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Signed Measure equivalent definition

Lebesgue inner measure formulaVariation of sum of measureSigned Measure IntegralVariation of a signed measure -Variation Signed Measure InequalityVariation of a signed measureDefinition of lebesgue integral with respect to measure $mu$Let $|E|_e$ = $infty$,E is Lebesgue measurable if and only if $|E|_i = |E|_e$ may be not holdProve that $mu cup nu $ is the largest signed measure that is less than or equal to both $mu,nu$.An expression for a Total Variation Measure of a Signed Measure

0

$begingroup$

Let $mu$ be a signed measure, define
$$
|mu|(E) = mu^+(E) +mu^-(E)
$$
Why is it that
$$
|mu|(E) = textsupsum_k=1^n|mu(E_k)|,
$$
where the supremum is taken over all finite disjoint families $E_k_k=1^n$ of measurable subsets of $X$.

analysis measure-theory signed-measures

share|cite|improve this question

edited Mar 29 at 18:10

Issacg628496

asked Mar 29 at 2:11

Issacg628496Issacg628496

416

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know the Hahn decomposition theorem?
  $endgroup$
  – Umberto P.
  Mar 29 at 19:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I figured it out.
  $endgroup$
  – Issacg628496
  Mar 30 at 17:37
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $mu$ be a signed measure, define
$$
|mu|(E) = mu^+(E) +mu^-(E)
$$
Why is it that
$$
|mu|(E) = textsupsum_k=1^n|mu(E_k)|,
$$
where the supremum is taken over all finite disjoint families $E_k_k=1^n$ of measurable subsets of $X$.

analysis measure-theory signed-measures

share|cite|improve this question

edited Mar 29 at 18:10

Issacg628496

asked Mar 29 at 2:11

Issacg628496Issacg628496

416

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know the Hahn decomposition theorem?
  $endgroup$
  – Umberto P.
  Mar 29 at 19:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I figured it out.
  $endgroup$
  – Issacg628496
  Mar 30 at 17:37
  

  
  
  
  

add a comment | 

$begingroup$

Let $mu$ be a signed measure, define
$$
|mu|(E) = mu^+(E) +mu^-(E)
$$
Why is it that
$$
|mu|(E) = textsupsum_k=1^n|mu(E_k)|,
$$
where the supremum is taken over all finite disjoint families $E_k_k=1^n$ of measurable subsets of $X$.

analysis measure-theory signed-measures

share|cite|improve this question

edited Mar 29 at 18:10

Issacg628496

asked Mar 29 at 2:11

Issacg628496Issacg628496

416

$endgroup$

share|cite|improve this question

edited Mar 29 at 18:10

Issacg628496

asked Mar 29 at 2:11

Issacg628496Issacg628496

416

share|cite|improve this question

edited Mar 29 at 18:10

Issacg628496

asked Mar 29 at 2:11

Issacg628496Issacg628496

416

asked Mar 29 at 2:11

Issacg628496Issacg628496

416

asked Mar 29 at 2:11

Issacg628496Issacg628496

416