Measurable space

Consider a set ${\displaystyle X}$ and a σ-algebra ${\displaystyle {\mathcal {A}}}$ on ${\displaystyle X}$. Then the tuple ${\displaystyle (X,{\mathcal {A}})}$ is called a measurable space.[2]

Note that in contrast to a measure space, no measure is needed for a measurable space.

Look at the set:

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

One possible ${\displaystyle \sigma }$-algebra would be:

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

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

${\displaystyle {\mathcal {A}}_{2}={\mathcal {P}}(X).}$

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

If ${\displaystyle X}$ is finite or countably infinite, the ${\displaystyle \sigma }$-algebra is most of the times the power set on ${\displaystyle X}$, so ${\displaystyle {\mathcal {A}}={\mathcal {P}}(X)}$. 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 ${\displaystyle {\mathcal {A}}={\mathcal {B}}(X)}$. This leads to the measurable space ${\displaystyle (X,{\mathcal {B}}(X))}$ that is common for all topological spaces such as the real numbers ${\displaystyle \mathbb {R} }$.

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]