An application of the General Lebesgue Dominated convergence theorem I came across the following problem in my self-study:
If $f_n, f$ are integrable and $f_n \rightarrow f$ a.e. on $E$,  then $\int_E |f_n  - f| \rightarrow 0$ iff $\int_E |f_n| \rightarrow \int_E |f|$. 
I am trying to prove (1) and the book I am using suggests that it follows from the Generalized Lebesgue Dominated Convergence Theorem:
Let $\{f_n\}_{n=1}^\infty$ be a sequence of measurable functions on $E$ that converge pointwise a.e. on $E$ to $f$.  Suppose there is a sequence $\{g_n\}$ of integrable functions on $E$ that converge pointwise a.e. on $E$ to $g$ such that $|f_n| \leq g_n$ for all $n \in \mathbb{N}$.  If $\lim\limits_{n \rightarrow \infty}$ $\int_E$ $g_n$ = $\int_E$ $g$, then $\lim\limits_{n \rightarrow \infty}$ $\int_E$ $f_n$ = $\int_E$ $f$.
I suspect that I need the right inequalities to help satisfy the hypothesis of the GLDCT, but I am not certain about what these inequalities should be.

Suppose that $$\int |f_n|\to |f|.$$ The function $$|f_n|+|f|- |f_n-f|$$ is no negative. The following is just a technique used in other answer on this site. By Fatou's Lemma
$$\begin{align*}
\liminf \int |f_n|+|f|- |f_n-f| &\leq \int\liminf(|f_n|+|f|- |f_n-f|)\\
\liminf\int |f_n|+\int |f|-\limsup\int |f_n-f| &\leq \liminf \int |f_n|+\int |f|-\int\limsup |f_n-f|\\
2\int |f|-\int 0 &\leq 2\int |f| - \int\limsup |f_n-f|,
\end{align*}$$
the last inequality leads
$$\limsup\int |f_n-f|\leq 0,$$
therefore $$\lim_{n\to\infty}\int |f_n-f|= 0.$$
For the other implication see Martin's answer.
