The Borel algebra (or Borel σ-algebra) on a topological space X with topology T is the smallest σ-algebra containing T. The existence and uniqueness of the Borel algebra is shown by noting that the intersection of all σ-algebras containing T is itself a σ-algebra, so this intersection is the Borel algebra. The elements of the Borel algebra are called Borel sets.
The Borel algebra may alternatively and equivalently defined as the smallest σ-algebra which contains all the closed subsets of X. A subset of X is a Borel set if and only if it can be obtained from open sets by using a countable series of the set operations union, intersection and complement.
A particularly important example is the Borel algebra on the set of real numbers. It underlies the Borel measure and also every probability distribution. The Borel algebra on the reals is the smallest sigma algebra on R which contains all the intervals.
