Fatou's Lemma is
Let $(\Omega, \Sigma, \mu)$ be a measure space and $X \in \Sigma$. Let $f_n; f_n : X \to [0, +\infty]$ be a sequence of $(\Sigma, \mathcal{B}_{\mathbb{R}_{\geq 0}})$-measurable functions. Define $f(x) = \lim\inf_{n\to\infty} f_n$. Then $f$ is $(\Sigma, \mathcal{B}_{\mathbb{R}_{\geq 0}})$-measurable and
$\begin{align} \int_X f d\mu \leq \liminf_{n\to\infty} \int_X f_n d\mu \end{align}$
Is this true if you generalize the codomain of $f$ and $f_n$ to be all of $\mathbb{R}$ and $(\Sigma, \mathcal{B}_{\mathbb{R}})$-measurable?
Is this true if you replace $\liminf$ with $\limsup$?
Provide counterexamples please if possible. No this is not a homework question.