Consider a set and a σ-algebra on . Then the tuple is called a measurable space.
Note that in contrast to a measure space, no measure is needed for a measurable space.
Look at the set:
One possible -algebra would be:
Then is a measurable space. Another possible -algebra would be the power set on :
With this, a second measurable space on the set is given by .
Common measurable spacesEdit
If is finite or countably infinite, the -algebra is most of the times the power set on , so . This leads to the measurable space .
If is a topological space, the -algebra is most commonly the Borel -algebra , so . This leads to the measurable space that is common for all topological spaces such as the real numbers .
Ambiguity with Borel spacesEdit
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 -algebra)