nLab Borel subset

Idea

Borel sets are certain subsets of a topological space. They form the Borel $\sigma$-algebra of the space, and they play an important role in measure theory.

Definition

Let $X$ be a topological space. Then there is a $\sigma$-algebra $\mathcal{B}$ on $X$ generated by the open subsets of $X$. Elements of $\mathcal{B}$ are called the Borel sets (or Borel subsets, or Borel-measurable sets, etc) of $X$, and $\mathcal{B}$ itself is called the Borel $\sigma$-algebra on $X$.

The Borel hierarchy

The preceding abstract definition can be made concrete (and predicative, at least over $\omega_1$, although most predicative mathematicians don't accept $\omega_1$) as follows:

• Start with the collection $\Sigma_0$ of open sets, and let $\Pi_0$ be the collection of the complements of the members of $\Sigma_0$ (the closed sets).
• Let $\Sigma_1$ be the collection of countably infinitary unions of sets in $\Pi_0$ (the $F_\sigma$-sets?), and let $\Pi_1$ be the collection of their complements (the $G_\delta$-sets, the countably infinitary intersections of sets in $\Sigma_0$); even $\Sigma_1 \cup \Pi_1$ is not in general a $\sigma$-algebra.
• Let $\Sigma_2$ be the collection of countably infinitary unions of sets in $\Pi_1$ (the $G_{\delta\sigma}$-sets), and let $\Pi_2$ be the collection of their complements (the $F_{\sigma\delta}$-sets, the countably infinitary intersections of sets in $\Sigma_1$).
• Continue, defining $\Sigma_n$ for all natural numbers $n$.
• Let $\Sigma_\omega$ be the union of the various $\Sigma_n$; although this is closed under complement, it is still not in general a $\sigma$-algebra.
• Continue, defining $\Sigma_\alpha$ for all countable ordinal numbers $\alpha$.
• Let $\Sigma_{\omega_1}$ be the union of the various $\Sigma_\alpha$; this is finally a $\sigma$-algebra.

So we need an uncountable number of steps, not just two.

This is only the beginning of descriptive set theory; our $\Sigma_\alpha$ are their $\Sigma^0_\alpha$ —except that for some reason they start with $\Sigma^0_1$ instead of $\Sigma^0_0$—, and the subject continues to higher values of the superscript.

(To remember the other symbols, you need to know two languages: French and German. The ‘$F$’ comes from French ‘fermé’ for ‘closed’, while ‘$G$’ is simply the next letter; the ‘$\sigma$’ and ‘$\delta$’ are from German ‘Summe’ for ‘union’ and ‘Durchschnitt’ for ‘intersection’ and are commonly used for countable such.)

Note that countable choice is essential here and elsewhere in measure theory, to show that a countable union of a countable union is a countable union. But the full axiom of choice is not; in fact, much of descriptive set theory (although this is irrelevant to the small portion above) works better with the axiom of determinacy instead.

The Borel $\delta$-ring and $\sigma$-ring

Sometimes one does measure theory with $\sigma$-rings or even $\delta$-rings, which are more general than $\sigma$-algebras. However, even the $\delta$-ring generated by a topology is in fact a $\sigma$-algebra.

Nevertheless, on a locally compact Hausdorff space, we may define the Borel $\delta$-ring and the Borel $\sigma$-ring to be the $\delta$-ring or $\sigma$-ring generated by the compact subsets of $X$. When $X$ is compact (such as the unit interval), then these both agree with the Borel $\sigma$-algebra; when $X$ is $\sigma$-compact (a countable union of compact subsets, such as the real line), then the Borel $\sigma$-ring still agrees with the Borel $\sigma$-algebra.

Related concepts

• projective set

• Borel measure