Standard Borel Space

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## Formal definition

## Properties

## Kuratowski's theorem

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Standard Borel Space

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 (*X*, ?) 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 ? is then the Borel ?-algebra.^{[1]}
Standard Borel spaces have several useful properties that do not hold for general measurable spaces.

- If (
*X*, ?) and (*Y*, ?) 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*, ?) and (*Y*, ?) 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.

**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.

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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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