Monotone Convergence Theorem

Theorem

Suppose $\{f_{n}\}$ is a sequence of non-negative measurable functions, $f_{n}:X\to [0,+\infty ]$ such that $f_{n-1}\leq f_{n}$ for all $n$. Furthermore, $\lim _{n\to +\infty }f_{n}=f(=\sup _{n}f_{n})$. Then

$\lim _{n\to +\infty }\int f_{n}=\int \lim _{n\to +\infty }f_{n}.$

Proof

First we prove that <math> \lim_{n\rightarrow +\infty} \int f_n \leq \int \lim_{n\rightarrow +\infty} f_n <\math>.

Since