Measurable space

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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.

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.^[2] 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 $σ$ -algebra would be: $ℱ_{1} = {X, \emptyset} .$ Then $(X, ℱ_{1})$ is a measurable space. Another possible $σ$ -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 $σ$ -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 $σ$ -algebra is most commonly the Borel $σ$ -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 $ℝ .$

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 $σ$ -algebra)^[3]

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Additionally, a Template:Em is a [[pi-system\|Template:Pi-system]] where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A Template:Em is a semiring where every complement $Ω ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ $A, B, A_{1}, A_{2}, \dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ \neq \emptyset .$

References

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[eommeasurablespace-1] 1.0 ^1.1 Template:SpringerEOM

[Klenke18-2] Template:Cite book

[Kallenberg15-3] Template:Cite book

[1]

[2]

[3]

Measurable space

Contents

Definition

Example

Common measurable spaces

Ambiguity with Borel spaces

See also

References

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Measurable space

Definition

Example

Common measurable spaces

Ambiguity with Borel spaces

See also

References

Navigation menu

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