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Proving integrability of a strange function

  • #1

Homework Statement



Hi guys. I'm really struggling with this problem. Any help is welcomed.

Suppose I have a function f(y) = [tex]\int[/tex]g(x)/(x^2) on the set [(y/2)^(1/2), [tex]\infty[/tex]]. g(x) is known to be integrable over all of R.

I want to show that f is integrable over [0,[tex]\infty[/tex]], and that the [tex]\int[/tex]f(y) on [0, [tex]\infty[/tex]] = 2*[tex]\int[/tex]f(x) on R.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,739
18
What are the limts on your integral defining f(y)
 
  • #3
sorry, I'm not great with typing these things in LaTex format.

I want to show that f(y) is integrable over [0,[tex]\infty[/tex]].

f(y) is defined as the function:
f(y) = [tex]\int[/tex][g(x)/(x^2)]dx with bounds [(y/2)^(1/2),[tex]\infty[/tex]].

apologies for the lack of clarity.
 
  • #4
hunt_mat
Homework Helper
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18
So f(y) is defined as:
[tex]
f(y)=\int_{\sqrt{\frac{y}{2}}}^{\infty}\frac{g(x)}{x^{2}}dx
[/tex]
 
  • #5
That's correct.
 
  • #6
hunt_mat
Homework Helper
1,739
18
First off f(y) is well defined on [0,inftinity). What theorems do you have at your disposal?

Oh are these Riemann integrals or Lebesgue integrals?
 
  • #7
Lebesgue. We have LDCT, Generalized LDCT, Monotone Convergence, etc.
 
  • #8
I think there must be some way to bound the function g(x). I'm just not sure how I can find an L1 function that serves an a.e. bound for g(x).
 
  • #9
berkeman
Mentor
57,287
7,273
Thread locked temporarily. This may be a question on a take-home exam.
 

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