Measurable space

Not to be confused with Measure 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.

It captures and generalises intuitive notions such as length, area, and volume with a set ${\displaystyle X}$ 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 ${\displaystyle X}$ and a σ-algebra ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X.}$ Then the tuple ${\displaystyle (X,{\mathcal {F}})}$ 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: $${\displaystyle {\mathcal {F}}_{1}=\{X,\varnothing \}.}$$ Then ${\displaystyle \left(X,{\mathcal {F}}_{1}\right)}$ is a measurable space. Another possible ${\displaystyle \sigma }$-algebra would be the power set on ${\displaystyle X}$: $${\displaystyle {\mathcal {F}}_{2}={\mathcal {P}}(X).}$$ With this, a second measurable space on the set ${\displaystyle X}$ is given by ${\displaystyle \left(X,{\mathcal {F}}_{2}\right).}$

Common measurable spaces

If ${\displaystyle X}$ is finite or countably infinite, the ${\displaystyle \sigma }$-algebra is most often the power set on ${\displaystyle X,}$ so ${\displaystyle {\mathcal {F}}={\mathcal {P}}(X).}$ 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 ${\displaystyle {\mathcal {F}}={\mathcal {B}}(X).}$ 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]

Families ${\displaystyle {\mathcal {F}}}$ of sets over ${\displaystyle \Omega }$ v t e
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F}}}$	${\displaystyle \varnothing \in {\mathcal {F}}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if ${\displaystyle A_{i}\searrow }$	only if ${\displaystyle A_{i}\nearrow }$
𝜆-system (Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if ${\displaystyle A_{i}\nearrow }$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Prefilter (Filter base)				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Filter subbase				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Open Topology							(even arbitrary ${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary ${\displaystyle \cap }$)				Never
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement ${\displaystyle B\setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ A semialgebra is a semiring where every complement ${\displaystyle \Omega \setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F}}}$ and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$

See also

• Borel set – Class of mathematical sets
• Measurable function – Kind of mathematical function
• Measure – Generalization of mass, length, area and volume
• Standard Borel space – Mathematical construction in topology
• Category of measurable spaces

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