Counting Measure
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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.[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 ?-finite if and only if the space is countable.[3]

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(x) = 1 for all x in X gives the counting measure.

References

  1. ^ a b Counting Measure at PlanetMath.org.
  2. ^ Schilling, René L. (2005). Measures, Integral and Martingales. Cambridge University Press. p. 27. ISBN 0-521-61525-9.
  3. ^ 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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