Proving integrability of a strange function

[jvalton1287]

Homework Statement

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

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

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

 

[hunt_mat]

What are the limts on your integral defining f(y)

 

[jvalton1287]

sorry, I'm not great with typing these things in LaTex format.

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

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

apologies for the lack of clarity.

 

[hunt_mat]

So f(y) is defined as:
$$f(y)=\int_{\sqrt{\frac{y}{2}}}^{\infty}\frac{g(x)}{x^{2}}dx$$

 

[jvalton1287]

That's correct.

 

[hunt_mat]

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?

 

[jvalton1287]

Lebesgue. We have LDCT, Generalized LDCT, Monotone Convergence, etc.

 

[jvalton1287]

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).

 

[berkeman]

Thread locked temporarily. This may be a question on a take-home exam.