counting measureCountingMeasure2002-02-18 02:33:332007-08-13 04:23:00Definition\usepackage{amssymb}
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\newcommand{\Linf}{\Lspace{\infty}}Let $(X,\borel)$ be a measurable space. The measure $\mu$ on $X$ defined by
\begin{equation*}
\mu(A) = \left\{
\begin{array}{ll}
n & \text{if}\, A\, \text{ has exactly }\, n\, \text{ elements} \\
\infty & \text{otherwise.}
\end{array}
\right.
\end{equation*}
for all $A\in\borel$ is called the \textit{counting measure} on $X$.
Usually this is applied when $X$ is countable, e.g. $\naturals$ or $\integers$.