# 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:

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

Then $(X,{\mathcal {A}}_{1})$  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 $(X,{\mathcal {A}}_{2})$ .

## Common measurable spaces

If $X$  is finite or countably infinite, the $\sigma$ -algebra is most of the times 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)