Beppo-Levi Theorem

The Beppo-Levi theorem is a result in measure theory that gives us conditions wherein we may then pass the integral through an infinite series of functions. That is to say, this theorem provides conditions under which the (possibly infinite) sum of the integrals is equal to the integral of the sums.

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

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