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Measure space

Set on which a generalization of volumes and integrals is defined

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Source updated June 28, 2026

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra), and the method that is used for measuring (the measure). One important example of a measure space is a probability space.

A measurable space consists of the first two components without a specific measure.

01Definition

A measure space is a triple $(X,{\mathcal {A}},\mu ),$ where

$X$ is a set
${\mathcal {A}}$ is a σ-algebra on the set $X$
$\mu$ is a measure on $(X,{\mathcal {A}})$

In other words, a measure space consists of a measurable space $(X,{\mathcal {A}})$ together with a measure on it.

02Example

Set $X=\{0,1\}$ . The ${\textstyle \sigma$ -algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by ${\textstyle \wp (\cdot ).$ Sticking with this convention, we set ${\mathcal {A}}=\wp (X)$

In this simple case, the power set can be written down explicitly: $\wp (X)=\{\varnothing ,\{0\},\{1\},\{0,1\}\}.$

As the measure, define ${\textstyle \mu$ by $\mu (\{0\})=\mu (\{1\})={\frac {1}{2}},$ so ${\textstyle \mu (X)=1$ (by additivity of measures) and ${\textstyle \mu (\varnothing )=0$ (by definition of measures).

This leads to the measure space ${\textstyle (X,\wp (X),\mu ).$ It is a probability space, since ${\textstyle \mu (X)=1.$ The measure ${\textstyle \mu$ corresponds to the Bernoulli distribution with ${\textstyle p={\frac {1}{2}},$ which is for example used to model a fair coin flip.

03Important classes of measure spaces

Most important classes of measure spaces are defined by the properties of their associated measures. This includes, in order of increasing generality:

Probability spaces, a measure space where the measure is a probability measure
Finite measure spaces, where the measure is a finite measure
$\sigma$ -finite measure spaces, where the measure is a $\sigma$ -finite measure

Another class of measure spaces are the complete measure spaces.

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Sources and credits

This article is adapted from the Wikipedia article “Measure space”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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