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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|28A20}}
  
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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{{TEX|done}}
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$\newcommand{\abs}[1]{\left|#1\right|}$
  
====References====
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===Definition===
<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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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 [[Algebra of sets|$\sigma$-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 $f^{-1} (]a, \infty[)$ is Borel for any $a\in\mathbb R$ (see for instance Exercise 26 of Chapter 3 in {{Cite|Ro}}). Consider two topological spaces $X$ and $Y$ and the corresponding Borel $\sigma$-algebras $\mathcal{B} (X)$ and $\mathcal{B} (Y)$. The Borel measurability of the function $f:X\to Y$ is then equivalent to the measurability of the map $f$ seen as map between the [[Measurable space|measurable spaces]] $(X, \mathcal{B} (X))$ and $(Y, \mathcal{B} (Y))$, see also [[Measurable mapping]].
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===Properties===
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As it is always the case for measurable real functions on any measurable space $X$, 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), see Sections 18, 19 and 20 of {{Cite|Hal}}.
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====Closure under composition====
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Moreover the compositions of Borel functions of one real variable are Borel functions. Indeed, if $X, Y$ and $Z$ are topological spaces and $f:X\to Y$, $g:Y\to Z$ Borel functions, then $g\circ f$ is a Borel function, as it follows trivially from the definition above.
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===Comparison with Lebesgue measurable 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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(the Lebesgue measurability of $f\circ g$ holds if we assume in addition that $f$ is continuous, whereas it fails if we assume the continuity of $g$ but only the Lebesgue measurability of $f$, see for instance Exercise 28d in Chapter 3 of {{Cite|Ro}}).
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All Borel real valued functions on the euclidean space are Lebesgue-measurable, but the converse is false. However, it follows easily from [[Luzin theorem|Lusin's Theorem]] that for any Lebesgue-measurable function $f$ there exists a Borel function $g$ which coincides with $f$ almost everywhere (with respect to the Lebesgue measure).
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===Comparison with Baire functions===
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Borel functions $f:\mathbb R\to \mathbb R$ are sometimes called Baire functions, since in this case 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}}). However, in the context of a general topological space $X$ the space of Baire functions is the smallest family of real-valued functions which is close under the operation of taking limits of pointwise converging sequences and which contains the continuous functions (see Section 51 of {{Cite|Hal}}). In a general topological space the class of Baire functions might be strictly smaller then the class of Borel functions.
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Borel real-valued functions of one real variable can be classified by the order of the Borel sets; the classes thus obtained are identical with the Baire classes.
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===Comments===
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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}}.
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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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|-
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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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|-
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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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|-
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan (1968) {{MR|0151555}} {{ZBL|0197.03501}}
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|-
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|}

Latest revision as of 17:50, 5 May 2024

2020 Mathematics Subject Classification: Primary: 28A20 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

Definition

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 $f^{-1} (]a, \infty[)$ is Borel for any $a\in\mathbb R$ (see for instance Exercise 26 of Chapter 3 in [Ro]). Consider two topological spaces $X$ and $Y$ and the corresponding Borel $\sigma$-algebras $\mathcal{B} (X)$ and $\mathcal{B} (Y)$. The Borel measurability of the function $f:X\to Y$ is then equivalent to the measurability of the map $f$ seen as map between the measurable spaces $(X, \mathcal{B} (X))$ and $(Y, \mathcal{B} (Y))$, see also Measurable mapping.

Properties

As it is always the case for measurable real functions on any measurable space $X$, 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), see Sections 18, 19 and 20 of [Hal].

Closure under composition

Moreover the compositions of Borel functions of one real variable are Borel functions. Indeed, if $X, Y$ and $Z$ are topological spaces and $f:X\to Y$, $g:Y\to Z$ Borel functions, then $g\circ f$ is a Borel function, as it follows trivially from the definition above.

Comparison with Lebesgue measurable 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 (the Lebesgue measurability of $f\circ g$ holds if we assume in addition that $f$ is continuous, whereas it fails if we assume the continuity of $g$ but only the Lebesgue measurability of $f$, see for instance Exercise 28d in Chapter 3 of [Ro]).

All Borel real valued functions on the euclidean space are Lebesgue-measurable, but the converse is false. However, it follows easily from Lusin's Theorem that for any Lebesgue-measurable function $f$ there exists a Borel function $g$ which coincides with $f$ almost everywhere (with respect to the Lebesgue measure).

Comparison with Baire functions

Borel functions $f:\mathbb R\to \mathbb R$ are sometimes called Baire functions, since in this case the set of all Borel functions is identical with the set of functions belonging to the Baire classes (Lebesgue's theorem, [Hau]). However, in the context of a general topological space $X$ the space of Baire functions is the smallest family of real-valued functions which is close under the operation of taking limits of pointwise converging sequences and which contains the continuous functions (see Section 51 of [Hal]). In a general topological space the class of Baire functions might be strictly smaller then the class of Borel functions.

Borel real-valued functions of one real variable can be classified by the order of the Borel sets; the classes thus obtained are identical with the Baire classes.

Comments

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 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 function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_function&oldid=16471
This article was adapted from an original article by V.A. Skvortsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article