showing that $C_{\mathrm{c}}\left(X\right)$ is not always a complete metric space with respect to with $\Vert \Vert$.

HW: Show that if $X={ℝ}^k$ and $\mu$ is Lebesgue measure then ${\Vert \Vert }_{\infty }=\Vert \Vert$ giving that $L^{\infty }\left(\mu \right)=C_0\left({ℝ}^k\right)$.

Let $X$ be a locally compact Hausdorff topological space. Let $\mu$ be a regular positive Borel measure on $X$.

(a)   The space $C_0\left(X\right)$ is the completion of $C_c\left(X\right)$ with respect to $\Vert \Vert$.

(b)   If $p\in {ℝ}_{\ge 1}$ then the space $L^p\left(\mu \right)$ is the completion of $C_c\left(X\right)$ with respect to ${\Vert \Vert }_p$.

HW: Give an example showing that The space $L^{\infty }\left(\mu \right)$ is not necessarily the completion of $C_c\left(X\right)$ with respect to ${\Vert \Vert }_{\infty }$.

Notes and References

These notes were written for a course in "Measure Theory" at the Masters level at University of Melbourne. This presentation follows [Ru, Chapters 3]. See [Ru, Chapt. 3 Ex 21] for the resolution of the issue of completion vs dense subsets.

References

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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