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 the latter name is reserved for the general result: Monotone Convergence Theorem (Measure Theory).