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.

Look at the set:

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 ${\displaystyle \mathbb {R} .}$ 

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]

• Borel set – Class of mathematical sets
• Measurable function – Function for which the preimage of a measurable set is measurable
• Measure – Generalization of mass, length, area and volume
• Standard Borel space – Mathematical construction in topology