Standard Borel space

In mathematics, a '''standard Borel space''' is the Borel space associated with a Polish space. Except in the case of discrete Polish spaces, the standard Borel space is unique, up to isomorphism of measurable spaces.

==Formal definition==

A measurable space <math>(X, \Sigma)</math> is said to be "standard Borel" if there exists a metric on <math>X</math> that makes it a complete separable metric space in such a way that <math>\Sigma</math> is then the Borel σ-algebra.<ref>Mackey, G.W. (1957): Borel structure in groups and their duals. Trans. Am. Math. Soc., 85, 134-165.</ref> Standard Borel spaces have several useful properties that do not hold for general measurable spaces.

==Properties==

* If <math>(X, \Sigma)</math> and <math>(Y, T)</math> are standard Borel then any bijective measurable mapping <math>f : (X, \Sigma) \to (Y, \Tau)</math> is an isomorphism (that is, the inverse mapping is also measurable). This follows from Souslin's theorem, as a set that is both analytic and coanalytic is necessarily Borel. * If <math>(X, \Sigma)</math> and <math>(Y, T)</math> are standard Borel spaces and <math>f : X \to Y</math> then <math>f</math> is measurable if and only if the graph of <math>f</math> is Borel. * The product and direct union of a countable family of standard Borel spaces are standard. * Every complete probability measure on a standard Borel space turns it into a standard probability space.

==Kuratowski's theorem==

'''Theorem'''. Let <math>X</math> be a Polish space, that is, a topological space such that there is a metric <math>d</math> on <math>X</math> that defines the topology of <math>X</math> and that makes <math>X</math> a complete separable metric space. Then <math>X</math> as a Borel space is Borel isomorphic to one of (1) <math>\R,</math> (2) <math>\Z</math> or (3) a finite discrete space. (This result is reminiscent of Maharam's theorem.)

It follows that a standard Borel space is characterized up to isomorphism by its cardinality,<ref>{{citation | last=Srivastava| first=S.M. | title=A Course on Borel Sets | year=1991 | publisher=Springer Verlag | isbn=0-387-98412-7}}</ref> and that any uncountable standard Borel space has the cardinality of the continuum.

Borel isomorphisms on standard Borel spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being only Borel measurable.

==See also==

* {{annotated link|Measurable space}}

==References==

Category:Descriptive set theory Category:General topology Category:Measure theory Category:Space (mathematics)