nLab
Borel subset

Contents

Idea

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

Definition

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

The Borel hierarchy

The preceding abstract definition can be made concrete (and predicative, at least over ω 1\omega_1, although most predicative mathematicians don't accept ω 1\omega_1) as follows:

  • Start with the collection Σ 0\Sigma_0 of open sets, and let Π 0\Pi_0 be the collection of the complements of the members of Σ 0\Sigma_0 (the closed sets).
  • Let Σ 1\Sigma_1 be the collection of countably infinitary unions of sets in Π 0\Pi_0 (the F σF_\sigma-sets?), and let Π 1\Pi_1 be the collection of their complements (the G δG_\delta-sets, the countably infinitary intersections of sets in Σ 0\Sigma_0); even Σ 1Π 1\Sigma_1 \cup \Pi_1 is not in general a σ\sigma-algebra.
  • Let Σ 2\Sigma_2 be the collection of countably infinitary unions of sets in Π 1\Pi_1 (the G δσG_{\delta\sigma}-sets), and let Π 2\Pi_2 be the collection of their complements (the F σδF_{\sigma\delta}-sets, the countably infinitary intersections of sets in Σ 1\Sigma_1).
  • Continue, defining Σ n\Sigma_n for all natural numbers nn.
  • Let Σ ω\Sigma_\omega be the union of the various Σ n\Sigma_n; although this is closed under complement, it is still not in general a σ\sigma-algebra.
  • Continue, defining Σ α\Sigma_\alpha for all countable ordinal numbers α\alpha.
  • Let Σ ω 1\Sigma_{\omega_1} be the union of the various Σ α\Sigma_\alpha; this is finally a σ\sigma-algebra.

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

This is only the beginning of descriptive set theory?; our Σ α\Sigma_\alpha are their Σ α 0\Sigma^0_\alpha —except that for some reason they start with Σ 1 0\Sigma^0_1 instead of Σ 0 0\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 ‘FF’ comes from French ‘fermé’ for ‘closed’, while ‘GG’ is simply the next letter; the ‘σ\sigma’ and ‘δ\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 δ\delta-ring and σ\sigma-ring

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

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

Revised on October 29, 2013 11:24:00 by Toby Bartels (64.89.53.150)