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.

Formal definition

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.[1] Standard Borel spaces have several useful properties that do not hold for general measurable spaces.

Properties

  • If (X, Σ) and (Y, Τ) are standard Borel then any bijective measurable mapping 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 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.

Kuratowski's theorem

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,[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.

References

  1. ↑ Mackey, G.W. (1957): Borel structure in groups and their duals. Trans. Am. Math. Soc., 85, 134-165.
  2. ↑ Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7
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