counting measure

Let $(X,\mathfrak{B})$ be a measurable space. The measure $\mu$ on $X$ defined by

$\mu(A)=\left\{\begin{array}[]{ll}n&\text{if}\,A\,\text{ has exactly }\,n\,% \text{ elements}\\ \infty&\text{otherwise.}\end{array}\right.$

for all $A\in\mathfrak{B}$ is called the counting measure on $X$ . Usually this is applied when $X$ is countable, e.g. $\mathbb{N}$ or $\mathbb{Z}$ .