Bounded integrable periodic function

1. Jan 16, 2010

mesarmath

hi,
i have a hard problem, i guess so,
i am looking for any help
Code (Text):

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

$$lim_{n\rightarrow \infty}\int_{\Re}f(x)g(nx) dx=(\int_{\Re}f(x)dx)((1/p){\int_{0}^{p}g(x) dx)$$
thanks for any help

2. Jan 16, 2010

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.

3. Jan 16, 2010

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 lenght 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 cant solve , then it becomes a huge pain :)

4. Jan 17, 2010

mathman

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

5. Jan 18, 2010

mesarmath

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?

6. Jan 18, 2010

mathman

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