Let $f:\mathbb{R} \to \mathbb{R}$ such that $f$ is integrable over $[-n,n]$ for every $n \in \mathbb{R}$ and assume that $$\lim_{n \to \infty} \int^n_{-n}fdm < \infty.$$
Proposition: $f$ is integrable over $\mathbb{R}$ and $$\lim_{n \to \infty} \int^n_{-n}fdm=\int fdm.$$
I'm having trouble with proving the integrability. Once that has been shown, I can complete the proof by applying the dominated convergence theorem and using the appropriate indicator function.