Borel function

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 $f^{-1} (]a, \infty[)$ 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)