|
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 is closed under countable-fold set operations (complement, union of countably many sets and intersection of countably many sets). By contrast, 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. 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 one possible σ-algebra on ''X'' is where ∅ is the empty set. However, a finite algebra is always a σ-algebra. If 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). ==Motivation== There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sigma-algebra」の詳細全文を読む スポンサード リンク
|