Borel set
2020 Mathematics Subject Classification: Primary: 28A05 [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). They are also called Borel-measurable sets.
Definition
Given a topological space $X$, the Borel σ-algebra of $X$ is the $\sigma$-algebra generated by the open sets (i.e. the smallest $\sigma$-algebra of subsets of $X$ containing the open sets of $X$), cp. with Section 7 of Chapter 2 in [Ro]. When $X$ is a locally compact Hausdorff space some authors define the Borel sets as the smallest $\sigma$-ring containing the compact sets, see [Hal]. Under suitable assumptions, for instance on a separable locally compact metric space, the two notions coincide.
The primary example are the Borel sets on the real line (or more generally of the euclidean space), which correspond to choosing as $X$ the space of real numbers $\mathbb R$ (resp. $\mathbb R^n$) with the usual topology. Borel sets of the real line (or more generally of a euclidean space) are Lebesgue measurable. Conversely every Lebesgue measurable subset of the euclidean space coincides with a Borel set up to a set of measure zero. More precisely (cp. with Proposition 15 of Chapter 3 in [Ro]):
Theorem For every Lebesgue measurable set $E\subset \mathbb R$ there are
- a $G_\delta$ set $U\supset E$ with $\lambda (U\setminus E) = 0$;
- an $F_\sigma$ set $F\subset E$ with $\lambda (E\setminus F) = 0$.
Order of a Borel set
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 (cp. with Section 7 of [Ro]. Borel sets of an arbitrary finite order are defined in a similar manner by induction.
Transfinite construction
Using transfinite numbers we can define Borel sets of order $\alpha$ for any countable ordinal $\alpha$: 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 $\alpha'<\alpha$. On the real line (more in general in any Hilbert space and in any Baire space) there exist Borel sets of all order. The Borel $\sigma$-algebra is the union of all Borel sets so constructed (i.e. of order $\alpha$ for all countable ordinal $\alpha$), cp. with the transfinite construction of the $\sigma$-algebra generated by a family of set $\mathcal{A}$ in Algebra of sets (see also Exercise 9 of Section 5 in [Hal]).
The procedure above 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 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 |
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