Measurable space

In mathematics, a measurable space or Borel space^[1] is a basic object in measure theory. It consists of a set and a $\sigma$ -algebra on this set and provides information about the sets that will be measured.

Definition

Given a nonempty set $X$ and a $\sigma$ -algebra ${\mathcal {A}}$ on $X$ . Then the tuple $(X,{\mathcal {A}})$ 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

X=\{1,2,3\}.

One possible $\sigma$ -Algebra would be

{\mathcal {A}}_{1}=\{X,\emptyset \}.

Then $(X,{\mathcal {A}}_{1})$ is a measurable space. Another possible $\sigma$ -algebra would be the power set on $X$ :

{\mathcal {A}}_{2}={\mathcal {P}}(X).

With this, a second measurable space on the set $X$ is given by $(X,{\mathcal {A}}_{2})$ .

Common measurable spaces

If $X$ is finite or countable infinite, the $\sigma$ -algebra is most of the times the power set on $X$ , so ${\mathcal {A}}={\mathcal {P}}(X)$ . This leads to the measurable space $(X,{\mathcal {P}}(X))$ .

If $X$ is a topological space, the $\sigma$ -algebra is most commonly the Borel $\sigma$ -algebra ${\mathcal {B}}$ , so ${\mathcal {A}}={\mathcal {B}}(X)$ . This leads to the measurable space $(X,{\mathcal {B}}(X))$ that is common for all topological spaces such as the real numbers $\mathbb {R}$ .

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 $\sigma$ -algebra)^[3]

References

1 2 Sazonov, V.V. (2001) [1994], "Measurable space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
↑ 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.

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[eommeasurablespace-1] 1 2 Sazonov, V.V. (2001) [1994], "Measurable space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

[Klenke18-2] Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[Kallenberg15-3] 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.