Lebesgue's Monotone Convergence Theorem

Definition

Say $X$ has a measure $\mu$, and let $f_n:X\to [0,\infty ]$ be measurable and $f_1\le f_2\le f_3\le \dots$.

Then $\int f_{m}d\mu \to \int {\lim }_{n\to \infty }{f}_{n}d\mu$ as $m\to \infty$.