Dgdrxrt

How can I prevent hyper evolved versions of regular creatures from wiping out their cousins?

How can I fix/modify my tub/shower combo so the water comes out of the showerhead?

Why "Having chlorophyll without photosynthesis is actually very dangerous" and "like living with a bomb"?

What mechanic is there to disable a threat instead of killing it?

Why doesn't H₄O²⁺ exist?

Memorizing the Keyboard

What is the most common color to indicate the input-field is disabled?

Why is consensus so controversial in Britain?

Facing a paradox: Earnshaw's theorem in one dimension

How could indestructible materials be used in power generation?

Should I tell management that I intend to leave due to bad software development practices?

90's TV series where a boy goes to another dimension through portal near power lines

SSH "lag" in LAN on some machines, mixed distros

How can saying a song's name be a copyright violation?

Why is Collection not simply treated as Collection<?>

A reference to a well-known characterization of scattered compact spaces

Has there ever been an airliner design involving reducing generator load by installing solar panels?

Why does Arabsat 6A need a Falcon Heavy to launch

Intersection of two sorted vectors in C++

Twin primes whose sum is a cube

Where does SFDX store details about scratch orgs?

Can one be a co-translator of a book, if he does not know the language that the book is translated into?

Theorems that impeded progress

Western buddy movie with a supernatural twist where a woman turns into an eagle at the end

The existence of conditional expectation, $(mathscrS,Sigma,mu) $ $sigma$-finite space

Radon–Nikodym derivative and “normal” derivativeUniqueness of product measure (non $sigma$-finite case)Radon–Nikodym theorem: special caseThe existence of conditional expectation with respect to a sub-$sigma$-algebraProperties of the Kernel from the measurable space $(X,mathscrA)$ to $(Y,mathscrB)$$mu$ is a $sigma-$finite measure on and $E_n$ measurable sets. When $nu(E)=sum mu(Ecap E_n)$, is $nu$ is $sigma$-finite?Weakening the “positive $mu$” condition in Radon-Nikodym theoremRadon-Nikodym Theorem for (positive) measures, chain ruleDoes Radon-Nikodym imply Riesz Representation Theorem?(Hint Needed) Real-Analysis Exam

0

$begingroup$

Let $(mathscrS,Sigma,mu)$ be a $sigma$-finite measure space, and let $f$ be $Sigma$-measurable and integrable over $mathscrS$. Let $Sigma_0$ be a $sigma$-algrbra satisfying $Sigma_0subseteq Sigma$. Show that there is a unique function $f_0$ which is $Sigma_0$-measurable such that $int fgdmu=int f_0gdmu$ for every $Sigma$-measurable $g$ for which the integrals are finite.
i

My attempt:

Define $phi(A)=int_A fdmu$ for $Ain Sigma_0$. Since $f$ is integrable, then $phi$ is a set function. So applying Radon-Nikodym theorem, there is unique $f_0$ $Sigma_0$- measurable such that $phi(A)=int_A f_0dmu$ for $Ain Sigma_0$.

Hence, $int_A fdmu=int_A f_0dmu$ for $Ain Sigma_0$.

Next, consider $g=mathcalX_E$ for $EinSigma_0$ is true. And simple function. For the nonnegative function use monotone convergence theorem. For general $g$ consider $g^+-g^-$.

Is it correct my approach?

real-analysis analysis

share|cite|improve this question

edited Mar 29 at 3:13

Zoe

asked Mar 29 at 3:02

ZoeZoe

134

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Shouldn't it be every $Sigma_0$-measurable $g$?
  $endgroup$
  – forgottenarrow
  Mar 29 at 4:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think you have the right idea. You apply the Radon-Nikodym derivative and then construct the integral from the resulting measure. I don't know if you left it out because this is just a proof sketch, but it's important to make sure that $phi(cdot)$ is absolutely continuous with respect to $mu_f(cdot) := int_cdot f,dmu$. However this should be simple.
  $endgroup$
  – forgottenarrow
  Mar 29 at 4:49
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $(mathscrS,Sigma,mu)$ be a $sigma$-finite measure space, and let $f$ be $Sigma$-measurable and integrable over $mathscrS$. Let $Sigma_0$ be a $sigma$-algrbra satisfying $Sigma_0subseteq Sigma$. Show that there is a unique function $f_0$ which is $Sigma_0$-measurable such that $int fgdmu=int f_0gdmu$ for every $Sigma$-measurable $g$ for which the integrals are finite.
i

My attempt:

Define $phi(A)=int_A fdmu$ for $Ain Sigma_0$. Since $f$ is integrable, then $phi$ is a set function. So applying Radon-Nikodym theorem, there is unique $f_0$ $Sigma_0$- measurable such that $phi(A)=int_A f_0dmu$ for $Ain Sigma_0$.

Hence, $int_A fdmu=int_A f_0dmu$ for $Ain Sigma_0$.

Next, consider $g=mathcalX_E$ for $EinSigma_0$ is true. And simple function. For the nonnegative function use monotone convergence theorem. For general $g$ consider $g^+-g^-$.

Is it correct my approach?

real-analysis analysis

share|cite|improve this question

edited Mar 29 at 3:13

Zoe

asked Mar 29 at 3:02

ZoeZoe

134

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Shouldn't it be every $Sigma_0$-measurable $g$?
  $endgroup$
  – forgottenarrow
  Mar 29 at 4:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think you have the right idea. You apply the Radon-Nikodym derivative and then construct the integral from the resulting measure. I don't know if you left it out because this is just a proof sketch, but it's important to make sure that $phi(cdot)$ is absolutely continuous with respect to $mu_f(cdot) := int_cdot f,dmu$. However this should be simple.
  $endgroup$
  – forgottenarrow
  Mar 29 at 4:49
  

  
  
  
  

add a comment | 

$begingroup$

Let $(mathscrS,Sigma,mu)$ be a $sigma$-finite measure space, and let $f$ be $Sigma$-measurable and integrable over $mathscrS$. Let $Sigma_0$ be a $sigma$-algrbra satisfying $Sigma_0subseteq Sigma$. Show that there is a unique function $f_0$ which is $Sigma_0$-measurable such that $int fgdmu=int f_0gdmu$ for every $Sigma$-measurable $g$ for which the integrals are finite.
i

My attempt:

Define $phi(A)=int_A fdmu$ for $Ain Sigma_0$. Since $f$ is integrable, then $phi$ is a set function. So applying Radon-Nikodym theorem, there is unique $f_0$ $Sigma_0$- measurable such that $phi(A)=int_A f_0dmu$ for $Ain Sigma_0$.

Hence, $int_A fdmu=int_A f_0dmu$ for $Ain Sigma_0$.

Next, consider $g=mathcalX_E$ for $EinSigma_0$ is true. And simple function. For the nonnegative function use monotone convergence theorem. For general $g$ consider $g^+-g^-$.

Is it correct my approach?

real-analysis analysis

share|cite|improve this question

edited Mar 29 at 3:13

Zoe

asked Mar 29 at 3:02

ZoeZoe

134

$endgroup$

share|cite|improve this question

edited Mar 29 at 3:13

Zoe

asked Mar 29 at 3:02

ZoeZoe

134

share|cite|improve this question

edited Mar 29 at 3:13

Zoe

asked Mar 29 at 3:02

ZoeZoe

134

asked Mar 29 at 3:02

ZoeZoe

134

asked Mar 29 at 3:02

ZoeZoe

134