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.
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
The counting measure is a special case of a more general construct. 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 sup of the sums over all finite subsets, i.e.,
Taking f(x)=1 for all x in X produces the counting measure.
- Schilling, René L. (2005). Measures, Integral and Martingales. Cambridge University Press.
- Hansen, Ernst (2009). Measure Theory, Fourth Edition. Department of Mathematical Science, University of Copenhagen.