See σ-algebra in All languages combined, or Wiktionary
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Lieb, Michael Loss, Analysis, American Mathematical Society, page 4", "text": "Consider all the sigma-algebras that contain #x5C;mathcal#x7B;F#x7D; and take their intersection, which we call #x5C;Sigma, i.e., a subset A#x5C;subset#x5C;Omega is in #x5C;Sigma if and only if A is in every sigma-algebra containing #x5C;mathcal#x7B;F#x7D;. It is easy to check that #x5C;Sigma is indeed a sigma-algebra.", "type": "quotation" }, { "ref": "2013, Alexandr A. Borovkov, Probability Theory, Springer, page 15", "text": "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).", "type": "quotation" }, { "ref": "2017 February 4, Marco Taboga, Lectures on Probability Theory and Mathematical Statistics, 2nd edition, San Bernardino, CA, USA, §10.5.1, page 75", "text": "Denote by #x5C;mathcal#x7B;F#x7D; the set of subsets of [the sample space] Ω which are considered events. #x5C;mathcal#x7B;F#x7D; is called the space of events. In rigorous probability theory, #x5C;mathcal#x7B;F#x7D; is required to be a sigma-algebra on Ω.", "type": "quotation" } ], "glosses": [ "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." ], "holonyms": [ { "word": "measurable space" } ], "hypernyms": [ { "sense": "collection of subsets that obeys certain conditions", "word": "field of sets" }, { "sense": "collection of subsets that obeys certain conditions", "word": "set algebra" } ], "hyponyms": [ { "sense": "collection of subsets that obeys certain conditions", "word": "Borel σ-algebra" }, { "sense": "collection of subsets that obeys certain conditions", "word": "μ-completion" } ], "id": "en-σ-algebra-en-noun-vBr0kDB-", "links": [ [ "mathematical analysis", "mathematical analysis" ], [ "collection", "collection" ], [ "empty set", "empty set" ], [ "complement", "complement" ], [ "countable", "countable" ], [ "union", "union" ] ], "meronyms": [ { "word": "measurable set" } ], "raw_glosses": [ "(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." ], "related": [ { "word": "σ-ring" } ], "synonyms": [ { "sense": "collection of subsets that obeys certain conditions", "word": "σ-field" }, { "word": "sigma-algebra" }, { "word": "Σ-algebra" } ], "topics": [ "mathematical-analysis", "mathematics", "sciences" ], "translations": [ { "code": "fr", "lang": "French", "sense": "Translations", "tags": [ "feminine" ], "word": "tribu" }, { "code": "fr", "lang": "French", "sense": "Translations", "tags": [ "feminine" ], "word": "σ-algèbre" }, { "code": "de", "lang": "German", "sense": "Translations", "tags": [ "feminine" ], "word": "σ-Algebra" }, { "code": "ru", "lang": "Russian", "roman": "σ-algebra", "sense": "Translations", "tags": [ "feminine" ], "word": "σ-алгебра" } ], "wikipedia": [ "Sigma-algebra" ] } ], "sounds": [ { "ipa": "/ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/" } ], "word": "σ-algebra" }
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