Standard Borel space

In mathematics, a standard Borel space is the Borel set#Standard [Borel spaces and Kuratowski theorems|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 is said to be "standard Borel" if there exists a metric on that makes it a metric space|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. 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
or a finite discrete 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.