Bounded integrable periodic function

[mesarmath]

hi,
i have a hard problem, i guess so,
i am looking for any help

g(x) is a bounded Lebesgue measurable function that is periodic
i.e.  [tex]g(x)=g(x+p)[/tex]. Then for every [tex]f \in L^1(\Re)[/tex]

[tex]lim_{n\rightarrow \infty}\int_{\Re}f(x)g(nx) dx=(\int_{\Re}f(x)dx)((1/p){\int_{0}^{p}g(x) dx)[/tex]

thanks for any help

 

[mathman]

You can use the fact that g(nx+p)=g(n(x+p/n)). Therefore the integral over shorter and shorter intervals will contain a full cycle of g leading to the result described. I'll let you work out the details.

 

[mesarmath]

after 4 hours working :)
i solved , here the steps
1-) change of u=nx and as n goes to infty [-n,n] goes to whole space
2-) after some operations and by dividing [-n,n] into 2n/p equal pieces with length p/n
3-) then again change of u=t+ip-p where -n^2/p < i < n^2/p,
4-) by riemann integration as n goes to infinity, we get right hand side

that was real fun to solve it
if i can't solve , then it becomes a huge pain :)

 

[mathman]

The main quibble I have with your approach is that the original problem was in terms of f being Lebesgue integrable, not Riemann.

 

[mesarmath]

The main quibble I have with your approach is that the original problem was in terms of f being Lebesgue integrable, not Riemann.

the set of continuous functions is dense in the set of Lebesgue integrable functions

So when we do the problem for a continuous function (i.e. riemann integrable)
we can extend it to any lebesgue integrable function.

am i wrong?

 

[mathman]

I'm pretty rusty on the subject, but you are probably right.