While reading this question here about the proof for Scheffe's Lemma, I was confused since someone said the proof in the question was not correct. I thought the argument was fine, and the author only needed to use the General Dominated Convergence theorem to finish the argument. Continuing form his/her work, we have that $$\lim_n \int f + f_n = \int 2f < \infty$$ which implies $\lim_n \int|f-f_n| = 0$.
General Lebesgue Dominated Convergence Theorem: Let $\{f_n\}$ be a sequence of measurable functions that converges pointwise a.e to $f$. Suppose there is a sequence $\{g_n\}$ of nonnegative measurable functions that converges pointwise a.e to $g$ and dominateds $f_n$ in the sense that $$|f_n| \leq g_n.$$
$$\text{If } \lim_n \int g_n = \int g < \infty, \text{ then } \lim_n \int f_n = \int f.$$
