Let $f$ be integrable over $E$.
(i) If $\{E_n\}$ is an ascending countable collection of measurable subsets of $E$, then $\int_{\cup E_n} f=\lim\limits_{n \to \infty} \int_{E_n}f $.
(ii) If $\{E_n\}$ is a descending countable collection of measurable subsets of $E$, then $\int_{\cap E_n} f = \lim\limits_{n \to \infty} \int_{E_n} f$.
Let $E_0=\emptyset$. Then let $F_n=E_n \setminus E_{n-1}$. Now we have $E=\cup E_n=\cup F_n$ and $F_n$'s are disjoint measurable sets. I wanted to apply the following theorem $\int_E f = \sum\limits_{n=1}^{\infty}\int_{F_n} f$. However, I'm stuck on incorporating the continuity of measure into a proof about the continuity of integration.
I'm stuck on proving part (1) as part (2) will follow from taking complements of things from part (1).