Series as a Lebesgue integral against counting measure
= Series as a Lebesgue integral against counting measure
For $f:\mathbb N\to[0,\infty]$,
$$
\int_{\mathbb N}f\,d\#
=\sum_{n=1}^{\infty}f(n)
:=\sup_{F\subseteq\mathbb N\text{ finite}}\sum_{n\in F}f(n).
$$
A complex-valued function is integrable for counting measure exactly when its series is absolutely convergent.