The full question from this practice qual:
True/False: Let $(X,M,\mu)$ be any measure space. If $f_n,f \in L^1(\mu)$ are measurable functions, $f_n \rightarrow f$ $\mu$-a.e. and $\lim \int f_n \rightarrow \int f$, then $f_n \rightarrow f$ in $L^1(\mu)$.
I recognize that the question is essentially asking if I can move the limit inside the integral. The three theorems I have which allow me to do this are: Monotone Convergence Theorem, Fatou's Lemma, and Dominated Convergence Theorem. Since I don't have any increasing sequences of functions or dominating function, I would think if the answer is true, I need to use Fatou's Lemma. But I don't see any way to use Fatou's Lemma to justify it is true (the fact the question says "any" measure space signals to me it may be false, but I've been unable to construct a counter example)