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 σ-algebra, which defines the subsets that will be measured.

DefinitionEdit

Consider a set   and a σ-algebra   on   Then the tuple   is called a measurable space.[2]

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

ExampleEdit

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 spacesEdit

If   is finite or countably infinite, the  -algebra is most often 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 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 [1]
  • a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel  -algebra)[3]

See alsoEdit

ReferencesEdit

  1. ^ a b Sazonov, V.V. (2001) [1994], "Measurable space", Encyclopedia of Mathematics, EMS Press
  2. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  3. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 15. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.