Standard Borel space explained
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
is said to be "standard Borel" if there exists a
metric on
that makes it a
complete separable metric space in such a way that
is then the
Borel σ-algebra.
[1] Standard Borel spaces have several useful properties that do not hold for general measurable spaces.
Properties
and
are standard Borel then any bijective
measurable mapping
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.
and
are standard Borel spaces and
then
is measurable if and only if the
graph of
is Borel.
Kuratowski's theorem
Theorem. Let
be a
Polish space, that is, a
topological space such that there is a
metric
on
that defines the topology of
and that makes
a complete separable metric space. Then
as a Borel space is
Borel isomorphic to one of(1)
(2)
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, 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.
Notes and References
- Mackey, G.W. (1957): Borel structure in groups and their duals. Trans. Am. Math. Soc., 85, 134-165.
