Standard Borel space
In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up to isomorphism of measurable spaces, only one standard Borel space.A measurable space is said to be "standard Borel" if there exists a metric on that makes it a complete separable metric space in such a way that is then the Borel σ-algebra.
Standard Borel spaces have several useful properties that do not hold for general measurable spaces.Properties
Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X that defines the topology of X and that makes X a complete separable metric space. Then X as a Borel space is Borel isomorphic to one of
R, Z or a finite space.
It follows that a standard Borel space is characterized up to isomorphism by its cardinality, 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.
