Measure space

Template:Short description 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|Template:Mvar-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.

Definition

A measure space is a triple <math>(X, \mathcal A, \mu),</math> where<ref name="Kosorok83"/><ref name="Klenke18" />

• <math>X</math> is a set
• <math>\mathcal A</math> is a [[σ-algebra|Template:Mvar-algebra]] on the set <math>X</math>
• <math>\mu</math> is a measure on <math>(X, \mathcal{A})</math>

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

Example

Set <math>X = \{0, 1\}</math>. The <math display=inline>\sigma</math>-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 <math display=inline>\wp(\cdot).</math> Sticking with this convention, we set <math display=block>\mathcal{A} = \wp(X)</math>

In this simple case, the power set can be written down explicitly: <math display=block>\wp(X) = \{\varnothing, \{0\}, \{1\}, \{0, 1\}\}.</math>

As the measure, define <math display=inline>\mu</math> by <math display=block>\mu(\{0\}) = \mu(\{1\}) = \frac{1}{2},</math> so <math display=inline>\mu(X) = 1</math> (by additivity of measures) and <math display=inline>\mu(\varnothing) = 0</math> (by definition of measures).

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

Important 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<ref name="Kosorok83"/>
• Finite measure spaces, where the measure is a finite measure<ref name="eommeasurespace"/>
• <math> \sigma</math>-finite measure spaces, where the measure is a <math> \sigma </math>-finite measure<ref name="eommeasurespace"/>

Another class of measure spaces are the complete measure spaces.<ref name="Klenke33" />

References

<references> <ref name="Kosorok83" >Template:Cite book</ref> <ref name="Klenke18" >Template:Cite book</ref> <ref name="Klenke33" >Template:Cite book</ref> <ref name="eommeasurespace">Template:SpringerEOM</ref> </references>