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Superposition of elementary Dirac measures: looking for reference or explanation

Integrals and Dirac delta measureslooking for reference for 2 trig factslooking for reference for integral inequalityAbout Dirac measures and their sums (e.g. on Borel sigma-algebra)Elementary Reference for Algebraic GroupsReference for Bochner space.Looking for a reference (textbook) for an elementary analysis problem on uncountable sumsDirac delta in Dirichlet Distribution explanationProbability measures and dirac measuresReference for Dirac Delta function as gaussian

0

$begingroup$

Reading a paper concerning transport of measures, I came accross the following sentence:

"...the fundamental fact that a generic measure $mu$ can be written as the superposition of elementary Dirac masses, that is:
$$ mu = int_textsupp mu delta_x , dmu (x), $$
where $textsupp mu$ belongs to an appropriate $sigma$-algebra and the representation has to be understood in the sense of Bochner integrals.
"

I conviced myself that this is reasonably true working with toy examples involving step functions (i.e, a finite number of Dirac measures in principle), but I cannot find a reference for the general case which - at least according to the authors - should be a well-known fact; neither I can find a proof on my own since the definition of Bochner integral is quite abstract. Any kind of help is very much appreciated.

integration measure-theory reference-request dirac-delta bochner-spaces

share|cite|improve this question

asked Mar 22 at 15:26

GaCGaC

1,736312

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  physic stack exchange maybe ?
  $endgroup$
  – Roddy MacPhee
  Mar 22 at 15:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you actually believe? I mean this sound raw and technical measure theory to me..
  $endgroup$
  – GaC
  Mar 22 at 15:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe you can find this result in "Vector Measures" by Diestel & Uhl. I don't have it by my side at the moment, but it has a lot of similar results related to Bochner integrals.
  $endgroup$
  – Fritz
  Mar 23 at 8:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I will have a look, thank you very much!
  $endgroup$
  – GaC
  Mar 23 at 16:28
  

  
  
  
  

add a comment | 

0

$begingroup$

Reading a paper concerning transport of measures, I came accross the following sentence:

"...the fundamental fact that a generic measure $mu$ can be written as the superposition of elementary Dirac masses, that is:
$$ mu = int_textsupp mu delta_x , dmu (x), $$
where $textsupp mu$ belongs to an appropriate $sigma$-algebra and the representation has to be understood in the sense of Bochner integrals.
"

I conviced myself that this is reasonably true working with toy examples involving step functions (i.e, a finite number of Dirac measures in principle), but I cannot find a reference for the general case which - at least according to the authors - should be a well-known fact; neither I can find a proof on my own since the definition of Bochner integral is quite abstract. Any kind of help is very much appreciated.

integration measure-theory reference-request dirac-delta bochner-spaces

share|cite|improve this question

asked Mar 22 at 15:26

GaCGaC

1,736312

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  physic stack exchange maybe ?
  $endgroup$
  – Roddy MacPhee
  Mar 22 at 15:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you actually believe? I mean this sound raw and technical measure theory to me..
  $endgroup$
  – GaC
  Mar 22 at 15:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Maybe you can find this result in "Vector Measures" by Diestel & Uhl. I don't have it by my side at the moment, but it has a lot of similar results related to Bochner integrals.
  $endgroup$
  – Fritz
  Mar 23 at 8:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I will have a look, thank you very much!
  $endgroup$
  – GaC
  Mar 23 at 16:28
  

  
  
  
  

add a comment | 

$begingroup$

Reading a paper concerning transport of measures, I came accross the following sentence:

"...the fundamental fact that a generic measure $mu$ can be written as the superposition of elementary Dirac masses, that is:
$$ mu = int_textsupp mu delta_x , dmu (x), $$
where $textsupp mu$ belongs to an appropriate $sigma$-algebra and the representation has to be understood in the sense of Bochner integrals.
"

I conviced myself that this is reasonably true working with toy examples involving step functions (i.e, a finite number of Dirac measures in principle), but I cannot find a reference for the general case which - at least according to the authors - should be a well-known fact; neither I can find a proof on my own since the definition of Bochner integral is quite abstract. Any kind of help is very much appreciated.

integration measure-theory reference-request dirac-delta bochner-spaces

share|cite|improve this question

asked Mar 22 at 15:26

GaCGaC

1,736312

$endgroup$

share|cite|improve this question

asked Mar 22 at 15:26

GaCGaC

1,736312

share|cite|improve this question

asked Mar 22 at 15:26

GaCGaC

1,736312

asked Mar 22 at 15:26

GaCGaC

1,736312

asked Mar 22 at 15:26

GaCGaC

1,736312