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.

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 ${\mathcal {F}}$ on $X.$ Then the tuple $(X,{\mathcal {F}})$ 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: $X=\{1,2,3\}.$ One possible $\sigma$ -algebra would be: ${\mathcal {F}}_{1}=\{X,\varnothing \}.$ Then $\left(X,{\mathcal {F}}_{1}\right)$ is a measurable space. Another possible $\sigma$ -algebra would be the power set on $X$ : ${\mathcal {F}}_{2}={\mathcal {P}}(X).$ With this, a second measurable space on the set $X$ is given by $\left(X,{\mathcal {F}}_{2}\right).$

Common measurable spaces

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

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

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 $\sigma$ -algebra)^[3]

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