Here is the problem: If $f$ is a nonnegative Lebesgue measurable function and $\{E_n\}_{n=1}^{\infty}$ is a sequence of Lebesgue measurable sets with $E_1\subset E_2\subset ...$, then
$\int_{\bigcup_{n=1}^{\infty}E_n}f{\rm d}\lambda = \lim_{n\to\infty}\int_{E_n}f{\rm d}\lambda$.
I know that $\int_{\bigcup \limits_{n=1}^{\infty} E_{n}} f \,d\lambda = \int \limits_{\Bbb R} f \chi_{\bigcup \limits_{n=1}^{\infty} E_{n}} \,d\lambda$.
I'm also thinking that I should apply the Monotone Convergence Theorem.
However, I don't know where to go from here. Any help would be appreciated.