In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.

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

and a σ-algebra 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.

Example

-algebra would be the power set

Common measurable spaces

-algebra is most often the power set

is a topological space-algebra is most commonly the \sigma-algebra

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

• a measurable space that is Borel isomorphic-algebra)

See also

• Category of measurable spaces

References