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 $f_{n}:X\rightarrow [0,+\infty ]$. Then, $\sum _{n=1}^{\infty }\int f_{n}d\mu =\int \sum _{n=1}^{\infty }f_{n}d\mu$

Proof

First, the result is proven for finite sums. Take $f,g:X\rightarrow [0,+\infty ]$ measurable functions. As such, consider two sequences of simple functions $\{\phi _{j}\}_{j=1}^{\infty }$ and $\{\psi _{j}\}_{j=1}^{\infty }$ so that they monotonically converge pointwise to $f$ and $g$ respectively.