Borel set

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

Borel sets were introduced by E. Borel [Bor]; they play an important role in the study of Borel functions (cf. Borel function).

Definition

Given a topological space $X$, the Borel σ-algebra of $X$ is the smallest $\sigma$-algebra of subsets of $X$ containing the open sets of $X$. The primary example are the Borel sets on the real line, which correspond to choosing as $X$ the space of real numbers $\mathbb R$ with the usual topology. Borel sets of the real line (or more generally of a euclidean space) are Lebesgue measurable.

Order and transfinite construction

Borel sets are also called Borel-measurable sets. Obviously open and closed sets are Borel and they are sometimes called Borel sets of order zero. Other special classes of Borel sets which are often used are the $G_\delta$ sets, i.e. sets which are countable intersections of open sets, and the $F_\sigma$, i.e. countable unions of closed sets. The elements of these classes which are neither open nor closed are Borel sets of order one. Analogously one can define the $G_{\delta\sigma}$ and the $F_{\sigma\delta}$ sets and Borel sets of order two. Borel sets of an arbitrary finite order are defined in a similar manner by induction. With the aid of the transfinite numbers up to the first uncountable ordinal this classification may be exhaustively extended to all Borel sets: if $\alpha$ is a countable ordinal, the Borel sets of order $\alpha$ are those sets which can be obtained as countable unions or countable intersections of Borel sets of order strictly smaller than $\alpha$, but which are not Borel sets of any order $\alpha'<\alpha$. Whether a class of Borel sets of a certain given order is empty or not will depend on the basic space under consideration. In Euclidean, Hilbert and Baire spaces there exist Borel sets of all classes. This procedure can be used to show that, for instance, the Borel $\sigma$-algebra of the real line has the cardinality of continuum. In particular, since the Lebesgue measurable subsets of $\mathbb R$ have larger cardinality, there are Lebesgue measurable sets which are not Borel. For the same reason all separable spaces having the cardinality of the continuum contain sets that are not Borel sets.

Relation to analytic sets

Borel sets are a special case of analytic sets. Suslin's criterion states that an analytic set is Borel if and only if its complement is also an analytic set.

Comments

In a more general sense a Borel set is a set in an arbitrary Borel system of sets generated by some system of sets. For notational issues, see Borel set of ambiguous class.

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
[Ku]	K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968)
[Ro]	H.L. Royden, "Real analysis" , Macmillan (1968)