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- Sigma-algebra abstract "In mathematical analysis and in probability theory, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a set X is a collection of subsets of X that is closed under countably many set operations (complement, union and intersection). On the other hand, an algebra is only required to be closed under finitely many set operations. That is, a σ-algebra is an algebra of sets, completed to include countably infinite operations. The pair (X, Σ) is also a field of sets, called a measurable space.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.If X = {a, b, c, d}, one possible σ-algebra on X is Σ = {∅, {a, b}, {c, d}, {a, b, c, d}}, where ∅ is the empty set. However, a finite algebra is always a σ-algebra.If {A1, A2, A3, …} is a countable partition of X then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy).".
- Sigma-algebra wikiPageID "29586".
- Sigma-algebra wikiPageRevisionID "604129436".
- Sigma-algebra hasPhotoCollection Sigma-algebra.
- Sigma-algebra id "p/a011400".
- Sigma-algebra title "Algebra of sets".
- Sigma-algebra subject Category:Boolean_algebra.
- Sigma-algebra subject Category:Measure_theory.
- Sigma-algebra subject Category:Probability_theory.
- Sigma-algebra subject Category:Set_families.
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- Sigma-algebra comment "In mathematical analysis and in probability theory, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a set X is a collection of subsets of X that is closed under countably many set operations (complement, union and intersection). On the other hand, an algebra is only required to be closed under finitely many set operations. That is, a σ-algebra is an algebra of sets, completed to include countably infinite operations.".
- Sigma-algebra label "Przestrzeń mierzalna".
- Sigma-algebra label "Sigma-algebra".
- Sigma-algebra label "Sigma-algebra".
- Sigma-algebra label "Sigma-algebra".
- Sigma-algebra label "Sigma-álgebra".
- Sigma-algebra label "Tribu (mathématiques)".
- Sigma-algebra label "Σ-Algebra".
- Sigma-algebra label "Σ-álgebra".
- Sigma-algebra label "Σ-代数".
- Sigma-algebra label "Сигма-алгебра".
- Sigma-algebra label "完全加法族".
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- Sigma-algebra sameAs Σ-Algebra.
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- Sigma-algebra sameAs Tribu_(mathématiques).
- Sigma-algebra sameAs Sigma-algebra.
- Sigma-algebra sameAs 完全加法族.
- Sigma-algebra sameAs 시그마-대수.
- Sigma-algebra sameAs Sigma-algebra.
- Sigma-algebra sameAs Przestrzeń_mierzalna.
- Sigma-algebra sameAs Sigma-álgebra.
- Sigma-algebra sameAs m.07b1x.
- Sigma-algebra sameAs Q217357.
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- Sigma-algebra wasDerivedFrom Sigma-algebra?oldid=604129436.
- Sigma-algebra isPrimaryTopicOf Sigma-algebra.