I was going through the proof of the Dominated Convergence Theorem.
Now if we have that $(f_n)$ is a sequence of measurable functions such that $\lvert f_n\rvert\le g$ for all $n$ where $g$ is integrable on $\Bbb R$. And if $f = \lim_n f_n$ almost everwhere.
We can show that $(g+f_n)$ and $(g-f_n)$ are sequences of non-negative measurable functions.
How can we show this for $(g-f_n)$? We do get that $f_n\le g$, but what if $f_n(x)$ is infinite for some $x$?
Then by Fatou's lemma, we have that $\int\liminf(g+f_n)dx\le\liminf\int(g+f_n)dx$. Now from here, we can obtain that
$$\int(g+f)dx \le \int gdx+\liminf \int f_ndx$$
How? How do we get the left hand side in this? I can show that $\int\liminf f_ndx = \int fdx$, but how to prove that $\liminf f_n$ is integrable to prove the former? Also I know that $\liminf(g+f_n) \geq\liminf g+\liminf(f_n)$... but how do we get the left hand side? Can I integrate it throughout but why are all these limit inferior integrable?