Commutativity of Limits on Integrals of Non Monotonic Measurable functions

Question

Let $\{f_n\}$ be a sequence of non-negative measurable functions and suppose $f=\lim_{n\to\infty}f_n$ exists. Assuming $\lim_{n\to\infty}\int f_n d\mu$ exists, show that $\int f_n d\mu \leq \lim_{n\to\infty}\int f_n d\mu$

There is no assumption that the measurable functions are monotonically increasing, so I'm not quite sure how to proceed.

Answer 1

Let $g_n=\inf \{f_k: k \geq n\}$. Verify that $g_n$ is non-negative, measurable and increases to $f$. By Monotone Convergence Theorem $\int g_n \to \int f$. Also $g_n \leq f_n$ so $\int f =\lim \int g_n \leq \lim \int f_n$.