I'm struggling with a prelim practice problem and need some help; note that E may have infinite measure.

Claim: Let $\displaystyle f_n$ be a sequence of integrable functions such that $\displaystyle f_n \rightarrow f $ a.e. Assume that $\displaystyle \lim_{n \rightarrow \infty}{\int{f_n}} = \int{f} < \infty$. Prove or disprove that for any measurable set E, $\displaystyle \lim_{n \rightarrow \infty}{\int_{E}{f_n}} = \int_{E}{f}$.

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 $\displaystyle f_n$ and $\displaystyle f$ are non-negative, you can use Fatou's Lemma to prove the result. If we can show that $\displaystyle \int{f_n^+} \rightarrow \int{f^+}$ 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.