Locally compact Hausdorff topological spaces

A locally compact topological space is a topological space $\left(X,𝒯\right)$ such that if $x\in X$ then there exists $U\in 𝒯$ such that $x\in U$ and $\stackrel{‾}{U}$ is compact, where $\stackrel{‾}{U}$ is the closure of $U$.

A Hausdorff topological space is a topological space $\left(X,𝒯\right)$ such that if $p,q\in X$ and $p\ne q$ then there exist $U,V\in 𝒯$ such that $p\in U$, $q\in V$ and $U\cap V=\varnothing$.

Let $k\in {ℤ}_{>0}$. The topological space ${ℝ}^{k}$ is a locally compact Hausdorff topological space.

(Heine-Borel) Let $k\in {ℤ}_{>0}$ and let $K\subseteq {ℝ}^{k}$. Then $K$ is compact if and only if $K$ is closed and bounded.

The Banach space ${C}_{0}\left(X\right)$

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.

(a)   ${C}_{0}\left(X\right)$ with $‖\phantom{\rule{.1em}{0ex}}‖$ is a Banach space.
(b)   ${C}_{0}\left(X\right)$ is the completion of ${C}_{c}\left(X\right)$ with respect to $‖\phantom{\rule{.1em}{0ex}}‖$.

Regular measures

Let $\left(X,ℳ\right)$ be a measurable space and let $\mu :ℳ\to ℂ$ be a complex measure. The total variation of $\mu$ is the positive measure $|\mu |:ℳ\to \left[0,\infty \right]$ given by

 $|\mu |\left(E\right)=\mathrm{sup}\left\{\sum _{i=1}^{\infty }|\mu \left({E}_{i}\right)|\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}{E}_{1},{E}_{2},\dots \in ℳ\phantom{\rule{0.5em}{0ex}}\text{partition}\phantom{\rule{0.5em}{0ex}}E\right\}$.

Let $X$ be a locally compact Hausdorff topological space with topology $𝒯$ and let $ℬ$ be the σ-algebra generated by $𝒯$. A regular positive Borel measure is a positive measure $\mu :ℬ\to \left[0,\infty \right]$ such that if $E\in ℬ$ then $μ (E) =sup{ μ (K) | K⊆E is compact } =inf { μ (U) | U⊇E is open }.$ A regular complex Borel measure is a complex measure $\mu :ℬ\to ℂ$ such that the total variation measure $|\mu |$ is regular. The norm of $\mu$ is

 $‖\mu ‖=|\mu |\left(X\right)$.

[Ru, Chapt. 6 Ex. 3] Let $X$ be a locally compact Hausdorff topological space. The space $M\left(X\right)$ of regular complex Borel measures on $X$ with $‖\phantom{\rule{.1em}{0ex}}‖$ is a Banach space.

(Reisz representation theorem) [Ru, Theorem 6.19] Let $X$ be a locally compact Hausdorff topological space. Let $\Phi :{C}_{0}\left(X\right)\to ℂ$ be a bounded linear functional.

(a)   There exists a unique regular complex Borel measure $\mu$$such that$
 if $f\in {C}_{0}\left(X\right)$     then     $\Phi \left(f\right)={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$.
(b)   If $\mu$ is as in (a) then
 $‖\Phi ‖=|\mu |\left(X\right)$,
where $|\mu |$ is the total variation measure corresponding to $\mu$.

Let $X$ be a locally compact Hausdorff topological space. Let $\mu :ℬ\to \left[0,\infty \right]$ be a regular positive Borel measure on $X$.

(a)   If $p\in {ℝ}_{\ge 1}$ then the space ${L}^{p}\left(\mu \right)$ is the completion(COMPLETION VS DENSE???) of ${C}_{c}\left(X\right)$ with respect to ${‖\phantom{\rule{.1em}{0ex}}‖}_{p}$.
(b)   The space ${L}^{\infty }\left(\mu \right)$ is not necessarily the completion of ${C}_{c}\left(X\right)$ with respect to ${‖\phantom{\rule{.1em}{0ex}}‖}_{\infty }$.

A positive linear functional on ${C}_{c}\left(X\right)$ is a linear functional $\mu :{C}_{c}\left(X\right)\to ℂ$ such that if $f:X\to ℂ$ and $f\left(X\right)\subseteq {ℝ}_{\ge 0}then\mu \left(f\right)\in {ℝ}_{\ge 0}.$

Distributions

A distribution on $X$ is a continuous linear functional $\mu :{C}_{c}\left(X\right)\to ℂ$. SAY WHAT THE TOPOLOGY ON ${C}_{c}\left(X\right)$ is. THIS NEEDS REFERENCES!

Parts (a) and (b) of the following theorem form the Reisz representation theorem.

[Ru, Theorems 2.14 and 6.19] Let $X$ be a locally compact Hausdorff topological space.

(a)   The map $\begin{array}{ccc}\left\{\text{regular complex Borel measures}\right\}& \to & \left\{\phantom{\rule{.1em}{0ex}}\text{bounded linear functionals on}\phantom{\rule{.1em}{0ex}}{C}_{c}\left(X\right)\right\}\end{array}given by$
 $\mu \left(f\right)={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$
is an isometry of Banach spaces.
(b)   The map $\begin{array}{ccc}\left\{\text{regular positive Borel measures}\right\}& \to & \left\{\phantom{\rule{.1em}{0ex}}\text{positive linear functionals on}\phantom{\rule{.1em}{0ex}}{C}_{c}\left(X\right)\right\}\end{array}given by$
 $\mu \left(f\right)={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$
is an isometry of Banach spaces.

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 1-6]. WHAT IS THE RIGHT REFERENCE FOR DISTRIBUTIONS???

References

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