"σ-algebra" meaning in All languages combined

See σ-algebra on Wiktionary

Noun [English]

IPA: /ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/ Forms: σ-algebras [plural]
Head templates: {{en-noun|σ-algebras|head=σ-algebra}} σ-algebra (plural σ-algebras)
  1. (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. Wikipedia link: Sigma-algebra Categories (topical): Mathematical analysis, Measure theory Synonyms: sigma-algebra, Σ-algebra Synonyms (collection of subsets that obeys certain conditions): σ-field Hypernyms (collection of subsets that obeys certain conditions): field of sets, set algebra Hyponyms (collection of subsets that obeys certain conditions): Borel σ-algebra, μ-completion Holonyms: measurable space Meronyms: measurable set Related terms: σ-ring Translations (Translations): tribu [feminine] (French), σ-algèbre [feminine] (French), σ-Algebra [feminine] (German), σ-алгебра (σ-algebra) [feminine] (Russian)

Inflected forms

{
  "forms": [
    {
      "form": "σ-algebras",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "σ-algebras",
        "head": "σ-algebra"
      },
      "expansion": "σ-algebra (plural σ-algebras)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "English terms containing italics",
          "parents": [
            "Terms containing italics",
            "Terms by orthographic property",
            "Terms by lexical property"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "English terms written in multiple scripts",
          "parents": [
            "Terms written in multiple scripts",
            "Terms by orthographic property",
            "Terms by lexical property"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with DISPLAYTITLE conflicts",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with French translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with German translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with Russian translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematical analysis",
          "orig": "en:Mathematical analysis",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Measure theory",
          "orig": "en:Measure theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "2001, Elliott H. Lieb, Michael Loss, Analysis, American Mathematical Society, page 4:",
          "text": "Consider all the sigma-algebras that contain #92;mathcal#123;F#125; and take their intersection, which we call #92;Sigma, i.e., a subset A#92;subset#92;Omega is in #92;Sigma if and only if A is in every sigma-algebra containing #92;mathcal#123;F#125;. It is easy to check that #92;Sigma is indeed a sigma-algebra.",
          "type": "quote"
        },
        {
          "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": "quote"
        },
        {
          "ref": "2017 February 4, Marco Taboga, Lectures on Probability Theory and Mathematical Statistics, 2nd edition, San Bernardino, CA, USA, →ISBN, §10.5.1, page 75:",
          "text": "Denote by #92;mathcal#123;F#125; the set of subsets of [the sample space] Ω which are considered events. #92;mathcal#123;F#125; is called the space of events. In rigorous probability theory, #92;mathcal#123;F#125; is required to be a sigma-algebra on Ω.",
          "type": "quote"
        }
      ],
      "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"
}
{
  "forms": [
    {
      "form": "σ-algebras",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "σ-algebras",
        "head": "σ-algebra"
      },
      "expansion": "σ-algebra (plural σ-algebras)",
      "name": "en-noun"
    }
  ],
  "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"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "meronyms": [
    {
      "word": "measurable set"
    }
  ],
  "pos": "noun",
  "related": [
    {
      "word": "σ-ring"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms containing italics",
        "English terms spelled with Σ",
        "English terms with quotations",
        "English terms written in multiple scripts",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with DISPLAYTITLE conflicts",
        "Pages with entries",
        "Terms with French translations",
        "Terms with German translations",
        "Terms with Russian translations",
        "Translation table header lacks gloss",
        "en:Mathematical analysis",
        "en:Measure theory"
      ],
      "examples": [
        {
          "ref": "2001, Elliott H. Lieb, Michael Loss, Analysis, American Mathematical Society, page 4:",
          "text": "Consider all the sigma-algebras that contain #92;mathcal#123;F#125; and take their intersection, which we call #92;Sigma, i.e., a subset A#92;subset#92;Omega is in #92;Sigma if and only if A is in every sigma-algebra containing #92;mathcal#123;F#125;. It is easy to check that #92;Sigma is indeed a sigma-algebra.",
          "type": "quote"
        },
        {
          "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": "quote"
        },
        {
          "ref": "2017 February 4, Marco Taboga, Lectures on Probability Theory and Mathematical Statistics, 2nd edition, San Bernardino, CA, USA, →ISBN, §10.5.1, page 75:",
          "text": "Denote by #92;mathcal#123;F#125; the set of subsets of [the sample space] Ω which are considered events. #92;mathcal#123;F#125; is called the space of events. In rigorous probability theory, #92;mathcal#123;F#125; is required to be a sigma-algebra on Ω.",
          "type": "quote"
        }
      ],
      "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."
      ],
      "links": [
        [
          "mathematical analysis",
          "mathematical analysis"
        ],
        [
          "collection",
          "collection"
        ],
        [
          "empty set",
          "empty set"
        ],
        [
          "complement",
          "complement"
        ],
        [
          "countable",
          "countable"
        ],
        [
          "union",
          "union"
        ]
      ],
      "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."
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Sigma-algebra"
      ]
    }
  ],
  "sounds": [
    {
      "ipa": "/ˈsɪɡ.mə ˈæl.dʒɪ.bɹə/"
    }
  ],
  "synonyms": [
    {
      "sense": "collection of subsets that obeys certain conditions",
      "word": "σ-field"
    },
    {
      "word": "sigma-algebra"
    },
    {
      "word": "Σ-algebra"
    }
  ],
  "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"
    }
  ],
  "word": "σ-algebra"
}

Download raw JSONL data for σ-algebra meaning in All languages combined (4.0kB)


This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2025-01-10 from the enwiktionary dump dated 2025-01-01 using wiktextract (df33d17 and 4ed51a5). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.

If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.