Let $(X,{\cal M},\mu)$ be a measure space and let $f_n,f\in L^1(X)$, that is, $\int_X |f_n| {\rm d}\mu < \infty$ and $\int_X |f| {\rm d}\mu < \infty$. Suppose $f_n\to f$ almost everywhere. Then $$ \int_X |f_n| {\rm d}\mu \to \int_X |f| {\rm d}\mu \iff \int_X |f_n-f| {\rm d}\mu \to 0. $$
The $(\Leftarrow)$ implication is trivial, since $$\left|\int_X |f_n| {\rm d}\mu - \int_X |f|{\rm d}\mu\right| \leq \int_X |f_n-f| {\rm d}\mu.$$
How about the other direction? I was trying to use some convergence theorem (monotone, dominated, Fatou's lemma), but I don't know how to proceed.
Any help would be appreciated. Thanks in advance.
EDIT: (Using @carmichael561's hint)
If we define $g_n=|f_n|+|f|-|f_n-f|$, by triangle inequality we have that $g_n \geq 0$ and it follows from Fatou's lemma that $$\int \liminf g_n \leq \liminf \int g_n.$$
On one hand, $g_n\to 2|f|$ a.e. since $f_n\to f$ a.e. On the other hand, \begin{equation*} \begin{split} \liminf\int g_n & = \liminf\left(\int |f_n| + \int |f| - \int |f _n-f|\right) \\ & = \int |f| + \liminf\left(\int |f_n| - \int |f _n-f| \right) \\ & = \int |f| + \int |f| + \liminf\left(-\int |f_n-f|\right), \end{split} \end{equation*} since $\int |f_n| \to \int |f|$ by hypothesis.
We get then $$ 0 \leq \liminf\left(-\int |f _n-f| \right),$$ which gives $\limsup\int|f_n-f| \leq 0$.
Now $\int|f_n-f| \geq 0$ for all $n\in\mathbb N$ implies $\liminf \int|f_n-f| \geq 0$, and therefore $$ 0 \leq \liminf \int |f_n-f| \leq \limsup \int |f_n-f| \leq 0,$$ from where the claim follows.
The trick with this problem and other similar ones is to find the right sequence of functions to apply Fatou's lemma to. In this case set $$ g_n=|f_n|+|f|-|f_n-f|.$$