Measurable space

From HandWiki

In mathematics, a measurable space or Borel space[1] is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.

Definition

Consider a set [math]\displaystyle{ X }[/math] and a σ-algebra [math]\displaystyle{ \mathcal A }[/math] on [math]\displaystyle{ X. }[/math] Then the tuple [math]\displaystyle{ (X, \mathcal A) }[/math] is called a measurable space.[2]

Note that in contrast to a measure space, no measure is needed for a measurable space.

Example

Look at the set: [math]\displaystyle{ X = \{1,2,3\}. }[/math] One possible [math]\displaystyle{ \sigma }[/math]-algebra would be: [math]\displaystyle{ \mathcal A_1 = \{X, \varnothing\}. }[/math] Then [math]\displaystyle{ \left(X, \mathcal A_1\right) }[/math] is a measurable space. Another possible [math]\displaystyle{ \sigma }[/math]-algebra would be the power set on [math]\displaystyle{ X }[/math]: [math]\displaystyle{ \mathcal A_2 = \mathcal P(X). }[/math] With this, a second measurable space on the set [math]\displaystyle{ X }[/math] is given by [math]\displaystyle{ \left(X, \mathcal A_2\right). }[/math]

Common measurable spaces

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

If [math]\displaystyle{ X }[/math] is a topological space, the [math]\displaystyle{ \sigma }[/math]-algebra is most commonly the Borel [math]\displaystyle{ \sigma }[/math]-algebra [math]\displaystyle{ \mathcal B, }[/math] so [math]\displaystyle{ \mathcal A = \mathcal B(X). }[/math] This leads to the measurable space [math]\displaystyle{ (X, \mathcal B(X)) }[/math] that is common for all topological spaces such as the real numbers [math]\displaystyle{ \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 [1]
  • a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel [math]\displaystyle{ \sigma }[/math]-algebra)[3]


See also

References

  1. 1.0 1.1 Hazewinkel, Michiel, ed. (2001), "Measurable space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Measurable_space 
  2. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_341. 
  3. Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. 77. Switzerland: Springer. p. 15. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.