"μ-completion" meaning in All languages combined

{
  "forms": [
    {
      "form": "μ-completions",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "μ-completions",
        "head": "μ-completion"
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      "expansion": "μ-completion (plural μ-completions)",
      "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"
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          "source": "w"
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        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematical analysis",
          "orig": "en:Mathematical analysis",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Measure theory",
          "orig": "en:Measure theory",
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            "Formal sciences",
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            "All topics",
            "Fundamental"
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        }
      ],
      "examples": [
        {
          "text": "Every σ-algebra has a μ-completion: if a σ-algebra is complete, then it is equal to its μ-completion, otherwise it is contained by its μ-completion.",
          "type": "example"
        }
      ],
      "glosses": [
        "A σ-algebra which is obtained as a \"completion\" of a given σ-algebra, which includes all subsets of the given measure space which simultaneously contain a member of the given σ-algebra and are contained by a member of the given σ-algebra, as long as the contained and containing measurable sets have the same measure, in which case the subset in question is assigned a measure equal to the common measure of its contained and containing measurable sets (so the measure is also being completed, in parallel with the σ-algebra)."
      ],
      "hypernyms": [
        {
          "word": "σ-algebra"
        }
      ],
      "id": "en-μ-completion-en-noun-2pq5Bonn",
      "links": [
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          "mathematical analysis",
          "mathematical analysis"
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        [
          "σ-algebra",
          "σ-algebra"
        ],
        [
          "measure space",
          "measure space"
        ],
        [
          "measurable sets",
          "measurable sets"
        ],
        [
          "measure",
          "measure"
        ]
      ],
      "raw_glosses": [
        "(mathematical analysis) A σ-algebra which is obtained as a \"completion\" of a given σ-algebra, which includes all subsets of the given measure space which simultaneously contain a member of the given σ-algebra and are contained by a member of the given σ-algebra, as long as the contained and containing measurable sets have the same measure, in which case the subset in question is assigned a measure equal to the common measure of its contained and containing measurable sets (so the measure is also being completed, in parallel with the σ-algebra)."
      ],
      "related": [
        {
          "word": "complete measure"
        }
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "μ-completion"
}

{
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    {
      "form": "μ-completions",
      "tags": [
        "plural"
      ]
    }
  ],
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  ],
  "hypernyms": [
    {
      "word": "σ-algebra"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "complete measure"
    }
  ],
  "senses": [
    {
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        "Pages with entries",
        "en:Mathematical analysis",
        "en:Measure theory"
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      "examples": [
        {
          "text": "Every σ-algebra has a μ-completion: if a σ-algebra is complete, then it is equal to its μ-completion, otherwise it is contained by its μ-completion.",
          "type": "example"
        }
      ],
      "glosses": [
        "A σ-algebra which is obtained as a \"completion\" of a given σ-algebra, which includes all subsets of the given measure space which simultaneously contain a member of the given σ-algebra and are contained by a member of the given σ-algebra, as long as the contained and containing measurable sets have the same measure, in which case the subset in question is assigned a measure equal to the common measure of its contained and containing measurable sets (so the measure is also being completed, in parallel with the σ-algebra)."
      ],
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        ],
        [
          "σ-algebra",
          "σ-algebra"
        ],
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          "measure space",
          "measure space"
        ],
        [
          "measurable sets",
          "measurable sets"
        ],
        [
          "measure",
          "measure"
        ]
      ],
      "raw_glosses": [
        "(mathematical analysis) A σ-algebra which is obtained as a \"completion\" of a given σ-algebra, which includes all subsets of the given measure space which simultaneously contain a member of the given σ-algebra and are contained by a member of the given σ-algebra, as long as the contained and containing measurable sets have the same measure, in which case the subset in question is assigned a measure equal to the common measure of its contained and containing measurable sets (so the measure is also being completed, in parallel with the σ-algebra)."
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "μ-completion"
}