Let $g$ be a non-negative integrable function over $E$ and suppose $\{f_n\}$ is a sequence of measurable functions on $E$ such that for each $n$, $|f_n| \leq g$ a.e. on $E$. Show that $$ \int \liminf f_n \leq \liminf \int f_n \leq \limsup \int f_n \leq \int \limsup f_n.$$
I know that this problem is an application of the Lebesgue dominated convergence theorem.
Any idea of how to go about it thanks, I am really having a hard time with this problem.