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. It captures and generalises intuitive notions such as length, area, and volume with a set X 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 X and a σ-algebra \mathcal F on X. Then the tuple is called a measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space.

Example

Look at the set: One possible \sigma-algebra would be: Then is a measurable space. Another possible \sigma-algebra would be the power set on X: With this, a second measurable space on the set X is given by

Common measurable spaces

If X is finite or countably infinite, the \sigma-algebra is most often the power set on X, so This leads to the measurable space If X is a topological space, the \sigma-algebra is most commonly the Borel \sigma-algebra \mathcal B, so This leads to the measurable space that is common for all topological spaces such as the real numbers \R.

Ambiguity with Borel spaces

The term Borel space is used for different types of measurable spaces. It can refer to