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In mathematics, a measurable space or Borel space[1] is a basic object in measure theory. It consists of a set and a -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 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 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]

ReferencesEdit

  1. ^ a b Sazonov, V.V. (2001) [1994], "Measurable space", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  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. 77. Switzerland: Springer. p. 15. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.