I am expected to use the Monotone convergence theorem for non-negative functions in order to solve this question. I.e. LEt $(X, \Sigma, \mu)$ me a measure space. Suppose $fn:X \rightarrow \bar{\mathbb{R}}$ is a sequence of non-decreasing, non-negative measurable functions. Then $\int \lim_{n\rightarrow \infty}f_nd\mu = \lim_{n\rightarrow \infty} \int f_n d\mu$.
Assume $g_n$ is a sequence of non-negative measurable functions satifying$\int g_n d\mu < \frac{1}{n^2}$ for each $n\geq 1$. Prove that $\Sigma_{n=1}^{\infty}g_n(x) \leq + \infty$.
My thinking was that $\Sigma_{n=0}^{\infty} \int g_n < \Sigma_{n=0}^{\infty} \frac{1}{n^2}$, but because the integral amounts to summation anway we could swap the sum and the integral around... This seems dodgy and it doesn't use the monotone convergence theorem though! Help!!!