Editing Measure space

{{confused|Measurable space}}

{{short description|Set on which a generalization of volumes and integrals is defined}}
A '''measure space''' is a basic object of [[measure theory]], a branch of [[mathematics]] that studies generalized notions of [[volume]]s. It contains an underlying set, the [[subset]]s of this set that are feasible for measuring (the [[σ-algebra|{{mvar|σ}}-algebra]]) and the method that is used for measuring (the [[Measure (mathematics)|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|{{mvar|σ}}-algebra]] on the set <math>X</math>
* <math>\mu</math> is a [[Measure (mathematics)|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 (mathematics)|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 space]]s, 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 [[sigma-finite measure|<math> \sigma </math>-finite measure]]<ref name="eommeasurespace"/>

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

==References==

<references>
<ref name="Kosorok83" >{{cite book |last1=Kosorok |first1=Michael R. |year=2008  |title=Introduction to Empirical Processes and Semiparametric Inference |location=New York |publisher=Springer |page=83|isbn=978-0-387-74977-8 }}</ref>
<ref name="Klenke18" >{{cite book |last1=Klenke |first1=Achim |year=2008  |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=18}}</ref>
<ref name="Klenke33" >{{cite book |last1=Klenke |first1=Achim |year=2008  |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=33}}</ref>
<ref name="eommeasurespace">{{SpringerEOM |title=Measure space |id=Measure_space  |author-last1=Anosov |author-first1=D.V.}}</ref>
</references>

{{Measure theory}}
{{Lp spaces}}

[[Category:Measure theory]]
[[Category:Space (mathematics)]]