The Radon-Nikodym and Riesz representation theorems

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

Last updates: 10 March 2011

The Radon-Nikodym and Riesz representation theorems

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

(a)   The function $\phi :ℳ\to [0,\infty ]$ given by

$\phi \left(E\right)={\displaystyle {\int }_{E}fd\mu }$

is a positive measure on $ℳ$.

(b)   If $g:X\to [0,\infty ]$ is measurable then

${\displaystyle {\int }_{X}gd\phi }={\displaystyle {\int }_{X}gfd\mu }$.

Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ 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 ${\lambda }_1\left(E\right)={\lambda }_1(A\cap E)$, and

(b)  If $E\in ℳ$ then ${\lambda }_2\left(E\right)={\lambda }_2(B\cap E)$.

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 $i\in {\mathbb{Z}}_{>0}$ then $\mu \left(E_i\right)<\infty$.

[Ru, Theorem 6.10] Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ 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

$\lambda ={\lambda }_a+{\lambda }_s,{\lambda }_a\ll \mu \text{and}{\lambda }_s\perp \mu .$

(b)   There is a unique $h\in L^1\left(\mu \right)$ such that

if $E\in ℳ$    then  ${\lambda }_a\left(E\right)={\displaystyle {\int }_{E}hd\mu }$.

Let $(X,ℳ)$ 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     $\Phi \left(f\right)={\displaystyle {\int }_{E}fgd\mu }$.

(b)   If $g$ is as in (a) then

$\Vert \Phi \Vert ={\Vert g\Vert }_{\infty }$.

Let $(X,ℳ)$ 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     $\Phi \left(f\right)={\displaystyle {\int }_{E}fgd\mu }$.

(b)   If $g$ is as in (a) then

$\Vert \Phi \Vert ={\Vert g\Vert }_q$.

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

if $f\in C_c\left(X\right)$     then     $\Lambda f={\displaystyle {\int }_{X}fd\mu }$.

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

(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 $\mu$.

NonExistential versions

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

$w={\sum }_{n=1}^{\infty }{w}_n$,     where     $w_n=\{\begin{array}{cc}{\displaystyle \frac{2^{-n}}{1+\mu \left(E_n\right)},}& \text{if}x\in E_n,\\ 0,& \text{if}x\notin E_n.\end{array}$

Define a positive measure $\phi :ℳ\to [0,\infty ]\mathrm{????}$ by

$\phi \left(E\right)={\int }_X{\chi }_{E}d\lambda +{\int }_X{\chi }_{E}wd\mu$.

Let

$\Phi :L^2\left(\phi \right)\to ℂ$    be given by    $\Phi \left(f\right)={\int }_{X}fd\lambda$.

Let

$A=\{x\in X|0\le g\left(x\right)<1\}$     and     $B=\{x\in X|g\left(x\right)=1\}$.

Define ${\lambda }_a:ℳ\to ℂ$ and ${\lambda }_s:ℳ\to ℂ$ by

${\lambda }_a\left(E\right)=\lambda (A\cap E)$     and     ${\lambda }_s\left(E\right)=\lambda (B\cap E)$.

Let $h:X\to ℂ$ be given by

$h=\underset{n\to \infty }{\lim }g(1+g+\cdots +g^n)w$.

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 ${\lambda }_a\left(E\right)={\displaystyle {\int }_{E}hd\mu }$.

(a)   If ${\nu }_a$ and ${\nu }_s$ are complex measures such that

$\nu ={\nu }_a+{\nu }_s,{\nu }_a\ll \mu \text{and}{\nu }_s\perp \mu .$

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  ${\lambda }_a\left(E\right)={\displaystyle {\int }_{E}h\prime d\mu }$

then $h=h\prime$.

Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ 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 $\lambda \left(E\right)={\int }_X{\chi }_{E}gd\mu$.

(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     $\Phi \left(f\right)={\displaystyle {\int }_{X}fgd\mu }$,

(e)   $\Vert \Phi \Vert ={\Vert g\Vert }_q$,

(f)   If $g\prime \in L^q\left(\mu \right)$ such that

if $f\in L^p\left(\mu \right)$     then     $\Phi \left(f\right)={\displaystyle {\int }_{E}fg\prime d\mu }$

then $g\prime =g$.

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

$\mu \left(V\right)=\sup \left\{\Lambda f\right|f\in C_c\left(X\right),0\le f\le 1,\mathrm{supp}f\subseteq V\}$,     for $V$ open,

and

$\mu \left(E\right)=\inf \left\{\mu \left(V\right)\right|E\subseteq V\text{and}V\text{is open}\}$,

Then

(a)   $\mu :ℬ\to [0,\infty ]$ is a positive regular Borel measure,

(b)   If $f\in C_c\left(X\right)$ then $\Lambda f={\displaystyle {\int }_{X}fd\mu }$.

(c)   If $\nu :ℬ\to [0,\infty ]$ is a positive regular Borel measure which satisfies

if $f\in C_c\left(X\right)$     then     $\Lambda f={\displaystyle {\int }_{X}fd\nu }$

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

$\Lambda f=\sup \left\{\right|\Phi \left(h\right)\left|\right|h\in C_c\left(X\right),\left|h\right|\le f\}$,     if $f:X\to {ℝ}_{\ge 0}$,

$\Lambda f=$ \Lambda f^{+}-$ \Lambda f^{-}$,\; if$ f:X\to ℝ$,$ f=f^{+}-f^{-}$,\; and$ \left|f\right|=f^{+}+f^{-}$,$$

$\Lambda f=$ \Lambda u+i\Lambda v$,\; if$ f:X\to ℂ$,$ f=u+ivwith$ u,vX\to ℝ$.$$

Use the Positive Reisz representation theorem to get a positive regular Borel measure $\lambda :ℬ\to [0,\infty ]$ such that

$\Lambda f={\int }_{X}fd\lambda$,     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

$\Phi f={\int }_{X}fgd\lambda$,     for $f\in L^1\left(\lambda \right)$.

Define $\mu :ℬ\to ℂ$ by

$\mu \left(E\right)={\int }_X{\chi }_{E}gd\lambda$.

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

(e)   $\Vert \Phi \Vert =\Vert \mu \Vert$,

(f)   If $\mu \prime$$is\; a\; regular\; complex\; Borel\; measure\; such\; that$

if $f\in C_0\left(X\right)$     then     $\Phi \left(f\right)={\displaystyle {\int }_{X}fd\mu \prime }$

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.

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