# 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 $\overline \R$.

## Proof

## Also known as

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

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the latter name is reserved for the general result: Monotone Convergence Theorem (Measure Theory).

## Source of Name

This entry was named for Beppo Levi.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $9.6$