Suppose that $f_{n}: \mathbb{R}^{d} \rightarrow \mathbb{R}$ is a sequence of measurable functions that converge to a measurable function $f$ mM-almost everywhere. In addition, suppose that there is a non-negative measurable function $F$ such that $\int_{\mathbb{R}^{d}}F dx < \infty$ and $|f_{n}| \leq F$ for all $n$.
Prove that $\int_{\mathbb{R}^{d}}\limsup f_{n}dx \geq \limsup\int_{\mathbb{R}}dx$.
Then give an example to show that the above conclusion may fail without the assumption of the existence of the integrable dominating function F.
For the first part, try applying Fatou's lemma to the non-negative sequnce $g_n=F-f_n$.
For the counterexample, consider a sequence of functions which "escapes to infinity", e.g. $f_n(x)=1_{[n,n+1]}(x)$.