Let $f$ be a integrable function on $E$. Show that for each $\epsilon > 0$, there is an $N \in \mathbb{N}$ for which $n \geq N$, then $|\int_{E_n} f|< \epsilon$ where $E_n=\{x \in E : |x| \geq n \}$.
I want to know if this is a proper way to prove this.
First note, by the continuity of measure, $\{E_n\}$ is a descending chain of sets. I was thinking this follows from the continuity of measure. But I'm not sure how to show that $|\int_{E_n}f|< \epsilon$. That is, I'm not sure how to show that I can make the measure over $E_n$ arbitrarily small.