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$begingroup$

Let $ lambda $ denote Lebesgue measure on $ mathbb R $. Let $ xin mathbb R $, and let $ f: xmapsto f(x)in mathbb R $ be Lebesgue measurable. For Borel sets $ Bsubsetmathbb R $, define
$$ mu(B)=lambda(x:f(x)in B) .$$
It should be clear that $ mu $ is a measure, show that
$$ int_mathbb Rg(y)dmu(y)=int_mathbb R(gcirc f)(x)dlambda(x) $$
for all $ g $ such that the integrals make sense.

I am confused that if we have to prove the statement by showing from characteristic functions, non-negative functions to measurable functions like what we did in constructing Lebesgue integral? Or there is another way to prove it, i.e., by applying some theorems that I don't know?

real-analysis lebesgue-integral

share|cite|improve this question

asked Mar 24 at 16:14

user549397user549397

1,6941618

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The way to go is, as you say, to first consider characteristic functions, which naturally extends to simple functions, and then to non-negative measurable functions by Monotone convergence, and finally to arbitrary measurable functions by decomposing the into the positive and negative part as usual. That is the standard proof as far as I know.
  $endgroup$
  – MisterRiemann
  Mar 24 at 16:20
  
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $ lambda $ denote Lebesgue measure on $ mathbb R $. Let $ xin mathbb R $, and let $ f: xmapsto f(x)in mathbb R $ be Lebesgue measurable. For Borel sets $ Bsubsetmathbb R $, define
$$ mu(B)=lambda(x:f(x)in B) .$$
It should be clear that $ mu $ is a measure, show that
$$ int_mathbb Rg(y)dmu(y)=int_mathbb R(gcirc f)(x)dlambda(x) $$
for all $ g $ such that the integrals make sense.

I am confused that if we have to prove the statement by showing from characteristic functions, non-negative functions to measurable functions like what we did in constructing Lebesgue integral? Or there is another way to prove it, i.e., by applying some theorems that I don't know?

real-analysis lebesgue-integral

share|cite|improve this question

asked Mar 24 at 16:14

user549397user549397

1,6941618

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The way to go is, as you say, to first consider characteristic functions, which naturally extends to simple functions, and then to non-negative measurable functions by Monotone convergence, and finally to arbitrary measurable functions by decomposing the into the positive and negative part as usual. That is the standard proof as far as I know.
  $endgroup$
  – MisterRiemann
  Mar 24 at 16:20
  
  

  
  
  
  

add a comment | 

$begingroup$

Let $ lambda $ denote Lebesgue measure on $ mathbb R $. Let $ xin mathbb R $, and let $ f: xmapsto f(x)in mathbb R $ be Lebesgue measurable. For Borel sets $ Bsubsetmathbb R $, define
$$ mu(B)=lambda(x:f(x)in B) .$$
It should be clear that $ mu $ is a measure, show that
$$ int_mathbb Rg(y)dmu(y)=int_mathbb R(gcirc f)(x)dlambda(x) $$
for all $ g $ such that the integrals make sense.

I am confused that if we have to prove the statement by showing from characteristic functions, non-negative functions to measurable functions like what we did in constructing Lebesgue integral? Or there is another way to prove it, i.e., by applying some theorems that I don't know?

real-analysis lebesgue-integral

share|cite|improve this question

asked Mar 24 at 16:14

user549397user549397

1,6941618

$endgroup$

share|cite|improve this question

asked Mar 24 at 16:14

user549397user549397

1,6941618

share|cite|improve this question

asked Mar 24 at 16:14

user549397user549397

1,6941618

asked Mar 24 at 16:14

user549397user549397

1,6941618

asked Mar 24 at 16:14

user549397user549397

1,6941618