Let $f_n, f : X \longrightarrow \bar{\mathbb R} $ be measurable functions and $f^-$ integrable $(\bar {\mathbb R} := \mathbb R \cup \{+\infty, - \infty \})$. Show that $$f_n \geq f \text{ almost everywhere} \Rightarrow \int _X \lim_{n \rightarrow \infty} \inf f_n \text{ d}\mu \leq \lim_{n \rightarrow \infty} \inf \int _X f_n \text{ d}\mu $$
My attempt:
$\text{Case } 1$: $f \geq 0$
I can use Fatou's Lemma here
$\text{Case } 1$: $f \lt 0$
Since $f_n \geq f \Longrightarrow -f_n \leq f \Longrightarrow (f_n -f)_{_n} \text{ is a sequence of nonnegative functions}$.
Then $$\int _X \lim \inf f_n + \int -f \leq \int _X \lim \inf (f_n-f) \leq \lim \inf \int _X (f_n-f) \leq \lim \inf \int _X f_n + \int_X -f$$
$\Longrightarrow \int _X \lim \inf f_n \leq \lim \inf \int _X f_n $
Problem:
Since I only know that $f^-$ is integrable can I follow that $|f|$ is integrable and therefore $f_n$ and $f$ are integrable since $f_n$ & $f$ are both measurable? Also I am not sure if my proof is correct so any help here is appreciated