Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
• Non-negativity: For all E in Σ, we have μ(E) ≥ 0.

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Some important measures are listed here.
• The counting measure is defined by μ(S) = number of elements in S.
• The Lebesgue measure on ℝ is a complete translation-invariantmeasure

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Let μ be a measure.
Monotonicity
If E1 and E2 are measurable sets with E1 ⊆ E2 then
Measure of countable

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Completeness
A measurable set X is called a null setif μ(X) = 0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called

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For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably

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• Abelian von Neumann algebra
• Almost everywhere
• Carathéodory's extension theorem

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If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradoxand

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