Counting measure

Example

Let $X$ be a set and let $\Sigma$ be its power set. For $A \subseteq X$, let $\mu(A)$ be the number of elements of $A$ if $A$ is finite, and $\mu(A) = \infty$ if $A$ is infinite. Then $\mu$ is a measure on $(X, \Sigma)$, the counting measure.

For the check, $\mu(\varnothing) = 0$. Let $A_1, A_2, \ldots$ be pairwise disjoint subsets of $X$ with union $A$. Suppose all $A_n$ are finite and only finitely many are nonempty. Then $A$ is finite, and its number of elements is the sum of those of the $A_n$. Otherwise $A$ is infinite, and the sum $\sum_n \mu(A_n)$ is $\infty$ too, since it has a term $\infty$ or infinitely many terms at least $1$.