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In mathematics, a s-algebra (or sigma-algebra) (sigma is a Greek letter, upper case S, lower case s) over a set X is a nonempty collection S of subsets of X 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,&_160;S) is also a field of sets, sometimes called a s-field or a measurable space. Thus, if X = {a, b, c, d}, one possible sigma algebra on X is S = {?Ø, {a, b}, {c, d}, {a, b, c, d}?}. The main use of s-algebras is in the definition of measures on X. The concept is important in mathematical analysis and probability theory. Formally, a subset S of the power set of a set X is a s-algebra if and only if it has the following properties
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