Counting measure
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 is finite, and ∞ if the subset is infinite.
Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω. Define a measure μ on this sigma algebra by setting μ(A) = |A| if A is a finite subset of Ω and μ(A) = ∞ if A is an infinite subset of Ω, where |A| denotes the cardinality of set A. Then (Ω, Σ, μ) is a measure space.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the counting measure is an intuitive way to put a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
on any set: the "size" of a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
is taken to be the number of elements in the subset, if the subset is finite, and ∞ if the subset is infinite.
Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω. Define a measure μ on this sigma algebra by setting μ(A) = |A| if A is a finite subset of Ω and μ(A) = ∞ if A is an infinite subset of Ω, where |A| denotes the cardinality of set A. Then (Ω, Σ, μ) is a measure space.