# Beppo-Levi Theorem

The Beppo-Levi theorem is a result in measure theory which gives a sufficient condition for interchanging an integral with an infinite series. The setting and result is essentially a particular case of the monotone convergence theorem, though one needs to be careful that all intermediary functions in the proof remain measurable so that monotone convergence may be applied.

## Statement

Let ${\displaystyle (X,\Sigma ,\mu )}$ be the underlying measure space and let ${\displaystyle \{f_{n}\}_{n=1}^{\infty }}$ be a sequence of measurable functions with ${\displaystyle f_{n}:X\rightarrow [0,+\infty ]}$ for each ${\displaystyle n\in \mathbb {N} }$. Then, ${\displaystyle \sum _{n=1}^{\infty }\int f_{n}d\mu =\int \sum _{n=1}^{\infty }f_{n}d\mu }$

## Proof

We know for any two non-negative measurable functions ${\displaystyle f,g:X\to [0,+\infty ]}$ that

${\displaystyle \int f+\int g=\int f+g.}$
Iterating this formula inductively, we find for all ${\displaystyle N\in \mathbb {N} }$ that
${\displaystyle \int \sum _{n=1}^{N}f_{n}=\sum _{n=1}\int f_{n}.}$
In addition, we know that the sum of two nonnegative measurable functions is again nonnegative and measurable, and induction implies that each ${\displaystyle \sum _{n=1}^{N}f_{n}}$ is again measurable and nonnegative.

The sequence of functions ${\displaystyle \left\{\sum _{n=1}^{N}f_{n}\right\}_{n\in \mathbb {N} }}$ is monotonically nondecreasing since each ${\displaystyle f_{n}}$ is nonnegative. By the monotone convergence theorem, we thus deduce

${\displaystyle \lim _{N\to \infty }\int \sum _{n=1}^{N}f_{n}=\int \lim _{N\to \infty }\sum _{n=1}^{N}f_{n}=\int \sum _{n=1}^{\infty }f_{n}.}$

## References

1. Folland, Gerald. B; "Real Analysis: Modern Techniques and Their Applications." Wiley. 2007.