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.
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 of measurable subsets to consist of all subsets of. 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 σ-finite if and only if the space is countable.Discussion
The counting measure is a special case of a more general construction. With the notation as above, any function defines a measure
on via
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 = 1 for all x in X gives the counting measure.
