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{{Short description|Class of mathematical sets}}
In [[mathematics]], the '''Borel sets''' of a [[topological space]] are a particular class of "well-behaved" subsets of that space.  For example, whereas an arbitrary subset of the [[real number]]s might fail to be [[Lebesgue measurable]], every Borel set of reals is [[universally measurable]].  Which sets are Borel can be specified in a number of equivalent ways.  Borel sets are named after [[Émile Borel]].

The most usual definition goes through the notion of a [[&sigma;-algebra]], which is a collection of subsets of a topological space <math>X</math> that contains both the empty set and the entire set <math>X</math>, and is closed under [[countable set|countable]] union and complement.

Then we can define the '''Borel &sigma;-algebra''' over <math>X</math> to be the smallest &sigma;-algebra containing all open sets of <math>X</math>.{{efn|Equivalently, all closed sets.}}  A Borel subset of <math>X</math> is then simply an element of this &sigma;-algebra.

Borel sets are important in [[measure theory]], since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a [[Borel measure]].  Borel sets and the associated [[Borel hierarchy]] also play a fundamental role in [[descriptive set theory]].

In some contexts, Borel sets are defined to be generated by the [[compact set]]s of the topological space, rather than the open sets. The two definitions are equivalent for many [[well-behaved]] spaces, including all [[Hausdorff space|Hausdorff]] [[σ-compact space]]s, but can be different in more [[pathological (mathematics)|pathological]] spaces.

== Generating the Borel algebra ==
{{main|Borel hierarchy}}
In the case that <math>X</math> is a [[metric space]], the Borel algebra in the first sense may be described ''generatively'' as follows.

For a collection <math>T</math> of subsets of <math>X</math> (that is, for any subset of the [[power set]] <math>\mathcal P</math><math>(X)</math> of <math>X</math>), let
* <math>T_\sigma </math> be all countable unions of elements of <math>T</math>
* <math>T_\delta </math> be all countable intersections of elements of <math>T</math>
* <math>T_{\delta\sigma} = (T_\delta)_\sigma.</math>

Now define by [[transfinite induction]] a sequence <math>G^m</math>, where <math>m</math> is an [[ordinal number]], in the following manner:
* For the base case of the definition, let <math> G^0</math> be the collection of open subsets of <math>X</math>.
* If <math>i</math> is not a [[limit ordinal]], then <math>i</math> has an immediately preceding ordinal <math>i-1</math>. Let <math display="block"> G^i = [G^{i-1}]_{\delta \sigma}.</math>
* If <math>i</math> is a limit ordinal, set <math display="block"> G^i = \bigcup_{j < i} G^j. </math>

The claim is that the Borel algebra is <math>G^{\omega_1}</math>, where <math>{\omega_1}</math> is the [[first uncountable ordinal|first uncountable ordinal number]]. That is, the Borel algebra can be ''generated'' from the class of open sets by iterating the operation
<math display="block"> G \mapsto G_{\delta \sigma} </math>
to the first uncountable ordinal.

To prove this claim, any open set in a metric space is the union of an increasing sequence of closed sets. In particular, complementation of sets maps <math>G^m</math> into itself for any limit ordinal <math>m</math>; moreover if <math>m</math> is an uncountable limit ordinal, <math>G^m</math> is closed under countable unions.

For each Borel set <math>B</math>, there is some countable ordinal <math>\alpha_B</math> such that <math>B</math> can be obtained by iterating the operation over <math>\alpha_B</math>. However, as <math>B</math> varies over all Borel sets, <math>\alpha_B</math> will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is <math>\omega_1</math>, the first uncountable ordinal.

The resulting sequence of sets is termed the [[Borel hierarchy]].

=== Example ===
An important example, especially in the [[probability theory|theory of probability]], is the Borel algebra on the set of [[real number]]s. It is the algebra on which the [[Borel measure]] is defined.  Given a [[Random variable#Real-valued random variables|real random variable]] defined on a [[probability space]], its  [[probability distribution]] is by definition also a measure on the Borel algebra.

The Borel algebra on the reals is the smallest σ-algebra on <math>\mathbb R</math> that contains all the [[interval (mathematics)|intervals]].

In the construction by transfinite induction, it can be shown that, in each step, the [[cardinality|number]] of sets is, at most, the [[cardinality of the continuum]]. So, the total number of Borel sets is less than or equal to
<math display="block">\aleph_1 \cdot 2 ^ {\aleph_0}\, = 2^{\aleph_0}.</math>

In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of [[Lebesgue measurable]] sets that exist, which is strictly larger and equal to <math>2^{2^{\aleph_0}}</math>).

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

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

[[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 <math>B</math> in <math>Y</math>, the set <math>f^{-1}(B)</math> is measurable in <math>X</math>.

'''Theorem'''.  Let <math>X</math> be a [[Polish space]], that is, a topological space such that there is a [[Metric (mathematics)|metric]] <math>d</math> on <math>X</math> that defines the topology of <math>X</math> and that makes <math>X</math> a complete [[separable space|separable]] metric space. Then ''<math>X</math>'' as a Borel space is [[isomorphic]] to one of
# '''<math>\mathbb R</math>''', 
# <math>\mathbb Z</math>, 
# a finite space.
(This result is reminiscent of [[Maharam's theorem]].)

Considered as Borel spaces, the real line '''<math>\mathbb R</math>,''' the union of '''<math>\mathbb R</math>''' with a countable set, and '''<math>\mathbb R^n</math>''' 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]].

== Non-Borel sets ==
{{anchor|counterexample}}
An example of a subset of the reals that is non-Borel, due to [[Nikolai Luzin|Lusin]],<ref>{{Citation | language=fr | last=Lusin | first=Nicolas | year=1927 | title=Sur les ensembles analytiques | journal=[[Fundamenta Mathematicae]] | volume=10 |url=https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/25/0/93222/sur-les-ensembles-analytiques-nuls|pages=Sect. 62, pages 76–78| doi=10.4064/fm-10-1-1-95 | doi-access=free }}</ref> is described below. In contrast, an example of a [[non-measurable set]] cannot be exhibited, although the existence of such a set is implied, for example, by the [[axiom of choice]].

Every [[irrational number]] has a unique representation by an infinite [[simple continued fraction]]

:<math>x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} </math>

where <math>a_0</math> is some [[integer]] and all the other numbers <math>a_k</math> are ''positive'' integers. Let <math>A</math> be the set of all irrational numbers that correspond to sequences <math>(a_0,a_1,\dots)</math> with the following property: there exists an infinite [[subsequence]] <math>(a_{k_0},a_{k_1},\dots)</math> such that each element is a [[divisor]] of the next element. This set <math>A</math> is not Borel. However, it is [[analytic set|analytic]] (all Borel sets are also analytic), and complete in the class of analytic sets. <ref>For more details see [[descriptive set theory]] and the book by [[Alexander S. Kechris|A. S. Kechris]], especially Exercise (27.2) on page 209, Definition (22.9) on page 169, Exercise (3.4)(ii) on page 14, and on page 196.</ref>

It's important to note, that while [[Zermelo–Fraenkel axioms]] (ZF) are sufficient to formalize the construction of <math>A</math>, it cannot be proven in ZF alone that <math>A</math> is non-Borel. In fact, it is consistent with ZF that <math>\mathbb{R}</math> is a countable union of countable sets,<ref>{{cite book |last=Jech |first=Thomas |author-link=Thomas Jech |date=2008 |title=The Axiom of Choice | pages=142| publisher=Courier Corporation.}}</ref> so that any subset of <math>\mathbb{R}</math> is a Borel set.

Another non-Borel set is an inverse image <math>f^{-1}[0]</math> of an [[infinite parity function]] <math>f\colon \{0, 1\}^{\omega} \to \{0, 1\}</math>. However, this is a proof of existence (via the axiom of choice), not an explicit example.

==Alternative non-equivalent definitions==

According to [[Paul Halmos]],<ref>{{harv|Halmos|1950|loc=page 219}}</ref> a subset of a [[locally compact]] Hausdorff topological space is called a ''Borel set'' if it belongs to the smallest [[Sigma-ring|σ-ring]] containing all compact sets.

Norberg and Vervaat<ref>Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: ''Probability and Lattices'', in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150</ref> redefine the Borel algebra of a topological space <math>X</math> as the <math>\sigma</math>-algebra generated by its open subsets and its compact [[Saturated set|saturated subset]]s. This definition is well-suited for applications in the case where <math>X</math> is not Hausdorff. It coincides with the usual definition if <math>X</math> is [[second countable]] or if every compact saturated subset is closed (which is the case in particular if <math>X</math> is Hausdorff).

== See also ==

* {{annotated link|Borel hierarchy}}
* {{annotated link|Borel isomorphism}}
* {{annotated link|Baire set}}
* {{annotated link|Cylindrical σ-algebra}}
* {{annotated link|Descriptive set theory}}
* {{annotated link|Polish space}}

== Notes ==
{{notelist}}

== References ==
{{reflist}}

== Further reading ==

* [[William Arveson]], ''An Invitation to C*-algebras'', Springer-Verlag, 1981. (See Chapter 3 for an excellent exposition of ''Polish topology'')
* [[Richard M. Dudley|Richard Dudley]], '' Real Analysis and Probability''. Wadsworth, Brooks and Cole, 1989
*{{cite book
|first=Paul R.
|last=Halmos
|author-link=Paul Halmos
|title=Measure theory
|publisher=D. van Nostrand Co
|year=1950}}  See especially Sect. 51 "Borel sets and Baire sets".
* [[Halsey Royden]], ''Real Analysis'', Prentice Hall, 1988
* [[Alexander S. Kechris]], ''Classical Descriptive Set Theory'', Springer-Verlag, 1995 (Graduate texts in Math., vol. 156)

==External links==
* {{springer|title=Borel set|id=p/b017120}}
* [https://web.archive.org/web/20130923121802/http://mws.cs.ru.nl/mwiki/prob_1.html#K12 Formal definition] of Borel Sets in the [[Mizar system]], and the [http://mmlquery.mizar.org/cgi-bin/mmlquery/emacs_search?input=(symbol+Borel_Sets+%7C+notation+%7C+constructor+%7C+occur+%7C+th)+ordered+by+number+of+ref list of theorems] {{Webarchive|url=https://web.archive.org/web/20200601022908/http://mmlquery.mizar.org/cgi-bin/mmlquery/emacs_search?input=(symbol+Borel_Sets+%7C+notation+%7C+constructor+%7C+occur+%7C+th)+ordered+by+number+of+ref |date=2020-06-01 }} that have been formally proved about it.
* {{MathWorld |title=Borel Set |id=BorelSet}}

{{Measure theory}}
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[[Category:Topology]]
[[Category:Descriptive set theory]]