Borel

Borel is a surname associated with several notable figures in mathematics, the most influential of whom is Émile Borel (1871–1956), a French mathematician and statesman who helped establish foundational work in measure theory and probability. In his name, several central concepts in analysis and probability are named.

In mathematics, the Borel sigma-algebra of a topological space is the sigma-algebra generated by its open sets, and Borel sets are the members of this sigma-algebra. A Borel measure is a measure defined on this sigma-algebra, forming the backbone of modern probability on standard spaces. The Borel-Cantelli lemmas provide criteria for almost sure convergence of sequences of events: if the sum of their probabilities is finite, then only finitely many occur almost surely; and, under independence, if the sum diverges, then infinitely many occur with probability one.

In analysis, the Borel transform and Borel summation are techniques for dealing with divergent series and asymptotic expansions. Borel's paradox is a counterintuitive result about conditional probability on the sphere that highlights measure-theoretic subtleties.

The surname also figures prominently in algebraic groups through Armand Borel (1923–2003), a Swiss-born French mathematician who helped develop the theory of algebraic and Lie groups; the Borel subgroup, a maximal connected solvable subgroup, is named after him.

The name Borel thus links foundational work in measure theory, probability, and algebraic groups, with Émile Borel and Armand Borel as its most celebrated bearers.