I posted an answer to this question here
http://www.mathhelpforum.com/math-he...ral-limit.html
This back when I was less careful, so you may want to wait for NCA or another senior member to verify it.
The basic gist is that since
on
and
almost everywhere (pointwise) then by Lebesgue's Dominated Convergence Theorem
and using the fact that
.
NOTE: Once again the original post should not have probably been done because I was using machinery a bit beyond my capabilities of yet so you should wait for confirmation...but I thought it may give another perspective to this problem.