Integrability of a limit of functions

Question

I'm having troubles with the following problem:

Let $\Omega \subset \mathbb{R^3}$ with finite Lebesgue measure, $f_n:\Omega \mapsto \mathbb{R^3}$ a sequence of measurable functions with the following properties:

1) $\int_\Omega |f_n(x)|^2 dx<8 $

2)${f_n}$ converges to $f$ almost everywhere.

I need to say if $f$ is measurable and integrable.

The first point seems easy since the limit of a sequence of measurable function should be a measurable function, but i don't know how to proceed with the second request. I know that convergence almost everywhere implies uniform convergence almost everywhere (Egorov's Theorem) and therefore convergence in measure, but i can't deduce $L^p$ convergence. I also know that since ${f_n}\in L^2(\Omega)$ we have ${f_n}\in L^1(\Omega)$, but i can't find any dominating function or any counterexample.

Answer 1

We have \begin{align*} \int_{\Omega}|f(x)|^{2}dx\leq\liminf_{n}\int|f_{n}|^{2}dx\leq 8, \end{align*} and \begin{align*} \int_{\Omega}|f(x)|dx\leq\|f\|_{L^{2}(\Omega)}|\Omega|^{1/2}<\infty. \end{align*}