Beppo Levi's Theorem

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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