Borelsigmaalgebran

The Borelsigmaalgebran, or Borel sigma-algebra, is the sigma-algebra generated by the open sets of a topological space X. In the standard real-line setting, it is denoted B(R) and is the smallest sigma-algebra that contains all open sets. It can be constructed as the intersection of all sigma-algebras that contain the open sets, or equivalently as the sigma-algebra generated by the collection of open intervals with rational endpoints, since these generate the open sets in R.

Borel sets are closed under countable unions, countable unions of complements, and countable intersections, making them include every open set, every closed set, and many derived sets such as F_sigma and G_delta sets. Not all subsets of R are Borel; there exist non-Borel sets with pathological constructions.

On the real line, B(R) is countably generated: there exists a countable basis, such as open intervals with rational endpoints, from which all Borel sets can be obtained through countable operations. This property aids measure-theoretic work, for example when defining measures on the Borel sigma-algebra.

The Lebesgue measure is defined on a larger sigma-algebra, the Lebesgue sigma-algebra, which is the completion of the Borel sigma-algebra with respect to Lebesgue measure; it includes all subsets of Lebesgue-null sets.

In probability theory, the Borel sigma-algebra provides the natural sigma-algebra for real-valued random variables, and for random vectors in Euclidean spaces. A function is Borel-measurable if the preimage of every Borel set is measurable.

The concept is named after Émile Borel and can be extended to any topological space.