Editing Borel set (section)

== Standard Borel spaces and Kuratowski theorems ==
<!-- This section is linked from [[Kazimierz Kuratowski]] -->
{{see also|Standard Borel space}}

Let ''X'' be a topological space. The '''Borel space''' associated to ''X'' is the pair (''X'',''B''), where ''B'' is the σ-algebra of Borel sets of ''X''.

[[George Mackey]] defined a Borel space somewhat differently, writing that it is "a set together with a distinguished σ-field of subsets called its Borel sets."<ref>{{citation | last=Mackey| first=G.W. | title=Ergodic Theory and Virtual Groups |  year=1966 | journal=[[Math. Ann.]]|volume=166|pages=187&ndash;207|issue=3|author-link=George Mackey|doi=10.1007/BF01361167| s2cid=119738592 |issn=0025-5831}}</ref> However, modern usage is to call the distinguished sub-algebra the ''measurable sets'' and such spaces [[measurable space|''measurable spaces'']]. The reason for this distinction is that the Borel sets are the σ-algebra generated by ''open'' sets (of a topological space), whereas Mackey's definition refers to a set equipped with an ''arbitrary'' σ-algebra. There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.<ref>[https://mathoverflow.net/q/87888 Jochen Wengenroth, Is every sigma-algebra the Borel algebra of a topology?]</ref>

Measurable spaces form a [[category (mathematics)|category]] in which the [[morphism]]s are [[measurable function]]s between measurable spaces. A function <math>f:X \rightarrow Y</math> is [[measurable function|measurable]] if it [[pullback|pulls back]] measurable sets, i.e., for all measurable sets ''B'' in ''Y'', the set <math>f^{-1}(B)</math> is measurable in ''X''.

'''Theorem'''.  Let ''X'' be a [[Polish space]], that is, a topological space such that there is a [[Metric (mathematics)|metric]] ''d'' on ''X'' that defines the topology of ''X'' and that makes ''X'' a complete [[separable space|separable]] metric space.  Then ''X'' as a Borel  space is [[isomorphic]] to one of
# '''R''', 
# '''Z'''{{clarification needed|date=October 2024}}, 
# a finite space. 
(This result is reminiscent of [[Maharam's theorem]].)

Considered as Borel spaces, the real line '''R''', the union of '''R''' with a countable set, and '''R'''<sup>n</sup> are isomorphic.

A '''[[standard Borel space]]''' is the Borel space associated to a [[Polish space]]. A standard Borel space is characterized up to isomorphism by its cardinality,<ref>{{citation | last=Srivastava| first=S.M. | title=A Course on Borel Sets |  year=1991 | publisher=[[Springer Verlag]] | isbn=978-0-387-98412-4}}</ref> and any uncountable standard Borel space has the cardinality of the continuum.

For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces.  Note however, that the range of a continuous noninjective map may fail to be Borel.  See [[analytic set]].

Every [[probability measure]] on a standard Borel space turns it into a [[standard probability space]].