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 infinity if the subset is infinite.[1]
The counting measure can be defined on any measurable space (that is, 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 that is, all subsets of are measurable sets. Then the counting measure on this measurable space is the positive measure defined by
The counting measure on is σ-finite if and only if the space is countable.[3]
Integration on with counting measure
Take the measure space , where is the set of all subsets of the naturals and the counting measure. Take any measurable . As it is defined on , can be represented pointwise as
Each is measurable. Moreover . Still further, as each is a simple function
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
See also
- Pip (counting) – Easily countable items
- Set function – Function from sets to numbers
References
- ^ a b Counting Measure at PlanetMath.
- ^ Schilling, René L. (2005). Measures, Integral and Martingales. Cambridge University Press. p. 27. ISBN 0-521-61525-9.
- ^ Hansen, Ernst (2009). Measure Theory (Fourth ed.). Department of Mathematical Science, University of Copenhagen. p. 47. ISBN 978-87-91927-44-7.
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