Folland's exercise 2.18, proving Fatou's lemma in a more general setting I'm trying to prove that Fatou's lemma works if $f_n$ is a sequence of measurable functions and $g$ is a non-negative integrable function such that $f_n\ge -g$. That is, it is to be proven that $\int \liminf f_n\le \liminf \int f_n$.
I tried to search a similar question on this site but couldn't find any.
Applying Fatou's lemma to $f_n+g$ gives $$\int(\lim\inf f_n+g)\le \liminf \int f_n+g\tag 1$$
Since $\lim\inf f_n$ need not be integrable, $\int(\lim\inf f_n+g)=\int \liminf f_n +\int g$ is not allowed.  The same can be said about RHS of $(1)$.
$f_n= f_n^+- f_n^-\implies \liminf f_n=-\limsup(f_n^--f_n^+)\ge -\limsup f_n^--\limsup (-f_n^+)=\liminf f_n^+-\limsup f_n^-$.
I'm not sure how to take it from here.
Please help. Thanks.

If $\liminf f_n$ is integrable easy. Suppose $\liminf_n$ is not integrable. Then either $\int(\liminf f_n)_+=\infty$ or $\int(\liminf f_n)_-=\infty$.
As $(\liminf_n f_n)+g\geq0$ and $g$ is integrable, $\int(\liminf f_n)_-\neq\infty$. The only alternative is that $\int(\liminf_nf_n)_+=\infty$ and $\int(\liminf_nf_n)_- <\infty$.

To make things simple suppose $f,g$ are measurable, $g\in L_1$ and $f+g\geq0$ ($g$ may take negative values). Of course  $f\in L_1$ iff $f+g\in L_1$ in which case
$$\int(f+g)=\int f +\int g$$
Assume that $f+g\not\in L_1$. Since $f+g\geq0$, $\int(f+g)$ exists and takes value on $\mathbb{R}\cup\{\infty\}$, hence $\int(f+g)=\infty$. Let $E=\{f\geq0\}$ so that
$$f+g=(f_+ + g\mathbb{1}_E)-(f_- - g\mathbb{1}_{E^c})$$
On $E^c$ one has $-f_- +g\geq0$ and so,  $0\leq f_-\leq \mathbb{1}_{E^c} g$. It follows that $f_-\in L_1$. Consequently,  $\int(f_+ +g\mathbb{1}_{E})=\infty$ and so, $\int f=\int f_+=\infty$. Therefore
$$\int (f+g)=\int f +\int g$$
