Borel algebra

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In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, the Borel sets are defined using compact sets and their complements rather than closed and open sets. These two definitions are equivalent for most typical spaces, including any locally compact, separable metric space (or more generally any σ-compact space), but are different for certain pathological spaces.


Generating the Borel algebra

In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows.

For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let

  • T_\sigma \quad be all countable unions of elements of T
  • T_\delta \quad be all countable intersections of elements of T
  •  T_{\delta\sigma}=(T_\delta)_\sigma.\,

Now define by transfinite induction a sequence Gm, where m is an ordinal number, in the following manner:

  • For the base case of the definition,

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