In mathematics, a measurable space or Borel space[1] is a basic object in measure theory.
It consists of a set and a σ-algebra, which defines the subsets that will be measured.
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.
Consider a set
is called a measurable space.
are called measurable sets within the measurable space.
Note that in contrast to a measure space, no measure is needed for a measurable space.
Look at the set:
is a measurable space.
-algebra would be the power set on
With this, a second measurable space on the set
is finite or countably infinite, the
-algebra is most often the power set on
This leads to the measurable space
is a topological space, the
-algebra is most commonly the Borel
This leads to the measurable space
that is common for all topological spaces such as the real numbers
The term Borel space is used for different types of measurable spaces.
It can refer to 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