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.
- If and are standard Borel then any bijective measurable mapping 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 and are standard Borel spaces and then is measurable if and only if the graph of 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.
Theorem. Let be a Polish space, that is, a topological space such that there is a metric on that defines the topology of and that makes a complete separable metric space. Then as a Borel space is Borel isomorphic to one of (1) (2) or (3) a finite space. (This result is reminiscent of Maharam's theorem.)
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.