Beppo Levi's Theorem

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

Let $\left({f_n}\right)_{n \in \N} \in \mathcal{M}_{\overline{\R}}^+$ be an increasing sequence of positive $\Sigma$-measurable functions.

Let $\displaystyle \sup_{n \in \N} f_n: X \to \overline{\R}$ be the pointwise supremum of $\left({f_n}\right)_{n \in \N}$, where $\overline{\R}$ denotes the extended real numbers.

Then:


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

where the supremum on the right is in the ordering on $\overline{\R}$.

Also known as
Some authors refer to this result as Beppo Levi's lemma, others call it the monotone convergence theorem.

This latter name is discouraged as it possibly leads to confusion with the Monotone Convergence Theorem for sequences of real numbers.