Measurable space

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In mathematics, a measurable space or Borel space<ref name="eommeasurablespace" /> is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets 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 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.

DefinitionEdit

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 is needed for a measurable space.

ExampleEdit

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 spacesEdit

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 <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 spacesEdit

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 isomorphic to a measurable subset of the real numbers (again with the Borel <math>\sigma</math>-algebra)<ref name="Kallenberg15" />

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See alsoEdit

ReferencesEdit

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<references> <ref name="eommeasurablespace" >Template:SpringerEOM</ref> <ref name="Kallenberg15" >Template:Cite book</ref> <ref name="Klenke18" >Template:Cite book</ref> </references>

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