Let $(E\_n)\_{n \in \mathbb{N}}$ be a sequence of measurable sets such that $\lim\_{n \to \infty} E\_n =E$ for some measurable set $E$. When does it hold that
$$ \lim\_{n \to \infty}\int\_{E\_n}f d \mu = \int\_{E}f d\mu$$
What if we replace $\lim$ with $\lim \sup$ or $\lim \inf$?