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Measurable space

From Wikipedia, the free encyclopedia

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.

It captures and generalises intuitive notions such as length, area, and volume with a set of 'points' in the space, but regions of the space are the elements of the σ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region.

Definition

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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.

Example

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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

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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 spaces

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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 also

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References

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  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.