In mathematics, a σ-algebra (or sigma-algebra) over a set X is a nonempty collection Σ of subsets of X (including X itself) that is closed under complementation and countable unions of its members. It is a Boolean algebra, completed to include countably infinite operations. The pair (X, Σ) is also a field of sets, sometimes called a σ-field or a measurable space.

Thus, if X = {a, b, c, d}, one possible sigma algebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }.

The main use of σ-algebras is in the definition of measures on X. The concept is important in mathematical analysis and probability theory.

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