Interchange of Integral and Discrete Sum

This theorem is an application of the Beppo Levi monotone convergence theorem as is clear from the proof.

Proof

Let $f_n=\sum _{k=1}^n{g}_k$. We will apply the Beppo Levi monotone convergence theorem to $f$, by first checking that the three criteria are satisfied.

First, $f_n$ is Lebesgue integrable as it is a finite sum of Lebesgue integrable functions. Secondly, the fact that $g_k\ge 0$ implies that as additional terms are added to the sum, the value of the sum is increasing, and thus $f_1\le f_2\le f_3\le \dots$. Finally, the fact that the sum is convergent: $\sum _{k=1}^{\mathrm{\infty }}{g}_k<\mathrm{\infty }$ implies that $f_n$ has a finite supremum.

Now, from the monotone convergence theorem: