In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Consider a set and a σ-algebra on . Then the tuple is called a measurable space.
Note that in contrast to a measure space, no measure is needed for a measurable space.
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 .
If is a topological space, the -algebra is most commonly the Borel -algebra , so . This leads to the measurable space that is common for all topological spaces such as the real numbers .
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)
- ^ a b 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.