##### Wiktionary

**counting measure**

n. (context analysis English) A positive measure which, for each subset of a given measure space, assigns a value equal to the number of points in that subset.

##### Wikipedia

**Counting measure**

In mathematics, the **counting measure** is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset has finitely many elements, and ∞ if the subset is infinite.

The counting measure can be defined on any measurable set, but is mostly used on countable sets.

In formal notation, we can make any set *X* into a measurable space by taking the sigma-algebra Σ of measurable subsets to consist of all subsets of *X*. Then the counting measure *μ* on this measurable space (*X*, Σ) is the positive measure Σ → [0, + ∞] defined by

$$\mu(A)=\begin{cases}
\vert A \vert & \text{if } A \text{ is finite}\\
+\infty & \text{if } A \text{ is infinite}
\end{cases}$$

for all *A* ∈ Σ, where |*A*| denotes the cardinality of the set *A*.

The counting measure on (*X*, Σ) is σ-finite if and only if the space *X* is countable.