Standard Borel space

In mathematics, a standard Borel space is the Borel space associated to a Polish space. Up to isomorphism, there is only one nontrivial standard Borel space.

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

Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which defines the topology of X and which 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, 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.

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