In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a -algebra on this set and provides information about the sets that will be measured.
Look at the set
One possible -Algebra would be
Then is a measurable space. Another possible -algebra would be the power set on :
With this, a second measurable space on the set is given by .
Common measurable spaces
If is finite or countable infinite, the -algebra is most of the times the power set on , so . This leads to the measurable space .
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 
- a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel -algebra)
- Sazonov, V.V. (2001) , "Measurable space", in Hazewinkel, Michiel (ed.), 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.