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

Then:


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

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, while others call it the monotone convergence theorem.

On ProofWiki, latter name is reserved for a more general result, see Monotone Convergence Theorem (Measure Theory) for details.