$\int_{\bigcup_{n=1}^{\infty}E_n}f=\sum_{n=1}^{\infty}\int_{E_n}f$ given $f$ positive and measurable I'm learning about measure theory (specifically Lebesgue intregation) and need help with the following problem: 

Let $f:\mathbb{R}\rightarrow[0,+\infty)$ be measurable and let $\{E_n\}$ be a collection of pairwise disjoint measurable sets. Prove that $\int_{\bigcup_{n=1}^{\infty}E_n}f=\sum_{n=1}^{\infty}\int_{E_n}f.$

For convenience I set $E=\bigcup_{n=1}^{\infty}E_n$. 
This problem looks like an application of the monotone convergence theorem but I'm having a hard time applying it. I need to find a sequence of functions that is positive an nondecreasing but I don't know how to define it. 

Set $f_N=\sum_{n=1}^{N} f\chi_{E_n}$.
As $f\chi_{E_n}\geq 0$ for each $n$,  $f_1\leq f_2 \leq f_3 \leq f_4....$.
Now observe that $f_n \rightarrow f \chi_{E}$ as $n\rightarrow \infty$, where $E=\bigcup_{n=1}^{\infty}E_n$. Thus by MCT we obtain the following equality:
$\lim_{n} \int f_n d\mu= \int f\chi_{E} d\mu$.
Which is the conclusion you seek.
