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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 Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under union or intersection of countably many subsets. The pair (X, Σ) is called a measurable space.A σ-algebra generalises the concept of an algebra of sets, which only needs to be closed under the union or intersection of finitely many subsets.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 a formal mathematical definition of sufficient statistic, 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 X is 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 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).".
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- Sigma-algebra wikiPageOutDegree "102".
- Sigma-algebra wikiPageRevisionID "705383037".
- Sigma-algebra wikiPageWikiLink Abuse_of_notation.
- Sigma-algebra wikiPageWikiLink Algebra_of_sets.
- Sigma-algebra wikiPageWikiLink Axiom_of_choice.
- Sigma-algebra wikiPageWikiLink Borel_hierarchy.
- Sigma-algebra wikiPageWikiLink Borel_set.
- Sigma-algebra wikiPageWikiLink Calligraphy.
- Sigma-algebra wikiPageWikiLink Category:Boolean_algebra.
- Sigma-algebra wikiPageWikiLink Category:Measure_theory.
- Sigma-algebra wikiPageWikiLink Category:Probability_theory.
- Sigma-algebra wikiPageWikiLink Category:Set_families.
- Sigma-algebra wikiPageWikiLink Category_(mathematics).
- Sigma-algebra wikiPageWikiLink Closed_set.
- Sigma-algebra wikiPageWikiLink Closure_(mathematics).
- Sigma-algebra wikiPageWikiLink Complement_(set_theory).
- Sigma-algebra wikiPageWikiLink Complete_measure.
- Sigma-algebra wikiPageWikiLink Conditional_expectation.
- Sigma-algebra wikiPageWikiLink Conditional_probability_distribution.
- Sigma-algebra wikiPageWikiLink Convergence_of_random_variables.
- Sigma-algebra wikiPageWikiLink Cumulative_distribution_function.
- Sigma-algebra wikiPageWikiLink Cylinder_set.
- Sigma-algebra wikiPageWikiLink De_Morgans_laws.
- Sigma-algebra wikiPageWikiLink Disjoint_sets.
- Sigma-algebra wikiPageWikiLink Dynkin_system.
- Sigma-algebra wikiPageWikiLink Empty_set.
- Sigma-algebra wikiPageWikiLink Euclidean_space.
- Sigma-algebra wikiPageWikiLink Filtration_(mathematics).
- Sigma-algebra wikiPageWikiLink Fraktur.
- Sigma-algebra wikiPageWikiLink Function_(mathematics).
- Sigma-algebra wikiPageWikiLink Image_(mathematics).
- Sigma-algebra wikiPageWikiLink Integral.
- Sigma-algebra wikiPageWikiLink Intersection_(set_theory).
- Sigma-algebra wikiPageWikiLink Interval_(mathematics).
- Sigma-algebra wikiPageWikiLink Join_(sigma_algebra).
- Sigma-algebra wikiPageWikiLink Lebesgue_integration.
- Sigma-algebra wikiPageWikiLink Lebesgue_measure.
- Sigma-algebra wikiPageWikiLink Lebesgue–Stieltjes_integration.
- Sigma-algebra wikiPageWikiLink Mathematical_analysis.
- Sigma-algebra wikiPageWikiLink Measurable_function.
- Sigma-algebra wikiPageWikiLink Measure_(mathematics).
- Sigma-algebra wikiPageWikiLink Metric_space.
- Sigma-algebra wikiPageWikiLink Monotone_class_theorem.
- Sigma-algebra wikiPageWikiLink Morphism.
- Sigma-algebra wikiPageWikiLink Open_set.
- Sigma-algebra wikiPageWikiLink Ordinal_number.
- Sigma-algebra wikiPageWikiLink Partition_of_a_set.
- Sigma-algebra wikiPageWikiLink Pi_system.
- Sigma-algebra wikiPageWikiLink PlanetMath:950.
- Sigma-algebra wikiPageWikiLink Power_set.
- Sigma-algebra wikiPageWikiLink Probability_measure.
- Sigma-algebra wikiPageWikiLink Probability_space.
- Sigma-algebra wikiPageWikiLink Probability_theory.
- Sigma-algebra wikiPageWikiLink Product_topology.
- Sigma-algebra wikiPageWikiLink Random_variable.
- Sigma-algebra wikiPageWikiLink Real_line.
- Sigma-algebra wikiPageWikiLink Real_number.
- Sigma-algebra wikiPageWikiLink Ring_of_sets.
- Sigma-algebra wikiPageWikiLink Sample_space.
- Sigma-algebra wikiPageWikiLink Separable_sigma_algebra.
- Sigma-algebra wikiPageWikiLink Set-theoretic_limit.
- Sigma-algebra wikiPageWikiLink Sigma-algebra.
- Sigma-algebra wikiPageWikiLink Sigma-ring.
- Sigma-algebra wikiPageWikiLink Sigma_additivity.
- Sigma-algebra wikiPageWikiLink Singleton_(mathematics).
- Sigma-algebra wikiPageWikiLink Statistics.
- Sigma-algebra wikiPageWikiLink Stochastic_process.
- Sigma-algebra wikiPageWikiLink Stopping_time.
- Sigma-algebra wikiPageWikiLink Subset.
- Sigma-algebra wikiPageWikiLink Sufficient_statistic.
- Sigma-algebra wikiPageWikiLink Topological_space.
- Sigma-algebra wikiPageWikiLink Transfinite_induction.
- Sigma-algebra wikiPageWikiLink Union_(set_theory).
- Sigma-algebra wikiPageWikiLink Universal_set.
- Sigma-algebra wikiPageWikiLink Vitali_set.
- Sigma-algebra wikiPageWikiLinkText "σ-algebra".
- Sigma-algebra wikiPageWikiLinkText "σ-algebras".
- Sigma-algebra wikiPageWikiLinkText "''σ''-algebra".
- Sigma-algebra wikiPageWikiLinkText "''σ''-algebra".
- Sigma-algebra wikiPageWikiLinkText "-algebra".
- Sigma-algebra wikiPageWikiLinkText "-algebras".
- Sigma-algebra wikiPageWikiLinkText "-field".
- Sigma-algebra wikiPageWikiLinkText "Sigma field".
- Sigma-algebra wikiPageWikiLinkText "Sigma-algebra".
- Sigma-algebra wikiPageWikiLinkText "Sigma-algebra#Product σ-algebra".
- Sigma-algebra wikiPageWikiLinkText "discrete sigma algebra".
- Sigma-algebra wikiPageWikiLinkText "generated sigma-algebra".
- Sigma-algebra wikiPageWikiLinkText "generated".
- Sigma-algebra wikiPageWikiLinkText "generates".
- Sigma-algebra wikiPageWikiLinkText "measurable space".
- Sigma-algebra wikiPageWikiLinkText "measurable spaces".
- Sigma-algebra wikiPageWikiLinkText "product -algebra".
- Sigma-algebra wikiPageWikiLinkText "sigma algebra".
- Sigma-algebra wikiPageWikiLinkText "sigma-algebra".
- Sigma-algebra wikiPageWikiLinkText "smallest σ-algebra".
- Sigma-algebra wikiPageWikiLinkText "sub-sigma-algebras".
- Sigma-algebra wikiPageWikiLinkText "σ-Algebra".