Borel function

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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  ). 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  .

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  ).

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,  ). 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  ). 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  ,  .

References

	E. Borel, "Leçons sur la théorie des fonctions" , Gauthier-Villars (1898) Template:ZBL
	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL
	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Template:ZBL
	F. Hausdorff, "Set theory", Chelsea (1978)MR0141601 Template:ZBL
	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Template:ZBL
	A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1956) MR0079843 Template:ZBL
	K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) MR0217751 MR0259836 Template:ZBL
	H.L. Royden, "Real analysis" , Macmillan (1968) MR0151555 Template:ZBL