Counting measure

Definition

$\#(A)=\{\begin{array}{ll}|A|& |A|<\infty \\ \infty & |A|=\infty \end{array}$

Facts

Consider a Measure Space $(\Omega ,\mathcal{F},\#)$ where $\Omega$ is a countable set, $\mathcal{F}=2^{\Omega }$ is a Sigma-Field defined on the set $\Omega$, and $\#$ is a counting measure on $\mathcal{F}$. Then $\int fd\#=\sum _{i\in \Omega }f(i)$