# Monotone Convergence Theorem

## Theorem

### Monotone Convergence Theorem (Real Analysis)

Let $\sequence {x_n}$ be a bounded monotone sequence sequence in $\R$.

Then $\sequence {x_n}$ is convergent.

### Monotone Convergence Theorem (Measure Theory)

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $u : X \to \overline \R_{\ge 0}$ be a positive $\Sigma$-measurable function.

Let $\sequence {u_n}_{n \mathop \in \N}$ be an sequence of positive $\Sigma$-measurable functions $u_n : X \to \overline \R_{\ge 0}$ such that:

$\map {u_i} x \le \map {u_j} x$ for all $i \le j$

and:

$\ds \map u x = \lim_{n \mathop \to \infty} \map {u_n} x$

hold for $\mu$-almost all $x \in X$.

Then:

$\ds \int u \rd \mu = \lim_{n \mathop \to \infty} \int u_n \rd \mu$