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.

Not to be confused with Measure space.

Definition

Consider a set ${\displaystyle X}$ and a σ-algebra ${\displaystyle {\mathcal {A}}}$ on ${\displaystyle X}$. Then the tuple ${\displaystyle (X,{\mathcal {A}})}$ is called a measurable space.[2]

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

Example

Look at the set

${\displaystyle X=\{1,2,3\}.}$

One possible ${\displaystyle \sigma }$-algebra would be

${\displaystyle {\mathcal {A}}_{1}=\{X,\emptyset \}.}$

Then ${\displaystyle (X,{\mathcal {A}}_{1})}$ is a measurable space. Another possible ${\displaystyle \sigma }$-algebra would be the power set on ${\displaystyle X}$:

${\displaystyle {\mathcal {A}}_{2}={\mathcal {P}}(X).}$

With this, a second measurable space on the set ${\displaystyle X}$ is given by ${\displaystyle (X,{\mathcal {A}}_{2})}$.

Common measurable spaces

If ${\displaystyle X}$ is finite or countable infinite, the ${\displaystyle \sigma }$-algebra is most of the times the power set on ${\displaystyle X}$, so ${\displaystyle {\mathcal {A}}={\mathcal {P}}(X)}$. This leads to the measurable space ${\displaystyle (X,{\mathcal {P}}(X))}$.

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

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 [1]
• a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel ${\displaystyle \sigma }$-algebra)[3]

References

1. 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. 77. Switzerland: Springer. p. 15. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.