The identity is $$ \lim\limits_{n\rightarrow \infty}\inf\int_{E^c}f_n=\int_\mathbb{R}f-\lim_{n\rightarrow \infty}\sup\int_Ef_n $$ Where we are given that $f_n\rightarrow f$ pointwise, the sequence of functions is measurable and nonnegative, $E$ is measurable and $$ \int_{\mathbb{R}}f=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f_n $$
I can make up some reasons why the equality makes sense; we are tightly undervaluing the integral on the subset of the whole area (on $\mathbb{R}$) $E^c$ and then subtracting a tight overestimate on the complement of the area from the whole.
Since that's just a lot of words and this seems like a useful technique, I could use some help seeing this better. Thank you!
You're overthinking it.
For any $n \in \mathbb N$, $$ \int_{E^c} f_n = \int_{\mathbb R} f_n - \int_{E} f_n.$$
Take $\lim_{n \to \infty} \inf$ of both sides.
[Note that, since $\lim_{n \to \infty} \int_{\mathbb R} f_n$ exists, $$\lim_{n \to \infty} \inf \int_{\mathbb R} f_n = \lim_{n \to \infty} \int_{\mathbb R} f_n.$$
Also note that $$\lim_{n \to \infty} \inf \left( - \int_E f_n \right) = - \lim_{n \to \infty} \sup \int_E f_n.$$ ]