How to go about showing that $\lim_{n\to\infty}\int_{-n}^{n}f=\lim_{n\to\infty}\int _{\{f\leq n\}}f$ given that $f$ is nonnegative and is a finite integral a.e.?
I thought about applying the monotone convergence theorem, but we aren't ensured that the sequence $f_n$ is increasing... (or rather there is no "$f_n$" here, just the limit itself)
Since I barely have anything, I would prefer a hint in the right direction over a complete proof. Thanks.
Observe that $\int_{-n}^n f=\int_{\mathbb{R}} \chi_{[−n,n]} f,\int_{\{f\leq n\}} f=\int_{\mathbb{R}}\chi_{\{f\leq n\}}f$ and set $f_n = \chi_{[-n,n]} f$ in the first case and $f_n = \chi_{\{f \leq n\}} f$ in the second case.