For E with finite measure I think you can just use
.
For the general case I would look at something like
where if , and then show that
I'm struggling with a prelim practice problem and need some help; note that E may have infinite measure.
Claim: Let be a sequence of integrable functions such that a.e. Assume that . Prove or disprove that for any measurable set E, .
I't's not clear that the conjecture is even true but I haven't been able to find a counter example if it's not. If and are non-negative, you can use Fatou's Lemma to prove the result. If we can show that then we can use the generalized Lebesque Dom Conv Thm. but I haven't been able to prove the limit. I also tried assuming that E has finite measure and applying Egoroff's theorem to no avail. I also tried assuming that the claim is true for E of finite measure and then proving the more general case - didn't get home on that approach either. I also played a little with signed measure ideas (positive and negative sets) but that just seemed to make the problem more abstract without adding any value. The way through may be in these somewhere in these ideas but I haven't been able to pull it out.
This remind's me Scheffé's lemma. The positiveness assumption (or the absolute values) involved in Scheffé's lemma suggests that the results fails otherwise, but I can't think of a counterexample either. I will give it further thoughts...
After further thoughts...
I considered the well-known counterexample to the bounded convergence theorem, namely the "traveling bump" (I don't know the English name) : define the sequence of functions (it equals 1 if and 0 otherwise). Any other smoother bump would do fine. We have for any , but for every . Of course, the convergence to 0 is not uniform and can't be dominated either.
Then I defined . This time, two opposite bumps travel toward . We have for all , and . This is the situation of the question. However, for every .