Show that
Hi problem,
We can get at the exercise using the Lebesgue Dominated Convergence Theorem. Let and note that for all so that for all
1) Since we're working on , always. Hence, which is integrable on i.e. is our "dominating function."
2) Since we're working on , as for all Furthermore, for all Hence, the pointwise limit of the sequence is for and . Now the set is of measure 0, so is actually irrelevant as far as the integral is concerned.
Now it is a matter of checking to see if we've satisfied the conditions of Lebesgue's DCT, applying it, then evaluating the integral of the limit function properly to get what we're after.
Does this get things going in the right direction? Let me know if anything is unclear. Good luck!