## The Radon-Nikodym and Riesz representation theorems

[Ru, Theorem 1.29] Let $X$ be a measurable space and let $\mu :ℳ\to \left[0,\infty \right]$ be a positive measure on $ℳ$. Let $f:X\to \left[0,\infty \right]$ be a measurable function.

(a)   The function $\phi :ℳ\to \left[0,\infty \right]$ given by
 $\phi \left(E\right)={\int }_{E}f\phantom{\rule{0.1em}{0ex}}d\mu$
is a positive measure on $ℳ$.
(b)   If $g:X\to \left[0,\infty \right]$ is measurable then
 ${\int }_{X}g\phantom{\rule{0.1em}{0ex}}d\phi ={\int }_{X}gf\phantom{\rule{0.1em}{0ex}}d\mu$.

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 measure $\lambda$ is absolutely continuous with respect to $\mu$, $\lambda \ll \mu$, if $\lambda$ satisfies

 if $E\in ℳ$ and $\mu \left(E\right)=0$ then $\lambda \left(E\right)=0$.
Two measures ${\lambda }_{1}$ and ${\lambda }_{2}$ are mutually singular, ${\lambda }_{1}\perp {\lambda }_{2}$ if there exist $A,B\in ℳ$ such that
(a)  $A\cap B=\varnothing$,
(b)  If $E\in ℳ$ then ${\lambda }_{1}\left(E\right)={\lambda }_{1}\left(A\cap E\right)$, and
(b)  If $E\in ℳ$ then ${\lambda }_{2}\left(E\right)={\lambda }_{2}\left(B\cap E\right)$.
A σ-finite positive measure is a positive measure $\mu$ on $X$ such that there exist ${E}_{1},{E}_{2},\dots \in ℳ$ such that
 $X=\bigcup _{i=1}^{\infty }{E}_{i}$     and     if $i\in {ℤ}_{>0}$ then $\mu \left({E}_{i}\right)<\infty$.

[Ru, Theorem 6.10] Let $\left(X,ℳ\right)$ be a measurable space and let $\mu :ℳ\to \left[0,\infty \right]$ be a σ-finite positive measure. Let $\lambda :ℳ\to ℂ$ be a complex measure.

(a)   There exist unique complex measures ${\lambda }_{a}:ℳ\to ℂ$ and ${\lambda }_{s}:ℳ\to ℂ$ such that
 $\lambda ={\lambda }_{a}+{\lambda }_{s},\phantom{\rule{2em}{0ex}}{\lambda }_{a}\ll \mu \phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\lambda }_{s}\perp \mu .$
(b)   There is a unique $h\in {L}^{1}\left(\mu \right)$ such that
 if $E\in ℳ$    then  ${\lambda }_{a}\left(E\right)={\int }_{E}h\phantom{\rule{0.1em}{0ex}}d\mu$.

Let $\left(X,ℳ\right)$ be a measurable space and let $\mu$ be a σ-finite positive measure on $X$. Let $\Phi :{L}^{1}\left(\mu \right)\to ℂ$ be a bounded linear functional on ${L}^{1}\left(\mu \right)$.

(a)   There exists a unique $g\in {L}^{\infty }\left(\mu \right)$ such that
 if $f\in {L}^{1}\left(\mu \right)$     then     $\Phi \left(f\right)={\int }_{E}fg\phantom{\rule{0.1em}{0ex}}d\mu$.
(b)   If $g$ is as in (a) then
 $‖\Phi ‖={‖g‖}_{\infty }$.

Let $\left(X,ℳ\right)$ be a measurable space and let $\mu$ be a σ-finite positive measure on $X$. Let $p\in {ℝ}_{>1}$ and let

 $q\in {ℝ}_{>1}$     be given by     $\frac{1}{p}+\frac{1}{q}=1$.
Let $\Phi :{L}^{p}\left(\mu \right)\to ℂ$ be a bounded linear functional.
(a)   There exists a unique $g\in {L}^{q}\left(\mu \right)$ such that
 if $f\in {L}^{p}\left(\mu \right)$     then     $\Phi \left(f\right)={\int }_{E}fg\phantom{\rule{0.1em}{0ex}}d\mu$.
(b)   If $g$ is as in (a) then
 $‖\Phi ‖={‖g‖}_{q}$.

(Positive Reisz representation theorem) Let $X$ be a locally compact Hausdorff topological space. Let $\Lambda :{C}_{c}\left(X\right)\to \left[0,\infty \right]$ be a positive linear functional. Then there exists a unique regular positive Borel measure $\mu :ℬ\to \left[0,\infty \right]$ such that

 if $f\in {C}_{c}\left(X\right)$     then     $\Lambda f={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$.

(Complex Reisz representation theorem) 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 $\mu :ℬ\to ℂ$ 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 $\mu$.

## NonExistential versions

[Ru, Theorem 6.10] Let $\left(X,ℳ\right)$ be a measurable space and let $\mu :ℳ\to \left[0,\infty \right]$ be a σ-finite positive measure. Let $\lambda :ℳ\to ℂ$ be a complex measure. Let

 $w={\sum }_{n=1}^{\infty }{w}_{n}$,     where     ${w}_{n}=\left\{\begin{array}{cc}\frac{{2}^{-n}}{1+\mu \left({E}_{n}\right)},& \text{if}\phantom{\rule{1em}{0ex}}x\in {E}_{n},\\ 0,& \text{if}\phantom{\rule{1em}{0ex}}x\notin {E}_{n}.\end{array}$
Define a positive measure $\phi :ℳ\to \left[0,\infty \right]\mathrm{????}$ by
 $\phi \left(E\right)={\int }_{X}{\chi }_{E}\phantom{\rule{0.2em}{0ex}}d\lambda +{\int }_{X}{\chi }_{E}w\phantom{\rule{0.2em}{0ex}}d\mu$.
Let
 $\Phi :{L}^{2}\left(\phi \right)\to ℂ$    be given by    $\Phi \left(f\right)={\int }_{X}f\phantom{\rule{0.2em}{0ex}}d\lambda$.
Let
 $A=\left\{x\in X\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}0\le g\left(x\right)<1\right\}$     and     $B=\left\{x\in X\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}g\left(x\right)=1\right\}$.
Define ${\lambda }_{a}:ℳ\to ℂ$ and ${\lambda }_{s}:ℳ\to ℂ$ by
 ${\lambda }_{a}\left(E\right)=\lambda \left(A\cap E\right)$     and     ${\lambda }_{s}\left(E\right)=\lambda \left(B\cap E\right)$.
Let $h:X\to ℂ$ be given by
 $h=\underset{n\to \infty }{\mathrm{lim}}g\left(1+g+\cdots +{g}^{n}\right)w$.
Then
(a)   ${\lambda }_{a}$ is a complex measure.
(b)   ${\lambda }_{s}$ is a complex measure.
(c)   $\lambda ={\lambda }_{a}+{\lambda }_{s}$,
(d)   ${\lambda }_{a}\ll \mu$,
(e)   ${\lambda }_{s}\perp \mu$,
(f)   $h\in {L}^{1}\left(\mu \right)$,
(g)   if $E\in ℳ$ then ${\lambda }_{a}\left(E\right)={\int }_{E}h\phantom{\rule{0.1em}{0ex}}d\mu$.
(a)   If ${\nu }_{a}$ and ${\nu }_{s}$ are complex measures such that
 $\nu ={\nu }_{a}+{\nu }_{s},\phantom{\rule{2em}{0ex}}{\nu }_{a}\ll \mu \phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\nu }_{s}\perp \mu .$
then ${\nu }_{a}={\lambda }_{a}$ and ${\nu }_{s}={\lambda }_{s}$.
(b)   If $h\prime \in {L}^{1}\left(\mu \right)$ such that
 if $E\in ℳ$    then  ${\lambda }_{a}\left(E\right)={\int }_{E}h\prime \phantom{\rule{0.1em}{0ex}}d\mu$
then $h=h\prime$.

Let $\left(X,ℳ\right)$ be a measurable space and let $\mu :ℳ\to \left[0,\infty \right]$ be a σ-finite positive measure on $X$. Let $p\in {ℝ}_{>1}$ and let

 $q\in {ℝ}_{>1}$     be given by     $\frac{1}{p}+\frac{1}{q}=1$.
Let $\Phi :{L}^{p}\left(\mu \right)\to ℂ$ be a bounded linear functional.
Let $\lambda :ℳ\to ℂ$ be given by $\lambda \left(E\right)=\Phi \left({\chi }_{E}\right)$.
Use Radon-Nikodym to produce $g\in {L}^{1}\left(\mu \right)$ such that $\lambda \left(E\right)={\int }_{X}{\chi }_{E}g\phantom{\rule{0.2em}{0ex}}d\mu$.
(a)   $\lambda$ is a complex measure,
(b)   $\lambda \ll \mu$,
(c)   $g\in {L}^{q}\left(\mu \right)$,
(d)   if $f\in {L}^{p}\left(\mu \right)$     then     $\Phi \left(f\right)={\int }_{X}fg\phantom{\rule{0.1em}{0ex}}d\mu$,
(e)   $‖\Phi ‖={‖g‖}_{q}$,
(f)   If $g\prime \in {L}^{q}\left(\mu \right)$ such that
 if $f\in {L}^{p}\left(\mu \right)$     then     $\Phi \left(f\right)={\int }_{E}fg\prime \phantom{\rule{0.1em}{0ex}}d\mu$
then $g\prime =g$.

(Positive Reisz representation theorem) Let $X$ be a locally compact Hausdorff topological space. Let $\Lambda :{C}_{c}\left(X\right)\to \left[0,\infty \right]$ be a bounded linear functional. Let $𝒫\left(X\right)$ be the set of all subsets of $X$ and let $\mu :𝒫\left(X\right)\to \left[0,\infty \right]$ be given by

 $\mu \left(V\right)=\mathrm{sup}\left\{\Lambda f\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\in {C}_{c}\left(X\right),0\le f\le 1,\mathrm{supp}f\subseteq V\right\}$,     for $V$ open,
and
 $\mu \left(E\right)=\mathrm{inf}\left\{\mu \left(V\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}E\subseteq V\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}V\phantom{\rule{0.5em}{0ex}}\text{is open}\right\}$,
Then
(a)   $\mu :ℬ\to \left[0,\infty \right]$ is a positive regular Borel measure,
(b)   If $f\in {C}_{c}\left(X\right)$ then $\Lambda f={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$.
(c)   If $\nu :ℬ\to \left[0,\infty \right]$ is a positive regular Borel measure which satisfies
 if $f\in {C}_{c}\left(X\right)$     then     $\Lambda f={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\nu$
then $\nu =\mu$.

(Complex Reisz representation theorem) Let $X$ be a locally compact Hausdorff topological space. Let $\Phi :{C}_{0}\left(X\right)\to ℂ$ be a bounded linear functional. Define $\Lambda :{C}_{c}\left(X\right)\to ℂ$ by

 $\Lambda f=\mathrm{sup}\left\{|\Phi \left(h\right)|\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}h\in {C}_{c}\left(X\right),|h|\le f\right\}$,     if $f:X\to {ℝ}_{\ge 0}$,
Use the Positive Reisz representation theorem to get a positive regular Borel measure $\lambda :ℬ\to \left[0,\infty \right]$ such that
 $\Lambda f={\int }_{X}f\phantom{\rule{0.2em}{0ex}}d\lambda$,     for $f\in {C}_{c}\left(X\right)$.
Use ${{L}^{1}\left(\lambda \right)}^{*}={L}^{\infty }\left(\lambda \right)$ to get $g\in {L}^{\infty }\left(\lambda \right)$ such that
 $\Phi f={\int }_{X}fg\phantom{\rule{0.2em}{0ex}}d\lambda$,     for $f\in {L}^{1}\left(\lambda \right)$.
Define $\mu :ℬ\to ℂ$ by
 $\mu \left(E\right)={\int }_{X}{\chi }_{E}g\phantom{\rule{0.2em}{0ex}}d\lambda$.
Then
(a)   $\Lambda :{C}_{c}\left(X\right)\to ℂ$ is a positive linear functional on ${C}_{c}\left(X\right)$,
(b)   $\Phi :{C}_{c}\left(X\right)\to ℂ$ extends to a bounded linear functional on $\Phi :{L}^{1}\left(\lambda \right)\to ℂ$,
(c)   $\mu$ is a regular complex Borel measure,
(d)   if $f\in {C}_{0}\left(X\right)$ then $\Phi \left(f\right)={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$.
(e)   $‖\Phi ‖=‖\mu ‖$,
(f)   If $\mu \prime$$is a regular complex Borel measure such that$
 if $f\in {C}_{0}\left(X\right)$     then     $\Phi \left(f\right)={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu \prime$
then $\mu \prime =\mu$.

## 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]. The nonExistential versions above need work: they don't cover the various cases clearly, and are slightly inaccurate in places. See [Ru, Chapters 1-6].

## References

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