Fatou's lemma requires that $f_{n}$ be non-negative. I know that there exist counterexamples such as $f_{n} = \mathbb {-1}_{[n,n+1]}$ which show that the non-negativity condition is critical. However, I don't know what role this constraint comes into play in the proof of this lemma. Where is it needed?
Here is the proof:
Let $g_{k}:=inf_{n \geq k} f_{n}$, then $\lim_{n} inf f_{n} = \uparrow \lim g_{k}$. For $n \geq k$,
$\begin{align}
f_{n} \geq g_{k} \Longrightarrow \int f_{n} \, d\mu \geq \int g_{k} \, d\mu \Longrightarrow \inf_{n \geq k} \int f_{n} \, d\mu \geq \int g_{k} \, d\mu
\end{align}
$. Therefore, $
\begin{align}
\int \lim
\inf f_{n} \, d\mu = \uparrow \lim_{k} \int g_{k} \, d\mu \leq \uparrow \lim_{k} \inf_{n \geq k} \int f_{n} \, d\mu = \lim_{n} \inf \int f_{n} \, d\mu
\end{align}
$, where in the first equation, we use Monotone Convergence Theorem.
Best Answer
Monotone Convergence theorem requires non-negativiness. Just consider the functions $f_{n} = \mathbb {-1}_{[n,n+1]}$. Note that $f_{n} \uparrow 0$ , but $ \int f_n = 1$, for all $n$.
Of course, if $f_n$ are measurable, you can work with $f_n^+$ and $f_n^-$. The issue here is that, if $f_n \uparrow f$, then $f_n^- \downarrow f^-$. So you can not apply Monotone Convergence Theorem to the negative part (you would need Dominated Convergence Theorem).
In the proof of Fatou's lemma, that is exactly where the proof breaks if you don't assume that $f_{n}$ are non-negative:
Let $g_{k}:=\inf_{n \geq k} f_{n}$, then, for the negative part we have $g_{k}^-=\sup_{n \geq k} f_{n}^-$ and $\lim_n \sup f_n^- = \downarrow \lim g_k^-$. (then you would need Dominated Converge Theorem).
To correct this, you might be tempted to write:
Let $g_{k}:=\inf_{n \geq k} f_{n}^+$ and $h_{k}:=\inf_{n \geq k} f_{n}^-$. We have then $\lim_{n} \inf f_{n}^+ = \uparrow \lim g_{k}$ and $\lim_{n} \inf f_{n}^- = \uparrow \lim h_{k}$, but now the issue is that, although $f =f^+ - f^-$, we have that $\lim_{n} \inf f_{n} \neq \lim_{n} \inf f_{n}^+ - \lim_{n} \inf f_{n}^-$. So, although you can apply Fatou to the positive and negative parts separately, you can not combine the two results for $f$.