Difference between revisions of "Borel function"
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− | {{MSC| | + | {{MSC|28A20}} |
[[Category:Classical measure theory]] | [[Category:Classical measure theory]] | ||
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$\newcommand{\abs}[1]{\left|#1\right|}$ | $\newcommand{\abs}[1]{\left|#1\right|}$ | ||
+ | ===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 $A$ | 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). | + | (recall that the [[Algebra of sets|$\sigma$-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 $a\in\mathbb R$. | + | 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]]. |
− | + | ===Properties=== | |
− | 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) | + | 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}}. |
+ | ====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 | 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. | + | 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 | + | (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}}). |
+ | |||
+ | 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). | ||
+ | |||
+ | ===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|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. | ||
+ | |||
+ | 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 {{Cite|Hal}}, {{Cite|Ko}}. | Borel functions have found use not only in set theory and function theory but also in probability theory, see {{Cite|Hal}}, {{Cite|Ko}}. | ||
− | + | ===References=== | |
{| | {| | ||
|- | |- | ||
− | |valign="top"|{{Ref|Bor}}|| E. Borel, "Leçons sur | + | |valign="top"|{{Ref|Bor}}|| E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1898) {{ZBL|29.0336.01}} |
|- | |- | ||
|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}} | |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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|valign="top"|{{Ref|Hal}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} | |valign="top"|{{Ref|Hal}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}} | ||
|- | |- | ||
− | |valign="top"|{{Ref|Hau}}|| F. Hausdorff, "Set theory", Chelsea (1978) | + | |valign="top"|{{Ref|Hau}}|| F. Hausdorff, "Set theory", Chelsea (1978){{MR|0141601}} {{ZBL|0488.04001}} |
|- | |- | ||
|valign="top"|{{Ref|He}}|| E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) {{MR|0188387}} {{ZBL|0137.03202}} | |valign="top"|{{Ref|He}}|| E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) {{MR|0188387}} {{ZBL|0137.03202}} | ||
|- | |- | ||
− | |valign="top"|{{Ref|Ko}}|| A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint ( | + | |valign="top"|{{Ref|Ko}}|| A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1956) {{MR|0079843}} {{ZBL|0074.12202}} |
|- | |- | ||
− | |valign="top"|{{Ref|Ku}}|| K. Kuratowski, "Topology" , '''1–2''' , Acad. Press (1966–1968) | + | |valign="top"|{{Ref|Ku}}|| K. Kuratowski, "Topology" , '''1–2''' , Acad. Press (1966–1968) {{MR|0217751}} {{MR|0259836}} {{ZBL|0158.40802}} |
|- | |- | ||
− | |valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis" , Macmillan (1968) | + | |valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis" , Macmillan (1968) {{MR|0151555}} {{ZBL|0197.03501}} |
|- | |- | ||
|} | |} |
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 |
Borel function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_function&oldid=27322