Standard Borel space

A measurable space (X, Σ) is said to be "standard Borel" if there exists a metric on X that makes it a complete separable metric space in such a way that Σ is then the Borel σ-algebra.[1] Standard Borel spaces have several useful properties that do not hold for general measurable spaces.

• If (X, Σ) and (Y, Τ) are standard Borel then any bijective measurable mapping ${\displaystyle f:(X,\Sigma )\rightarrow (Y,\mathrm {T} )}$ is an isomorphism (i.e., 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 (X, Σ) and (Y, Τ) are standard Borel spaces and ${\displaystyle f:X\rightarrow Y}$ then f is measurable if and only if the graph of f 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 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 (1) R, (2) Z 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,[2] 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.