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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b0171202.png" />-set''
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{{MSC|28A05}}
  
A set which may be obtained as the result of not more than a countable number of operations of union and intersection of closed and open sets in a topological space. More exactly, a Borel set is an element of the smallest countably-additive class of sets containing the closed sets, and which is closed with respect to complementation. Borel sets are also called Borel-measurable sets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b0171204.png" />-measurable sets. Open and closed sets are said to be Borel sets of order zero. Borel sets of order one are sets of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b0171205.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b0171206.png" /> which are, respectively, countable sums of closed sets and countable intersections of open sets. Borel sets of the second order are sets of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b0171207.png" /> (the intersection of a countable number of sets of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b0171208.png" />) and sets of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b0171209.png" /> (the union of a countable number of sets of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712010.png" />). Borel sets of an arbitrary finite order are defined in a similar manner by induction. With the aid of transfinite numbers of the second class (cf. [[Transfinite number|Transfinite number]]) this classification may be exhaustively extended to all Borel sets. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712011.png" /> be an arbitrary transfinite number of the second class; the Borel sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712012.png" /> will include any Borel set of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712013.png" /> that is not a Borel set of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712014.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712015.png" />. Whether the classes of Borel sets are empty or not will depend on the basic space under consideration. In Euclidean, Hilbert and Baire spaces there exist Borel sets of all classes.
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[[Category:Classical measure theory]]
  
Borel sets are a special case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712016.png" />-sets. For an [[A-set|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712017.png" />-set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712018.png" /> to be a Borel set it is necessary and sufficient that the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712019.png" /> also is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017120/b01712020.png" />-set (Suslin's criterion). In spaces with a [[Lebesgue measure|Lebesgue measure]] all Borel sets are Lebesgue measurable. The converse proposition is not true. All separable spaces having the cardinality of the continuum contain sets that are not Borel sets.
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{{TEX|done}}
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$\newcommand{\abs}[1]{\left|#1\right|}$
  
Borel sets were introduced by E. Borel [[#References|[1]]]; they play an important role in the study of Borel functions (cf. [[Borel function|Borel function]]).
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Borel sets were introduced by E. Borel {{Cite|Bor}}; they play an important role in the study of Borel functions (cf. [[Borel function|Borel function]]).
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They are also called Borel-measurable sets.
  
In a more general sense a Borel set is a set in an arbitrary [[Borel system of sets|Borel system of sets]] generated by some system of sets. The Borel sets in a topological space are generated by the system of closed subsets of this space.
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====Definition====
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Given a topological space $X$, the Borel [[Algebra of sets|σ-algebra]] of $X$ is the $\sigma$-algebra generated by the open sets (i.e. the smallest $\sigma$-algebra of
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subsets of $X$ containing the open sets of $X$), cp. with Section 7 of Chapter 2 in {{Cite|Ro}}. When $X$ is a locally compact Hausdorff space some authors define the Borel sets as the smallest [[Ring of sets|$\sigma$-ring]] containing the compact sets, see {{Cite|Hal}}. Under suitable assumptions, for instance on a separable locally compact metric space, the two notions coincide.  
  
====References====
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The primary example are the Borel sets on the real line (or more generally of the euclidean space), which
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Borel,  "Leçons sur les fonctions discontinues" , Gauthier-Villars  (1898)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''1–2''' , Acad. Press  (1966–1968) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956) (Translated from Russian)</TD></TR></table>
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correspond to choosing as $X$ the space of real numbers $\mathbb R$ (resp. $\mathbb R^n$) with the usual topology. Borel sets of the real line
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(or more generally of a euclidean space) are [[Measurable set|Lebesgue measurable]]. Conversely every Lebesgue measurable subset of the euclidean space coincides with a Borel set up to a set of measure zero. More precisely (cp. with Proposition 15 of Chapter 3 in {{Cite|Ro}}):
  
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'''Theorem'''
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For every Lebesgue measurable set $E\subset \mathbb R$ there are
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* a $G_\delta$ set $U\supset E$ with $\lambda (U\setminus E) = 0$;
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* an $F_\sigma$ set $F\subset E$ with $\lambda (E\setminus F) = 0$.
  
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====Order of a Borel set====
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Obviously open and closed sets are Borel and they are sometimes
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called Borel sets of order zero. Other special classes of Borel sets which are often used are the [[G-delta|$G_\delta$]] sets,
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i.e. sets which are countable intersections of open sets, and the [[F-sigma|$F_\sigma$]], i.e. countable unions of
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closed sets . The elements of these classes which are neither open nor closed are Borel sets of order one.
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Analogously one can define the $G_{\delta\sigma}$ and the $F_{\sigma\delta}$ sets and Borel sets of order two (cp. with Section 7 of {{Cite|Ro}}. Borel sets of an arbitrary finite order are defined in a similar manner by induction.
  
====Comments====
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====Transfinite construction====
For notational questions, see [[Borel set of ambiguous class|Borel set of ambiguous class]].
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Using [[Transfinite number|transfinite numbers]] we can define Borel sets of order $\alpha$ for any countable ordinal $\alpha$: if $\alpha$ is a countable ordinal,
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the Borel sets of order $\alpha$ are those sets which can be obtained  as countable unions or countable intersections of Borel sets of order strictly smaller than $\alpha$, but which are not Borel sets of any order $\alpha'<\alpha$. On the real line (more in general in any [[Hilbert space]] and in any [[Baire space]]) there exist Borel sets of all order.
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The Borel $\sigma$-algebra is the union of all Borel sets so constructed (i.e. of order $\alpha$ for all countable ordinal $\alpha$), cp. with the transfinite construction of the $\sigma$-algebra generated by a family of set $\mathcal{A}$ in [[Algebra of sets]] (see also Exercise 9 of Section 5 in {{Cite|Hal}}).
  
====References====
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The procedure above can be used to show that, for instance, the Borel $\sigma$-algebra of the real line has the cardinality of continuum. In particular, since the Lebesgue measurable subsets of $\mathbb R$ have larger cardinality, there are Lebesgue measurable sets which are not Borel. For the same reason all separable spaces having the cardinality of the continuum contain sets that are not Borel sets.
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Kuratowski,  "Introduction to set theory and topology" , Pergamon (1961)  (Translated from Polish)</TD></TR></table>
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====Relation to analytic sets====
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Borel sets are a special case of [[A-set|analytic sets]]. [[Suslin theorem|Suslin's criterion]] states that an analytic set is Borel if and only if its complement is also an analytic set.
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===Comments===
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In a more general sense a Borel set is a set in an arbitrary [[Borel system of sets|Borel system of sets]] generated by some system of sets.
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For notational issues, see [[Borel set of ambiguous class|Borel set of ambiguous class]].
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===References===
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{|
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|-
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|valign="top"|{{Ref|Bor}}|| E. Borel,  "Leçons sur la théorie des fonctions" , Gauthier-Villars  (1898) {{ZBL|29.0336.01}}
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|-
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|valign="top"|{{Ref|Bou}}||      N. Bourbaki, "Elements of mathematics. Integration" ,  Addison-Wesley    (1975) pp. Chapt.6;7;8 (Translated from French)  {{MR|0583191}}    {{ZBL|1116.28002}} {{ZBL|1106.46005}}  {{ZBL|1106.46006}}    {{ZBL|1182.28002}} {{ZBL|1182.28001}}  {{ZBL|1095.28002}}    {{ZBL|1095.28001}} {{ZBL|0156.06001}}
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|-
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|valign="top"|{{Ref|Hal}}|| P.R. Halmos,  "Measure theory" , v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|-
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|valign="top"|{{Ref|Hau}}|| F. Hausdorff, "Set theory", Chelsea (1978){{MR|0141601}} {{ZBL|0488.04001}}
 +
|-
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|valign="top"|{{Ref|He}}||    E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" ,  Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}}
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|-
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|valign="top"|{{Ref|Ko}}||  A.N. Kolmogorov,  "Foundations of the theory of probability" ,  Chelsea, reprint  (1956) {{MR|0079843}} {{ZBL|0074.12202}} 
 +
|-
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|valign="top"|{{Ref|Ku}}|| K. Kuratowski,  "Topology" , '''1–2''' , Acad. Press (1966–1968) {{MR|0217751}} {{MR|0259836}} {{ZBL|0158.40802}}
 +
|-
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan (1968) {{MR|0151555}} {{ZBL|0197.03501}}
 +
|-
 +
|}

Latest revision as of 19:29, 23 May 2024

2020 Mathematics Subject Classification: Primary: 28A05 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

Borel sets were introduced by E. Borel [Bor]; they play an important role in the study of Borel functions (cf. Borel function). They are also called Borel-measurable sets.

Definition

Given a topological space $X$, the Borel σ-algebra of $X$ is the $\sigma$-algebra generated by the open sets (i.e. the smallest $\sigma$-algebra of subsets of $X$ containing the open sets of $X$), cp. with Section 7 of Chapter 2 in [Ro]. When $X$ is a locally compact Hausdorff space some authors define the Borel sets as the smallest $\sigma$-ring containing the compact sets, see [Hal]. Under suitable assumptions, for instance on a separable locally compact metric space, the two notions coincide.

The primary example are the Borel sets on the real line (or more generally of the euclidean space), which correspond to choosing as $X$ the space of real numbers $\mathbb R$ (resp. $\mathbb R^n$) with the usual topology. Borel sets of the real line (or more generally of a euclidean space) are Lebesgue measurable. Conversely every Lebesgue measurable subset of the euclidean space coincides with a Borel set up to a set of measure zero. More precisely (cp. with Proposition 15 of Chapter 3 in [Ro]):

Theorem For every Lebesgue measurable set $E\subset \mathbb R$ there are

  • a $G_\delta$ set $U\supset E$ with $\lambda (U\setminus E) = 0$;
  • an $F_\sigma$ set $F\subset E$ with $\lambda (E\setminus F) = 0$.

Order of a Borel set

Obviously open and closed sets are Borel and they are sometimes called Borel sets of order zero. Other special classes of Borel sets which are often used are the $G_\delta$ sets, i.e. sets which are countable intersections of open sets, and the $F_\sigma$, i.e. countable unions of closed sets . The elements of these classes which are neither open nor closed are Borel sets of order one. Analogously one can define the $G_{\delta\sigma}$ and the $F_{\sigma\delta}$ sets and Borel sets of order two (cp. with Section 7 of [Ro]. Borel sets of an arbitrary finite order are defined in a similar manner by induction.

Transfinite construction

Using transfinite numbers we can define Borel sets of order $\alpha$ for any countable ordinal $\alpha$: if $\alpha$ is a countable ordinal, the Borel sets of order $\alpha$ are those sets which can be obtained as countable unions or countable intersections of Borel sets of order strictly smaller than $\alpha$, but which are not Borel sets of any order $\alpha'<\alpha$. On the real line (more in general in any Hilbert space and in any Baire space) there exist Borel sets of all order. The Borel $\sigma$-algebra is the union of all Borel sets so constructed (i.e. of order $\alpha$ for all countable ordinal $\alpha$), cp. with the transfinite construction of the $\sigma$-algebra generated by a family of set $\mathcal{A}$ in Algebra of sets (see also Exercise 9 of Section 5 in [Hal]).

The procedure above can be used to show that, for instance, the Borel $\sigma$-algebra of the real line has the cardinality of continuum. In particular, since the Lebesgue measurable subsets of $\mathbb R$ have larger cardinality, there are Lebesgue measurable sets which are not Borel. For the same reason all separable spaces having the cardinality of the continuum contain sets that are not Borel sets.

Relation to analytic sets

Borel sets are a special case of analytic sets. Suslin's criterion states that an analytic set is Borel if and only if its complement is also an analytic set.

Comments

In a more general sense a Borel set is a set in an arbitrary Borel system of sets generated by some system of sets. For notational issues, see Borel set of ambiguous class.

References

[Bor] E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1898) Zbl 29.0336.01
[Bou] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[Hal] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Hau] F. Hausdorff, "Set theory", Chelsea (1978)MR0141601 Zbl 0488.04001
[He] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[Ko] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1956) MR0079843 Zbl 0074.12202
[Ku] K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) MR0217751 MR0259836 Zbl 0158.40802
[Ro] H.L. Royden, "Real analysis" , Macmillan (1968) MR0151555 Zbl 0197.03501
How to Cite This Entry:
Borel set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_set&oldid=15948
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article