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Measurable space - Wikipedia

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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.

It captures and generalises intuitive notions such as length, area, and volume with a set X {\displaystyle 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.

Consider a set X {\displaystyle X} and a σ-algebra F {\displaystyle {\mathcal {F}}} on X . {\displaystyle X.} Then the tuple ( X , F ) {\displaystyle (X,{\mathcal {F}})} is called a measurable space.^[2] The elements of F {\displaystyle {\mathcal {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.

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

Common measurable spaces[edit]

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

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

Ambiguity with Borel spaces[edit]

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

Families F {\displaystyle {\mathcal {F}}} of sets over Ω {\displaystyle \Omega }

Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon }
or, is F {\displaystyle {\mathcal {F}}} closed under: Directed by ⊇ {\displaystyle \,\supseteq } A ∩ B {\displaystyle A\cap B} A ∪ B {\displaystyle A\cup B} B ∖ A {\displaystyle B\setminus A} Ω ∖ A {\displaystyle \Omega \setminus A} A 1 ∩ A 2 ∩ ⋯ {\displaystyle A_{1}\cap A_{2}\cap \cdots } A 1 ∪ A 2 ∪ ⋯ {\displaystyle A_{1}\cup A_{2}\cup \cdots } Ω ∈ F {\displaystyle \Omega \in {\mathcal {F}}} ∅ ∈ F {\displaystyle \varnothing \in {\mathcal {F}}} F.I.P. π-system Semiring Never Semialgebra (Semifield) Never Monotone class only if A i ↘ {\displaystyle A_{i}\searrow } only if A i ↗ {\displaystyle A_{i}\nearrow } 𝜆-system (Dynkin System) only if
A ⊆ B {\displaystyle A\subseteq B} only if A i ↗ {\displaystyle A_{i}\nearrow } or
they are disjoint Never Ring (Order theory) Ring (Measure theory) Never δ-Ring Never 𝜎-Ring Never Algebra (Field) Never 𝜎-Algebra (𝜎-Field) Never Dual ideal Filter Never Never ∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} Prefilter (Filter base) Never Never ∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} Filter subbase Never Never ∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} Open Topology
(even arbitrary ∪ {\displaystyle \cup } ) Never Closed Topology
(even arbitrary ∩ {\displaystyle \cap } ) Never Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon }
or, is F {\displaystyle {\mathcal {F}}} closed under: directed downward finite intersections finite unions relative complements complements in Ω {\displaystyle \Omega } countable intersections countable unions contains Ω {\displaystyle \Omega } contains ∅ {\displaystyle \varnothing } Finite Intersection Property

Additionally, a semiring is a π-system where every complement B ∖ A {\displaystyle B\setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.}
A semialgebra is a semiring where every complement Ω ∖ A {\displaystyle \Omega \setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.}
A , B , A 1 , A 2 , … {\displaystyle A,B,A_{1},A_{2},\ldots } are arbitrary elements of F {\displaystyle {\mathcal {F}}} and it is assumed that F ≠ ∅ . {\displaystyle {\mathcal {F}}\neq \varnothing .}

Borel set – Class of mathematical sets
Measurable function – Kind of mathematical function
Measure – Generalization of mass, length, area and volume
Standard Borel space – Mathematical construction in topology
Category of measurable spaces

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