Counting measure
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.^{[1]}
The counting measure can be defined on any measurable set, but is mostly used on countable sets.^{[1]}
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 .^{[2]}
The counting measure on is σ-finite if and only if the space is countable.^{[3]}
Discussion
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.
Notes
- ^ ^{a} ^{b} Counting Measure at PlanetMath.org.
- ^ Schilling (2005), p.27
- ^ Hansen (2009) p.47
References
- 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.