It captures and generalises intuitive notions such as length, area, and volume with a set of 'points' in the space, but regions of the space are the elements of the σ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region.
Definition
Consider a set and a σ-algebra on Then the tuple 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:
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 spaces
If is finite or countably infinite, the -algebra is most often 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 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 -algebra)[3]
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in
A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in are arbitrary elements of and it is assumed that