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

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

References

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