Measurable space


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.

Definition

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.

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 countable 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 spaces

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