Monotone Convergence Theorem (Measure Theory)

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\left({u_n}\right)_{n \in \N} \in \mathcal{L}^1 \left({\mu}\right)$, $u_n: X \to \R$ be a increasing sequence of $\mu$-integrable functions.

Let $\displaystyle \sup_{n \mathop \in \N} u_n: X \to \overline{\R}$ be the pointwise supremum of the $u_n$.

Then $\displaystyle \sup_{n \mathop \in \N} u_n$ is $\mu$-integrable iff:


 * $\displaystyle \sup_{n \mathop \in \N} \int u_n \, \mathrm d \mu < +\infty$

and, in that case:


 * $\displaystyle \int \sup_{n \mathop \in \N} u_n \, \mathrm d \mu = \sup_{n \mathop \in \N} \int u_n \, \mathrm d \mu$