What is "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.

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.