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

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

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

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

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