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.
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.
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 .
The counting measure on is ?-finite if and only if the space is countable.
The counting measure is a special case of a more general construction. With the notation as above, any function defines a measure
where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, i.e.,
Taking f(x) = 1 for all x in X gives the counting measure.