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| In [[mathematics]], a '''Borel set''' is any set in a [[topological space]] that can be formed from [[open set]]s (or, equivalently, from [[closed set]]s) through the operations of [[countable]] [[union (set theory)|union]], countable [[intersection (set theory)|intersection]], and [[relative complement]]. Borel sets are named after [[Émile Borel]].
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| For a topological space ''X'', the collection of all Borel sets on ''X'' forms a [[sigma-algebra|σ-algebra]], known as the '''Borel algebra''' or '''Borel σ-algebra'''. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).
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| 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]].
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| 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.
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| == Generating the Borel algebra ==
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| In the case ''X'' is a [[metric space]], the Borel algebra in the first sense may be described ''generatively'' as follows.
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| For a collection ''T'' of subsets of ''X'' (that is, for any subset of the [[power set]] P(''X'') of ''X''), let
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| * <math>T_\sigma \quad </math> be all countable unions of elements of ''T''
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| * <math>T_\delta \quad </math> be all countable intersections of elements of ''T''
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| * <math> T_{\delta\sigma}=(T_\delta)_\sigma.\, </math>
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| Now define by [[transfinite induction]] a sequence ''G<sup>m</sup>'', where ''m'' is an [[ordinal number]], in the following manner:
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| * For the base case of the definition, let <math> G^0</math> be the collection of open subsets of ''X''.
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| * If ''i'' is not a [[limit ordinal]], then ''i'' has an immediately preceding ordinal ''i − 1''. Let
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| *: <math> G^i = [G^{i-1}]_{\delta \sigma}.</math>
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| * If ''i'' is a limit ordinal, set
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| *:<math> G^i = \bigcup_{j < i} G^j. </math>
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| The claim is that the Borel algebra is ''G''<sup>ω<sub>1</sub></sup>, where ω<sub>1</sub> 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
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| :<math> G \mapsto G_{\delta \sigma}. </math>
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| to the first uncountable ordinal.
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| To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets. In particular, it is easy to show that complementation of sets maps ''G<sup>m</sup>'' into itself for any limit ordinal ''m''; moreover if ''m'' is an uncountable limit ordinal, ''G<sup>m</sup>'' is closed under countable unions.
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| Note that for each Borel set ''B'', there is some countable ordinal α<sub>''B''</sub> such that ''B'' can be obtained by iterating the operation over α<sub>''B''</sub>. However, as ''B'' varies over all Borel sets, α<sub>''B''</sub> will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω<sub>1</sub>, the first uncountable ordinal.
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| === Example ===
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| 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 real random variable defined on a [[probability space]], its [[probability distribution]] is by definition also a measure on the Borel algebra.
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| The Borel algebra on the reals is the smallest σ-algebra on '''R''' which contains all the [[interval (mathematics)|intervals]].
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| In the construction by transfinite induction, it can be shown that, in each step, the [[cardinality|number]] of sets is, at most, the [[power of the continuum]]. So, the total number of Borel sets is less than or equal to
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| :<math>\aleph_1 \times 2 ^ {\aleph_0}\, = 2^{\aleph_0}.\,</math>
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| ==Standard Borel spaces and Kuratowski theorems==<!-- This section is linked from [[Kazimierz Kuratowski]] -->
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| [[George_Mackey | Mackey]] writes that a '''Borel space''' 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. Annalen.]]}}</ref> However, more modern terminology is to call such spaces ''[[measurable space]]s''. The reason for this distinction is that the Borel σ-algebra is 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 which are ''not'' Borel spaces in this more restricted topological sense.<ref>[http://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topology Jochen Wengenroth (mathoverflow.net/users/21051), Is every sigma-algebra the Borel algebra of a topology?, http://mathoverflow.net/questions/87888 (version: 2012-02-09)]</ref>
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| 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'', <math>f^{-1}(B)</math> is a measurable set in ''X''.
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| '''Theorem'''. Let ''X'' be a [[Polish space]], that is, a topological space such that there is a [[Metric (mathematics)|metric]] ''d'' on ''X'' which defines the topology of ''X'' and which makes ''X'' a complete [[separable space|separable]] metric space. Then ''X'' as a Borel space is [[isomorphic]] to one of
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| (1) '''R''', (2) '''Z''' or (3) a finite space. (This result is reminiscent of [[Maharam's theorem]].)
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| Considered as Borel spaces, the real line '''R''' and the union of '''R''' with a countable set are isomorphic.
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| A '''standard Borel space''' is the Borel space associated to a [[Polish space]].
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| Any standard Borel space is defined (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=0-387-98412-7}}</ref> and any uncountable standard Borel space has the cardinality of the continuum.
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| For subsets of Polish spaces, Borel sets can be characterized as those sets which 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]].
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| Every [[probability measure]] on a standard Borel space turns it into a [[standard probability space]].
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| == Non-Borel sets ==
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| {{anchor|counterexample}}
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| An example of a subset of the reals which is non-Borel, due to [[Nikolai Luzin|Lusin]]<ref>{{Citation | last=Lusin | first=Nicolas | year=1927 | title=Sur les ensembles analytiques | journal=Fundamenta Mathematicae | publisher=Institute of mathematics, Polish academy of sciences | volume=10 | pages=1–95 | }}.</ref> (see Sect. 62, pages 76–78), is described below. In contrast, an example of a [[non-measurable set]] cannot be exhibited, though its existence can be proved.
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| Every [[irrational number]] has a unique representation by a [[continued fraction]]
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| :<math>x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} </math>
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| 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. In fact, it is [[analytic set|analytic]], and complete in the class of analytic sets. For more details see [[descriptive set theory]] and the book by [[Alexander S. Kechris|Kechris]], especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.
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| Another non-Borel set is an inverse image <math>f^{-1}[0]</math> of an [[Parity function#Infinite parity function|infinite parity function]] <math>f\colon \{0, 1\}^{\omega} \to \{0, 1\}</math>. However, this is a proof of existence (via the choice axiom), not an explicit example.
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| ==Alternative non-equivalent definitions==
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| According to [[Halmos]] {{harv|Halmos|1950|loc=page 219}}, a subset of a locally compact Hausdorff topological space is called a ''Borel set'' if it belongs to the smallest σ–ring containing all compact sets.
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| ==See also==
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| * [[Baire set]]
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| * [[Cylindrical σ-algebra]]
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| * [[Polish space]]
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| * [[Descriptive set theory]]
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| * [[Borel hierarchy]]
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| == References ==
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| An excellent exposition of the machinery of ''Polish topology'' is given in Chapter 3 of the following reference:
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| * [[William Arveson]], ''An Invitation to C*-algebras'', Springer-Verlag, 1981
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| * [[Richard Dudley]], '' Real Analysis and Probability''. Wadsworth, Brooks and Cole, 1989
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| *{{cite book
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| |first=Paul R.
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| |last=Halmos
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| |author-link=Paul Halmos
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| |title=Measure theory
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| |publisher=D. van Nostrand Co
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| |year=1950}} See especially Sect. 51 "Borel sets and Baire sets".
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| * [[Halsey Royden]], ''Real Analysis'', Prentice Hall, 1988
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| * [[Alexander S. Kechris]], ''Classical Descriptive Set Theory'', Springer-Verlag, 1995 (Graduate texts in Math., vol. 156)
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| {{reflist}}
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| ==External links==
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| * {{springer|title=Borel set|id=p/b017120}}
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| * [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] that have been formally proved about it.
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| * {{MathWorld |title=Borel Set |id=BorelSet}}
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| [[Category:Topology]]
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| [[Category:Descriptive set theory]]
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| [[el:Σ-άλγεβρα#σ-άλγεβρα Borel]]
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