BorelSigmaAlgebra

Borel sigma-algebra on a topological space X, denoted B(X), is the sigma-algebra generated by the open sets of X. On the real line R with the standard topology, B(R) is the sigma-algebra generated by the open intervals (a,b). It is the smallest sigma-algebra that makes every open set measurable.

Construction and basic facts: B(X) is obtained by closing the collection of open sets under countable unions, intersections, and complements. Equivalently, it can be generated by various equivalent families, such as the open intervals with rational endpoints, or the closed sets, or the class of F_sigma and G_delta sets. On R, B(R) is countably generated because the basis of open intervals with rational endpoints suffices.

Properties and examples: Borel sets include all open and closed sets, as well as countable unions of closed sets (F_sigma) and countable intersections of open sets (G_delta). The sigma-algebra is closed under countable unions, intersections, and complements. It is strictly smaller than the power set of R; there exist subsets of R that are not Borel.

Relations to measures: The Lebesgue measure is defined on the Lebesgue sigma-algebra, which is the completion of B(R) with respect to Lebesgue measure. Equivalently, every Lebesgue measurable set can be expressed as a Borel set union a subset of a null set.

Applications and significance: Borel sets form the natural domain of measurability in many areas of analysis and probability. They provide a robust framework for integration, probability measures, and descriptive set theory, and serve as the standard starting point for studying measurable functions.