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 discrete space. (This result is reminiscent of Maharam's theorem.)
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.
- Measurable space – Basic object in measure theory; set and a sigma-algebra