Algebra generated by semialgebra vs sigma algebra generated by semialgebra
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niversal algebra.
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes Xitself, is closed under complement, and is closed under countable unions.
The definition implies that it also includes the empty subset and that it is closed under countable intersections.
The pair (X, Σ) is called a measurable space or Borel space.
A σ-algebra is a type of algebra of sets. An algebra of sets needs only to be closed under the union or intersection of finitely many subsets, which is a weaker condition.[1]
The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.
In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic,[2] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.
If X = {a, b, c, d}, one possible σ-algebra on Xis Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is the empty set. In general, a finite algebra is always a σ-algebra.
If {A1, A2, A3, …} is a countable partition of Xthen the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes Xitself, is closed under complement, and is closed under countable unions.
The definition implies that it also includes the empty subset and that it is closed under countable intersections.
The pair (X, Σ) is called a measurable space or Borel space.
A σ-algebra is a type of algebra of sets. An algebra of sets needs only to be closed under the union or intersection of finitely many subsets, which is a weaker condition.[1]
The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.
In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic,[2] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.
If X = {a, b, c, d}, one possible σ-algebra on Xis Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is the empty set. In general, a finite algebra is always a σ-algebra.
If {A1, A2, A3, …} is a countable partition of Xthen the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.
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