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 {\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.
Definition 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. 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.
Example 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 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 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 σ {\displaystyle \sigma } -algebra)
See also 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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