I'm trying to prove / disprove the following:
If $f_n \geq 0$ is a sequence of integrable functions and $f_n \to f$ a.e., then $\lim \int_E f_n d\mu$ exists and $\int_E fd\mu \leq \lim \int_E f_n d\mu$. All limits here are as $n \to \infty$.
What I have so far: As $f_n \to f$, we have that $\liminf f_n = f$. If $(\int_Ef_nd\mu)$ converges, it is equal to $\liminf (\int_E f_n d\mu)$, and so by Fatou's Lemma, $$\int_E fd\mu = \int_E\left(\liminf f_n\right)d\mu \leq \liminf \left(\int_E f_nd\mu\right) = \lim\left(\int_Efd\mu\right).$$
However, I'm stuck on showing that $\lim (\int_E f_n d\mu)$ exists, if at all.