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A question about L1 function

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0

$begingroup$

Suppose $fin L^1(mathbbR^d)$ (that is, $f$ is real or complex valued and the Lebesgue measure $int|f|<infty$), and suppose $lambda>0$ and $1leq nleq d$. Define $g$ as: for any $xinmathbbR^d$:
$$xmapsto f(x_1,...,lambda x_n,...,x_d)$$
Question: is it true that $gin L^1(mathbbR^d)$ ? If true, what is its integral?

I feel that I need to use Fubini’s theorem but am unsure about how to do it. Any thoughts? Thanks!

measure-theory

share|cite|improve this question

asked Mar 22 at 0:39

Hank ChenHank Chen

215

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think that a change of variables should do the work. It also holds for the Lebesgue integral, as one can see on this question: math.stackexchange.com/questions/152338/…
  $endgroup$
  – Affonso
  Mar 22 at 0:48
  

  
  
  
  

add a comment | 

0

$begingroup$

Suppose $fin L^1(mathbbR^d)$ (that is, $f$ is real or complex valued and the Lebesgue measure $int|f|<infty$), and suppose $lambda>0$ and $1leq nleq d$. Define $g$ as: for any $xinmathbbR^d$:
$$xmapsto f(x_1,...,lambda x_n,...,x_d)$$
Question: is it true that $gin L^1(mathbbR^d)$ ? If true, what is its integral?

I feel that I need to use Fubini’s theorem but am unsure about how to do it. Any thoughts? Thanks!

measure-theory

share|cite|improve this question

asked Mar 22 at 0:39

Hank ChenHank Chen

215

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think that a change of variables should do the work. It also holds for the Lebesgue integral, as one can see on this question: math.stackexchange.com/questions/152338/…
  $endgroup$
  – Affonso
  Mar 22 at 0:48
  

  
  
  
  

add a comment | 

$begingroup$

Suppose $fin L^1(mathbbR^d)$ (that is, $f$ is real or complex valued and the Lebesgue measure $int|f|<infty$), and suppose $lambda>0$ and $1leq nleq d$. Define $g$ as: for any $xinmathbbR^d$:
$$xmapsto f(x_1,...,lambda x_n,...,x_d)$$
Question: is it true that $gin L^1(mathbbR^d)$ ? If true, what is its integral?

I feel that I need to use Fubini’s theorem but am unsure about how to do it. Any thoughts? Thanks!

measure-theory

share|cite|improve this question

asked Mar 22 at 0:39

Hank ChenHank Chen

215

$endgroup$

share|cite|improve this question

asked Mar 22 at 0:39

Hank ChenHank Chen

215

share|cite|improve this question

asked Mar 22 at 0:39

Hank ChenHank Chen

215

asked Mar 22 at 0:39

Hank ChenHank Chen

215

asked Mar 22 at 0:39

Hank ChenHank Chen

215