Measures (measure theory)

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 infinity if the subset is infinite. The counting measure can be defined on any measurable space (that is, any set along with a sigma-algebra) but is mostly used on countable sets. In formal notation, we can turn any set into a measurable space by taking the power set of as the sigma-algebra that is, all subsets of are measurable sets. Then the counting measure on this measurable space is the positive measure defined by for all where denotes the cardinality of the set The counting measure on is σ-finite if and only if the space is countable. (Wikipedia).

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Related pages

Infinity symbol | Power set | Mathematics | Set (mathematics) | Measure (mathematics) | Cardinality | Infinite set | Measurable space | Subset