I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies \displaystyle\int_{E} |f^{\epsilon}|dx \leq \eta$$
We have lemma: Let $\Omega$ an open bounded in $\mathbb{R}^n$. If $$ \begin{cases} &f^{\epsilon} \to f \qquad \mbox{a.e. in} \quad \Omega\\ & f^{\epsilon},f \in L^1(\Omega)\\ & f^{\epsilon} \quad \mbox{bounded in} \quad L^1(\Omega)\\ & f^{\epsilon} \quad \mbox{is equi integrable}\\ \end{cases} $$
then, $f^{\epsilon} \to f$ in $L^{1}(\Omega)$ strong.
To prove this lemma, we use the lemma Fatou with the Egorov theorem, but I haven't idea how we can use it to write the proof of this lemma.
Fix a positive $\eta$ and consider the corresponding $\delta$ in the definition of equi-integrability. By Egoroff's theorem, there exists some $A$ such that $$\lim_{n\to\infty}\sup_{x\in A}|f_n(x)-f(x)|=0\quad\mbox{and}\quad \lambda(\Omega\setminus A)\lt \delta.$$ We thus obtain, using integrability of $f$, $$\int_\Omega|f_n-f|dx\leqslant \lambda(\Omega)\sup_{x\in A}|f_n(x)-f(x)| +2\eta.$$