Counting measure

In mathematics, specifically measure theory, 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.[1]

The counting measure can be defined on any measurable space (i.e. any set along with a sigma-algebra) but is mostly used on countable sets.[1]

In formal notation, we can turn any set into a measurable space by taking the power set of as the sigma-algebra , i.e. all subsets of are measurable. Then the counting measure on this measurable space is the positive measure defined by

for all , where denotes the cardinality of the set .[2]

The counting measure on is σ-finite if and only if the space is countable.[3]