Counting measure - Alchetron, The Free Social Encyclopedia

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.

The counting measure can be defined on any measurable set, but is mostly used on countable sets.

In formal notation, we can make any set X into a measurable space by taking the sigma-algebra Σ of measurable subsets to consist of all subsets of X . Then the counting measure μ on this measurable space ( X , Σ ) is the positive measure Σ → [ 0 , + ∞ ] defined by

μ ( A ) = { | A | if A is finite + ∞ if A is infinite

for all A ∈ Σ , where | A | denotes the cardinality of the set A .

The counting measure on ( X , Σ ) is σ-finite if and only if the space X is countable.

Discussion

The counting measure is a special case of a more general construct. With the notation as above, any function f : X → [ 0 , ∞ ) defines a measure μ on ( X , Σ ) via

μ ( A ) := ∑ a ∈ A f ( a ) ∀ A ⊆ X ,

where the possibly uncountable sum of real numbers is defined to be the sup of the sums over all finite subsets, i.e.,

∑ y ∈ Y ⊆ R y := sup F ⊆ Y , | F | < ∞ { ∑ y ∈ F y } .

Taking f(x)=1 for all x in X produces the counting measure.

References

Counting measure Wikipedia

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