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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b0170802.png" />-function''
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{{MSC|28A33}}
  
A function for which all subsets of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b0170803.png" /> in its domain of definition are Borel sets (cf. [[Borel set|Borel set]]). Such functions are also known as Borel-measurable functions, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b0170805.png" />-measurable functions. The operations of addition, multiplication and limit transition — as in the general case of measurable functions — do not take one outside the class of Borel functions, but, unlike in the general case, the superposition of two Borel functions does also not lead outside the class of Borel functions. Moreover, [[#References|[1]]], if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b0170806.png" /> is a measurable function on an arbitrary space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b0170807.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b0170808.png" /> is a Borel function on the space of real numbers, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b0170809.png" /> is measurable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b01708010.png" />. All Borel functions are Lebesgue-measurable (cf. [[Measurable function|Measurable function]]). The converse proposition is not true. However, for any Lebesgue-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b01708011.png" /> there exists a Borel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b01708012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b01708013.png" /> almost-everywhere [[#References|[1]]]. Borel functions are sometimes called Baire functions, since the set of all Borel functions is identical with the set of functions belonging to the [[Baire classes|Baire classes]] (Lebesgue's theorem, [[#References|[2]]]). Borel functions can be classified by the order of the Borel sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b01708014.png" />; the classes thus obtained are identical with the Baire classes.
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[[Category:Classical measure theory]]
  
The concept of a Borel function has been generalized to include functions with values in an arbitrary metric space [[#References|[3]]]. One then also speaks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017080/b01708016.png" />-measurable mappings. Borel functions have found use not only in set theory and function theory but also in probability theory [[#References|[1]]], [[#References|[4]]].
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$\newcommand{\abs}[1]{\left|#1\right|}$
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A map $f:X\to Y$ between two topological spaces is called Borel (or Borel measurable) if $f^{-1} (A)$ is a [[Borel set]] for any open set $A$
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(recall that the $\sigma$-algebra of Borel sets of $X$ is the smallest $\sigma$ algebra containing the open sets).
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When the target $Y$ is the real line, it suffices to assume that $f^{-1} (]0, \infty[)$ is Borel for any $a\in\mathbb R$.
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The space of Borel real-valued functions over a given topological space is a vector space and it is closed under the operation of taking
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pointwise limits of sequences (i.e. if a sequence of Borel functions $f_n$ converges everywhere to a function $f$, then $f$ is also a Borel function). Moreover the compositions of Borel functions are Borel functions.
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The latter property is false for real-valued Lebesgue measurable functions on $\mathbb R$ (cf. [[Measurable function]]): there are pairs of Lebesgue
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measurable functions $f,g: \mathbb R\to\mathbb R$ such that $f\circ g$ is not Lebesgue measurable.  
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All Borel real valued functions on the real line are Lebesgue-measurable, but the converse is false. However, for any Lebesgue-measurable function $f$ there exists a Borel function $g$ which coincides with $f$ almost everywhere (with respect to the Lebesgue measure). Borel functions are sometimes called Baire functions, since the set of all Borel functions is identical with the set of functions belonging to the [[Baire classes|Baire classes]] (Lebesgue's theorem, {{Cite|Hau}}. Borel functions can be classified by the order of the Borel sets; the classes thus obtained are identical with the Baire classes).
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Borel functions have found use not only in set theory and function theory but also in probability theory, see {{Cite|Hal}}, {{Cite|Ko}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top"K. Kuratowski,  "Topology" , '''1–2''' , Acad. Press  (1966–1968) (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Kolmogorov,  "Foundations of the theory of probability" , Chelsea, reprint  (1950) (Translated from German)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Bor}}|| E. Borel,  "Leçons sur les fonctions discontinues" , Gauthier-Villars  (1898)
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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)
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|-
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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  (1950)
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|-
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|valign="top"|{{Ref|Ku}}|| K. Kuratowski,  "Topology" , '''1–2''' , Acad. Press  (1966–1968)  
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|-
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan (1968)
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|-
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|}

Revision as of 12:57, 1 August 2012

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

A map $f:X\to Y$ between two topological spaces is called Borel (or Borel measurable) if $f^{-1} (A)$ is a Borel set for any open set $A$ (recall that the $\sigma$-algebra of Borel sets of $X$ is the smallest $\sigma$ algebra containing the open sets). When the target $Y$ is the real line, it suffices to assume that $f^{-1} (]0, \infty[)$ is Borel for any $a\in\mathbb R$.

The space of Borel real-valued functions over a given topological space is a vector space and it is closed under the operation of taking pointwise limits of sequences (i.e. if a sequence of Borel functions $f_n$ converges everywhere to a function $f$, then $f$ is also a Borel function). Moreover the compositions of Borel functions are Borel functions.

The latter property is false for real-valued Lebesgue measurable functions on $\mathbb R$ (cf. Measurable function): there are pairs of Lebesgue measurable functions $f,g: \mathbb R\to\mathbb R$ such that $f\circ g$ is not Lebesgue measurable. All Borel real valued functions on the real line are Lebesgue-measurable, but the converse is false. However, for any Lebesgue-measurable function $f$ there exists a Borel function $g$ which coincides with $f$ almost everywhere (with respect to the Lebesgue measure). Borel functions are sometimes called Baire functions, since the set of all Borel functions is identical with the set of functions belonging to the Baire classes (Lebesgue's theorem, [Hau]. Borel functions can be classified by the order of the Borel sets; the classes thus obtained are identical with the Baire classes).

Borel functions have found use not only in set theory and function theory but also in probability theory, see [Hal], [Ko].

References

[Bor] E. Borel, "Leçons sur les fonctions discontinues" , Gauthier-Villars (1898)
[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)
[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 (1950)
[Ku] K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968)
[Ro] H.L. Royden, "Real analysis" , Macmillan (1968)
How to Cite This Entry:
Borel function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_function&oldid=27319
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article