Counting measure

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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

  1. ^ a b Counting Measure at PlanetMath.org.
  2. ^ Schilling (2005), p.27
  3. ^ 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.
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