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.

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 ${ℱ}$ on $X.$ Then the tuple $(X,{ℱ})$ is called a measurable space. The elements of ${ℱ}$ 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

Look at the set: $$X=\{1,2,3\}.$$ One possible $\sigma$-algebra would be: $${{ℱ}}_1=\{X,\varnothing \}.$$ Then $(X,{{ℱ}}_1)$ is a measurable space. Another possible $\sigma$-algebra would be the power set on $X$: $${{ℱ}}_2={𝒫}(X).$$ With this, a second measurable space on the set $X$ is given by $(X,{{ℱ}}_2).$

Common measurable spaces

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

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

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
• a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel $\sigma$-algebra)

See also

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• Category of measurable spaces