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Measurable space

## Summary

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.

## 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]