# Measurable space

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 ${\mathcal {F}}$ on $X.$ Then the tuple $(X,{\mathcal {F}})$ is called a measurable space. The elements of ${\mathcal {F}}$ are called measurable sets within the measurable space.
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
* a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel $\sigma$-algebra)
Families ${\mathcal {F}}$ of sets over $\Omega$ vte :
| 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 ${\mathcal {F}}\neq \varnothing .$ |||||||||||