General Lebesgue Dominated Convergence Theorem

Since $|f_n| \leq g_n$ for all $n$ and $f_n$ ($g_n$ respectively) converge pointwise a.e. on $E$ to $f$ ($g$ respectively), we have $|f|\leq g$ pointwise a.e. on $E$. Therefore, for all $n$ we have $$|f_n-f|\leq g_n+g$$ pointwise a.e. on $E$. Now apply Fatou Lemma to the nonegative function $g_n+g-|f_n-f|$, we have $$\liminf_{n\rightarrow\infty}\int_E(g_n+g-|f_n-f|)\geq\int_E\liminf_{n\rightarrow\infty}(g_n+g-|f_n-f|).$$ The right hand side is equal to $$\int_E\liminf_{n\rightarrow\infty}(g_n+g-|f_n-f|)=2\int_Eg,$$ since $f_n$ ($g_n$ respectively) converge pointwise a.e. on $E$ to $f$ ($g$ respectively). On the other hand, the left hand side is equal to $$\liminf_{n\rightarrow\infty}\int_E(g_n+g-|f_n-f|)=2\int_Eg-\limsup_{n\rightarrow\infty}\int_E|f_n-f|$$ since $\displaystyle\lim_{n \rightarrow \infty}\int_Eg_n=\int_Eg$ by assumption. Now putting all these together, we obtain $$0\geq\limsup_{n\rightarrow\infty}\int_E|f_n-f|.$$ Since $\displaystyle\int_E|f_n-f|\geq\Big|\int_Ef_n-f\Big|$, by the above inequality we have $$0\geq\limsup_{n\rightarrow\infty}\Big|\int_E(f_n-f)\Big|\geq\liminf_{n\rightarrow\infty}\Big|\int_Ef_n-f\Big|\geq 0.$$ By the above equality, $\displaystyle\limsup_{n\rightarrow\infty}\Big|\int_E(f_n-f)\Big|=\liminf_{n\rightarrow\infty}\Big|\int_E(f_n-f)\Big|$, i.e. $\displaystyle\lim_{n\rightarrow\infty}\Big|\int_E(f_n-f)\Big|$ exists. Moreover, by the above equality again, $\displaystyle\lim_{n\rightarrow\infty}\Big|\int_E(f_n-f)\Big|=0$, which implies $$\lim_{n\rightarrow\infty}\int_Ef_n=\int_Ef,$$ as required.

You made a mistake: $$\liminf \int (g_n-f_n) = \int g-\limsup \int f_n$$ not $$\liminf \int (g_n-f_n) = \int g-\liminf \int f_n.$$

Here is the proof:

$$\int (g-f)\leq \liminf \int (g_n-f_n)=\int g -\limsup \int f_n$$

which means that

$$\limsup \int f_n\leq \int f$$

Also

$$\int (g+f)\leq \liminf \int(g_n+f_n)=\int g + \liminf \int f_n$$

which means that

$$\int f\leq \liminf \int f_n$$

i.e.

$$\limsup \int f_n\leq \int f\leq \liminf\int f_n\leq \limsup \int f_n$$

So they are all equal.