Measurable space explained

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.

It captures and generalises intuitive notions such as length, area, and volume with a set

X 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

Consider a set

X and a σ-algebra l F on X. Then the tuple (X, l F) is called a measurable space.

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

Example

Look at the set:$$X\; =\; \backslash .$$One possible

\sigma -algebra would be:$$\backslash mathcal\; \_1\; =\; \backslash .$$Then \left(X, l{F} _{1} \right) is a measurable space. Another possible \sigma -algebra would be the power set on X :$$\backslash mathcal\_2\; =\; \backslash mathcal\; P(X).$$With this, a second measurable space on the set X is given by \left(X, l F _{2\right).}

Common measurable spaces

If

X is finite or countably infinite, the \sigma -algebra is most often the power set on X, so l{F} = l P(X). This leads to the measurable space (X, l P(X)).

If

X is a topological space, the \sigma -algebra is most commonly the Borel \sigma -algebra l B, so l{F} = l B(X). This leads to the measurable space (X, l B(X)) that is common for all topological spaces such as the real numbers \R.

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

\sigma -algebra)

See also

• • • • • Category of measurable spaces

References

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Notes and References

1. Book: Kallenberg. Olav. Olav Kallenberg. 2017. Random Measures, Theory and Applications. 77. Switzerland. Springer. 15. 10.1007/978-3-319-41598-7. 978-3-319-41596-3. Probability Theory and Stochastic Modelling.
2. Book: Klenke. Achim. 2008. Probability Theory. limited. Berlin. Springer. 10.1007/978-1-84800-048-3. 978-1-84800-047-6. 18.