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Measurable space

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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 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.

Consider a set and a σ-algebra on Then the tuple is called a measurable space.[2] The elements of are called measurable sets within the 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

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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]
Families of sets over
Is necessarily true of
or, is closed under: 		Directed
by 		F.I.P.
π-system Yes Yes No No No No No No No No
Semiring Yes Yes No No No No No No Yes Never
Semialgebra (semifield) Yes Yes No No No No No No Yes Never
Monotone class No No No No No only if only if No No No
𝜆-system (Dynkin system) Yes No No only if
 Yes No only if or
they are disjoint		 Yes Yes Never
Ring (order theory) Yes Yes Yes No No No No No No No
Ring (measure theory) Yes Yes Yes Yes No No No No Yes Never
δ-ring Yes Yes Yes Yes No Yes No No Yes Never
𝜎-ring Yes Yes Yes Yes No Yes Yes No Yes Never
Algebra (field) Yes Yes Yes Yes Yes No No Yes Yes Never
𝜎-algebra (𝜎-field) Yes Yes Yes Yes Yes Yes Yes Yes Yes Never
Filter Yes Yes Yes No No No Yes Yes No No
Proper filter Yes Yes Yes Never Never No Yes Yes Never Yes
Prefilter (filter base) Yes No No No No No No No No Yes
Filter subbase No No No No No No No No No Yes
Open topology Yes Yes Yes No No No
(even arbitrary )		 Yes Yes Never
Closed topology Yes Yes Yes No No
(even arbitrary )		 No Yes Yes Never
Is necessarily true of
or, is closed under: 		directed
downward 		finite
intersections 		finite
unions 		relative
complements 		complements
in 		countable
intersections 		countable
unions 		contains contains Finite
intersection
property

Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in
A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in
are arbitrary elements of and it is assumed that

  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.