Measurable space

Short description: Basic object in measure theory; set and a sigma-algebra

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.

Definition

Consider a set $\displaystyle{ X }$ and a σ-algebra $\displaystyle{ \mathcal F }$ on $\displaystyle{ X. }$ Then the tuple $\displaystyle{ (X, \mathcal F) }$ 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: $\displaystyle{ X = \{1,2,3\}. }$ One possible $\displaystyle{ \sigma }$-algebra would be: $\displaystyle{ \mathcal {F}_1 = \{X, \varnothing\}. }$ Then $\displaystyle{ \left(X, \mathcal{F}_1 \right) }$ is a measurable space. Another possible $\displaystyle{ \sigma }$-algebra would be the power set on $\displaystyle{ X }$: $\displaystyle{ \mathcal{F}_2 = \mathcal P(X). }$ With this, a second measurable space on the set $\displaystyle{ X }$ is given by $\displaystyle{ \left(X, \mathcal F_2\right). }$

Common measurable spaces

If $\displaystyle{ X }$ is finite or countably infinite, the $\displaystyle{ \sigma }$-algebra is most often the power set on $\displaystyle{ X, }$ so $\displaystyle{ \mathcal{F} = \mathcal P(X). }$ This leads to the measurable space $\displaystyle{ (X, \mathcal P(X)). }$

If $\displaystyle{ X }$ is a topological space, the $\displaystyle{ \sigma }$-algebra is most commonly the Borel $\displaystyle{ \sigma }$-algebra $\displaystyle{ \mathcal B, }$ so $\displaystyle{ \mathcal{F} = \mathcal B(X). }$ This leads to the measurable space $\displaystyle{ (X, \mathcal B(X)) }$ that is common for all topological spaces such as the real numbers $\displaystyle{ \R. }$

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 $\displaystyle{ \sigma }$-algebra)[3]