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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 Σ{displaystyle Sigma } of measurable subsets to consist of all subsets of X{displaystyle X}. Then the counting measure μ{displaystyle mu } on this measurable space (X,Σ){displaystyle (X,Sigma )} is the positive measure Σ→[0,+∞]{displaystyle Sigma rightarrow [0,+infty ]} defined by

μ(A)={|A|if A is finite+∞if A is infinite{displaystyle mu (A)={begin{cases}vert Avert &{text{if }}A{text{ is finite}}\+infty &{text{if }}A{text{ is infinite}}end{cases}}}

for all A∈Σ{displaystyle Ain Sigma }, where |A|{displaystyle vert Avert } denotes the cardinality of the set A{displaystyle A}.^[2]

The counting measure on (X,Σ){displaystyle (X,Sigma )} is σ-finite if and only if the space X{displaystyle X} is countable.^[3]

Discussion

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

μ(A):=∑a∈Af(a)∀A⊆X,{displaystyle mu (A):=sum _{ain A}f(a),forall Asubseteq 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⊆Ry:=supF⊆Y,|F|<∞{∑y∈Fy}.{displaystyle sum _{yin Ysubseteq mathbb {R} }y:=sup _{Fsubseteq Y,|F|<infty }left{sum _{yin F}yright}.}

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

Notes

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.