Borel function

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