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 $\lim _{n\rightarrow +\infty }\int f_{n}\leq \int \lim _{n\rightarrow +\infty }f_{n}$.

Since $f_{n}\leq f_{n+1}$ for all $n\in \mathbb {N}$, we have $f_{n}\leq \lim _{n\rightarrow +\infty }f_{n}$ and further $\int f_{n}\leq \int \lim _{n\rightarrow +\infty }f_{n}$.

Sending $n\rightarrow +\infty$ on LHS gives us the result.

Then we only need to prove that $\lim _{n\rightarrow +\infty }\int f_{n}\geq \int \lim _{n\rightarrow +\infty }f_{n}$.