## The Banach spaces ${L}^{p}\left(\mu \right)$, ${L}^{\infty }\left(\mu \right)$, and ${C}_{0}\left(X\right)$

Let $\left(X,ℳ\right)$ be a measurable space and let $\mu :ℳ\to \left[0,\infty \right]$ 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

 ${‖f‖}_{p}={\left({\int }_{X}{|f|}^{p}\phantom{\rule{0.1em}{0ex}}d\mu \right)}^{1/p}$.
Define
 ${L}^{p}\left(\mu \right)=\left\{f:X\to ℂ\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}\text{is measurable and}\phantom{\rule{0.2em}{0ex}}{‖f‖}_{p}<\infty \right\}$

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

 $\mathrm{essup}f=\mathrm{inf}\left\{\alpha \in ℝ\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}\mu \left({f}^{-1}\left(\left(\alpha ,\infty \right]\right)\right)=0\right\}$
with

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

 ${‖f‖}_{\infty }=\mathrm{essup}\phantom{\rule{0.2em}{0ex}}|f|$.
The essentially bounded measurable functions are the elements of
 ${L}^{\infty }\left(\mu \right)=\left\{f:X\to ℂ\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}\text{is measurable and}\phantom{\rule{0.2em}{0ex}}{‖f‖}_{\infty }<\infty \right\}$

Let $\left(X,ℳ\right)$ be a measurable space and let $\mu :ℳ\to \left[0,\infty \right]$ 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 ${‖\phantom{\rule{0.2em}{0ex}}‖}_{p}$.
(b)   ${L}^{\infty }\left(\mu \right)$ is a complete metric space with respect to ${‖\phantom{\rule{0.2em}{0ex}}‖}_{\infty }$.

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

 $\mathrm{supp}\left(f\right)=\stackrel{‾}{\left\{x\in X\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}f\left(x\right)\ne 0\right\}}$,
the closure of $\left\{x\in X\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}f\left(x\right)\ne 0\right\}$. Define
 ${C}_{c}\left(X\right)=\left\{f:X\to ℂ\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}\text{is continuous and}\phantom{\rule{0.2em}{0ex}}\mathrm{supp}\left(f\right)\phantom{\rule{.2em}{0ex}}\text{is compact}\right\}$
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 $|f\left(x\right)|<\epsilon$.
Define
 ${C}_{0}\left(X\right)=\left\{f:X\to ℂ\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}\text{is continuous and}\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{.2em}{0ex}}\text{vanishes at infinity}\right\}$
Define $‖\phantom{\rule{.1em}{0ex}}‖:{C}_{0}\left(X\right)\to {ℝ}_{>0}$ by
 $‖f‖=\mathrm{sup}\left\{|f\left(x\right)|\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}x\in X\right\}$.

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

(c)   ${C}_{0}\left(X\right)$ is a complete metric space with respect to with $‖\phantom{\rule{.1em}{0ex}}‖$.

HW: Give an example showing that ${C}_{c}\left(X\right)$ is not always a complete metric space with respect to with $‖\phantom{\rule{.1em}{0ex}}‖$.

HW: Show that if $X={ℝ}^{k}$ and $\mu$ is Lebesgue measure then ${‖\phantom{\rule{.1em}{0ex}}‖}_{\infty }=‖\phantom{\rule{.1em}{0ex}}‖$ 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 $‖\phantom{\rule{.1em}{0ex}}‖$.
(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 ${‖\phantom{\rule{.1em}{0ex}}‖}_{p}$.

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 ${‖\phantom{\rule{.1em}{0ex}}‖}_{\infty }$.

## 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.