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
• a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel ${\displaystyle \sigma }$-algebra)