Editing Measurable space

{{Short description|Basic object in measure theory; set and a sigma-algebra}}
{{confused|Measure space}}

In [[mathematics]], a '''measurable space''' or '''Borel space'''<ref name="eommeasurablespace" /> is a basic object in [[measure theory]]. It consists of a [[Set (mathematics)|set]] and a [[Sigma-algebra|σ-algebra]], which defines the [[subset]]s that will be measured.

It captures and generalises intuitive notions such as length, area, and volume with a set <math>X</math> of 'points' in the space, but ''regions'' of the space are the elements of the [[Sigma-algebra|σ-algebra]], since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region.

==Definition==

Consider a set <math>X</math> and a [[σ-algebra]] <math>\mathcal F</math> on <math>X.</math> Then the tuple <math>(X, \mathcal F)</math> is called a measurable space.<ref name="Klenke18" /> The elements of <math>\mathcal F</math> are called '''measurable sets''' within the measurable space.

Note that in contrast to a [[measure space]], no [[Measure (mathematics)|measure]] is needed for a measurable space.

==Example==

Look at the set:
<math display=block>X = \{1,2,3\}.</math>
One possible <math>\sigma</math>-algebra would be:
<math display="block">\mathcal {F}_1 = \{X, \varnothing\}.</math>
Then <math>\left(X, \mathcal{F}_1 \right)</math> is a measurable space. Another possible <math>\sigma</math>-algebra would be the [[power set]] on <math>X</math>:
<math display="block">\mathcal{F}_2 = \mathcal P(X).</math>
With this, a second measurable space on the set <math>X</math> is given by  <math>\left(X, \mathcal F_2\right).</math>

==Common measurable spaces==

If <math>X</math> is finite or countably infinite, the <math>\sigma</math>-algebra is most often the [[power set]] on <math>X,</math> so <math>\mathcal{F} = \mathcal P(X).</math> This leads to the measurable space <math>(X, \mathcal P(X)).</math>

If <math>X</math> is a [[topological space]], the <math>\sigma</math>-algebra is most commonly the [[Borel sigma algebra|Borel <math>\sigma</math>-algebra]] <math>\mathcal B,</math> so <math>\mathcal{F} = \mathcal B(X).</math> This leads to the measurable space <math>(X, \mathcal B(X))</math> that is common for all topological spaces such as the real numbers <math>\R.</math>

==Ambiguity with Borel spaces==

The term Borel space is used for different types of measurable spaces. It can refer to
* any measurable space, so it is a synonym for a measurable space as defined above <ref name="eommeasurablespace" />
* a measurable space that is [[Borel isomorphism|Borel isomorphic]] to a measurable subset of the real numbers (again with the Borel <math>\sigma</math>-algebra)<ref name="Kallenberg15" />

{{Families of sets}}

==See also==

* {{annotated link|Borel set}}
* {{annotated link|Measurable function}}
* {{annotated link|Measure (mathematics)|Measure}}
* {{annotated link|Standard Borel space}}
* [[Category of measurable spaces]]

==References==

{{reflist|group=note}}

<references>
<ref name="eommeasurablespace" >{{SpringerEOM|title=Measurable space|id=Measurable_space|author-last1=Sazonov|author-first1=V.V.}}</ref>
<ref name="Kallenberg15" >{{cite book|last1=Kallenberg|first1=Olav|author-link1=Olav Kallenberg|year=2017|title=Random Measures, Theory and Applications|volume=77|location= Switzerland|publisher=Springer|page=15|doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3|series=Probability Theory and Stochastic Modelling}}</ref>
<ref name="Klenke18" >{{cite book|last1=Klenke|first1=Achim|year=2008|title=Probability Theory|url=https://archive.org/details/probabilitytheor00klen_341|url-access=limited|location=Berlin|publisher=Springer|doi=10.1007/978-1-84800-048-3|isbn=978-1-84800-047-6|page=[https://archive.org/details/probabilitytheor00klen_341/page/n28 18]}}</ref>
</references>

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

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