Standard Borel space

In mathematics, a standard Borel space is the Borel space associated with a Polish space. Except in the case of discrete Polish spaces, the standard Borel space is unique, up to isomorphism of measurable spaces.

Formal definition

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

Properties

• If ${\displaystyle (X,\Sigma )}$ and ${\displaystyle (Y,T)}$ are standard Borel then any bijective measurable mapping ${\displaystyle f:(X,\Sigma )\to (Y,\mathrm {T} )}$ 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 ${\displaystyle (X,\Sigma )}$ and ${\displaystyle (Y,T)}$ are standard Borel spaces and ${\displaystyle f:X\to Y}$ then ${\displaystyle f}$ is measurable if and only if the graph of ${\displaystyle f}$ 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 ${\displaystyle X}$ be a Polish space, that is, a topological space such that there is a metric ${\displaystyle d}$ on ${\displaystyle X}$ that defines the topology of ${\displaystyle X}$ and that makes ${\displaystyle X}$ a complete separable metric space. Then ${\displaystyle X}$ as a Borel space is Borel isomorphic to one of (1) ${\displaystyle \mathbb {R} ,}$ (2) ${\displaystyle \mathbb {Z} }$ or (3) a finite discrete 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.

See also

• Measurable space – Basic object in measure theory; set and a sigma-algebra

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