Distributions

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

Last updates: 19 March 2011

Locally compact Hausdorff topological spaces

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

A Hausdorff topological space is a topological space $(X,𝒯)$ 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 $k\in {\mathbb{Z}}_{>0}$. The topological space ${ℝ}^k$ is a locally compact Hausdorff topological space.

(Heine-Borel) Let $k\in {\mathbb{Z}}_{>0}$ 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

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

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.

(a)   $C_0\left(X\right)$ with $\Vert \Vert$ is a Banach space.

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

Regular measures

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

${\displaystyle \left|\mu \right|\left(E\right)=\sup \left\{\sum _{i=1}^{\infty }\left|\mu \left(E_i\right)\right|\right|E_1,E_2,\dots \in ℳ\text{partition}E\}}$.

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 $\mu :ℬ\to [0,\infty ]$ such that if $E\in ℬ$ then $$\mu \left(E\right)=\sup \{\mu \left(K\right)|K\subseteq E\text{is compact}\}=\inf \{\mu (U\left)\right|U\supseteq E\text{is open}\}.$$ A regular complex Borel measure is a complex measure $\mu :ℬ\to ℂ$ such that the total variation measure $\left|\mu \right|$ is regular. The norm of $\mu$ is

$\Vert \mu \Vert =\left|\mu \right|(X)$.

[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 $\Vert \Vert$ 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     $\Phi \left(f\right)={\displaystyle {\int }_{X}fd\mu }$.

(b)   If $\mu$ is as in (a) then

$\Vert \Phi \Vert =\left|\mu \right|\left(X\right)$,

where $\left|\mu \right|$ is the total variation measure corresponding to $\mu$.

Let $X$ be a locally compact Hausdorff topological space. Let $\mu :ℬ\to [0,\infty ]$ 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 ${\Vert \Vert }_p$.

(b)   The space $L^{\infty }\left(\mu \right)$ is not necessarily the completion of $C_c\left(X\right)$ with respect to ${\Vert \Vert }_{\infty }$.

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 $\begin{array}{ccc}\left\{\text{regular complex Borel measures}\right\}& \to & \left\{\text{bounded linear functionals on}{C}_c\right(X\left)\right\}\end{array}given\; by$

$\mu \left(f\right)={\displaystyle {\int }_{X}fd\mu }$

is an isometry of Banach spaces.

(b)   The map $\begin{array}{ccc}\left\{\text{regular positive Borel measures}\right\}& \to & \left\{\text{positive linear functionals on}{C}_c\right(X\left)\right\}\end{array}given\; by$

$\mu \left(f\right)={\displaystyle {\int }_{X}fd\mu }$

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.

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