Let (E,(E)) be a measurable spaceMathworldPlanetmathPlanetmath. A measureMathworldPlanetmath on (E,(E)) is a function μ:(E){} with values in the extended real numbers such that:

  1. 1.

    μ(A)0 for A(E), with equality if A=

  2. 2.

    μ(i=0Ai)=i=0μ(Ai) for any sequence of pairwise disjoint sets Ai(E).

Occasionally, the term positive measure is used to distinguish measures as defined here from more general notions of measure which are not necessarily restricted to the non-negative extended reals.

The second property above is called countable additivityMathworldPlanetmath, or σ-additivity. A finitely additive measure μ has the same definition except that (E) is only required to be an algebra and the second property above is only required to hold for finite unions. Note the slight abuse of terminology: a finitely additive measure is not necessarily a measure.

The triple (E,(E),μ) is called a measure space. If μ(E)=1, then it is called a probability space, and the measure μ is called a probability measure.

Lebesgue measureMathworldPlanetmath on n is one important example of a measure.

Title measure
Canonical name Measure
Date of creation 2013-03-22 11:57:33
Last modified on 2013-03-22 11:57:33
Owner djao (24)
Last modified by djao (24)
Numerical id 19
Author djao (24)
Entry type Definition
Classification msc 60A10
Classification msc 28A10
Related topic LpSpace
Related topic SigmaFinite
Related topic Integral2
Related topic DistributionPlanetmathPlanetmath
Related topic LebesgueMeasure
Defines measure space
Defines probability space
Defines probability measure
Defines countably additive
Defines finitely additive
Defines σ-additive
Defines positive measure