On the theorem regarding continuity of integration:
Let $f$ be integrable over $E$.
If $\{E_{n}\}^{\infty}_{n=1}$ is an ascending countable collection of measurable subsets on $E$, then
$$\int_{\cup^{\infty}_{n=1} E_{n}} f = \lim_{n\to\infty} \int_{E_{n}} f,$$
I have seen a proof which used this result:
Let $f$ be integrable over $E$ and $\{E_{n}\}^{\infty}_{n=1}$ a disjoint countable collection of measurable subsets of $E$ whise union is $E$, then
$$\int_{E} f = \sum_{n=1}^{\infty} \int_{E_{n}} f$$
by "disjoint-ifying" the ascending sequence $\{E_{n}\}^{\infty}_{n=1}$.
Now, to prove the continuity of integration, is it necessary to define the function $f$ in terms $f_n = f \chi_n$, where $\chi_n$ is the characteristic function on the union of the ascending sequence?
After thinking about this one, it seems that the answer is that there is no need to define $f$ in that way since the integral is taken not on the entire measurable set $E$.