Lebesgue Integrable Function question

[TaylorWatts]

Homework Statement

Let f be a Lebesgue integrable function in the interval [a,b]

Show that:

lim integral from a to b (f(x)*|cosnx|) = 2/pi * integral from a to b (f(x))
n->infinity

Homework Equations

Every measurable function can be approximated arbitrarily close by simple functions.

The Attempt at a Solution

I've solved part i of the problem (which was to show the same setup except f*cosnx instead of f*|cosnx|) has an integral equal to zero.

I'm pretty stuck on this part - I'm sure I have to find a simple function but I'm not sure what a good simple function would be.

 

[Billy Bob]

Can you prove this is true for the special case when f is the characteristic function of an interval I=[r,s], where [r,s] is a subset of [a,b]?