$\{f_n\}$ are nonnegative monotonic increasing functions. Show that $$ \int_X \sum_{n=1}^\infty f_n \, d\mu = \sum_{n=1}^\infty \int_X f_n \, d\mu $$
Can someone give me a hint on how to show this?
I know that I can use the monotone convergence theorem, but I dont know how.
The $f_n$ are non-negative, so $g_n = \sum_{k=1}^{n} f_k$ is monotonic increasing. Now apply the MCT: $$ \int_X \sum_{k=1}^{\infty} f_k \, d\mu = \int_X \lim_{n \to \infty} \sum_{k=1}^{n} f_k \, d\mu = \int_X \lim_{n \to \infty} g_n \, d\mu = \lim_{n \to \infty} \int_X g_n \, d\mu = \lim_{n \to \infty} \sum_{k=1}^n \int_X f_k \, d\mu, $$ where the last equality uses the (finite) linearity of the integral.