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.

Definition

Consider a set $X$  and a σ-algebra ${\mathcal {A}}$  on $X.$  Then the tuple $(X,{\mathcal {A}})$  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:

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

One possible $\sigma$ -algebra would be:
${\mathcal {A}}_{1}=\{X,\varnothing \}.$

Then $\left(X,{\mathcal {A}}_{1}\right)$  is a measurable space. Another possible $\sigma$ -algebra would be the power set on $X$ :
${\mathcal {A}}_{2}={\mathcal {P}}(X).$

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

Common measurable spaces

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