σ-algebra
English edit
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Pronunciation edit
- IPA^{(key)}: /ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/
Noun edit
σ-algebra (plural σ-algebras)
- (mathematical analysis) A collection of subsets of a given set, such that the empty set is part of this collection, the collection is closed under complements (with respect to the given set) and the collection is closed under countable unions.
- 2001, Elliott H. Lieb, Michael Loss, Analysis, American Mathematical Society, page 4:
- Consider all the sigma-algebras that contain and take their intersection, which we call , i.e., a subset is in if and only if is in every sigma-algebra containing . It is easy to check that is indeed a sigma-algebra.
- 2013, Alexandr A. Borovkov, Probability Theory, Springer, page 15:
- Consider all the σ-algebras on [0,1] containing all intervals from that segment (there is at least one such σ-algebra, for the collection of all the subsets of a given set clearly forms a σ-algebra).
- 2017 February 4, Marco Taboga, Lectures on Probability Theory and Mathematical Statistics^{[1]}, 2nd edition, San Bernardino, CA, USA, →ISBN, §10.5.1, page 75:
- Denote by the set of subsets of [the sample space] Ω which are considered events. is called the space of events. In rigorous probability theory, is required to be a sigma-algebra on Ω.
Synonyms edit
- (collection of subsets that obeys certain conditions): σ-field
Hypernyms edit
- (collection of subsets that obeys certain conditions): field of sets, set algebra
Hyponyms edit
- (collection of subsets that obeys certain conditions): Borel σ-algebra, μ-completion
Meronyms edit
Holonyms edit
Translations edit
See also edit
Further reading edit
- Algebra of sets on Wikipedia.Wikipedia
- Sigma-ring on Wikipedia.Wikipedia
- sigma-algebra on nLab
- Algebra of sets on Encyclopedia of Mathematics
- Sigma-algebra (Computer Science) on Encyclopedia of Mathematics