{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:6923d159-9120-4de8-aad5-aaf9baa64b37",
  "labels": {
    "zh-CN": "生成的西格玛代数",
    "en": "generated sigma-algebra"
  },
  "alternative_labels": [],
  "scope_note": "",
  "identity_status": "active",
  "identity_revision": "3a07ed4501bbaebc500179599fb8f28ab6626e9344aae92a22b8bb27d46e06ae",
  "occurrences": [
    {
      "concept_id": "urn:uuid:6923d159-9120-4de8-aad5-aaf9baa64b37",
      "book_id": "szekeres",
      "edition_id": "source-snapshot-bc3404156f6e",
      "lang": "zh-CN",
      "route": "zh/chapter-11/01-measurable-spaces-and-functions",
      "anchor": "first-5139e4371bda3b6512b3-zh",
      "chapter": "11",
      "section": "11.1",
      "offset": 2500,
      "range": [
        7164,
        7416
      ],
      "source_revision": "0e5801479b5859d6fa0081d22792f6e5063856edace04ac2dfc1dab208a32811",
      "source_category": "translation",
      "review_status": "unreviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "生成的西格玛代数(generated sigma-algebra)",
      "context": {
        "range": [
          6601,
          7719
        ],
        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-11-1\">例 11.1</a> \n>\n> 给定任意集合 $X$，集合族 ${ \\mathcal { M } } = \\{ \\emptyset , X \\}$ 显然是一个西格玛代数。这是可能的最小西格玛代数。相比之下，最大的西格玛代数是所有子集构成的集合 $2 ^ { X }$。所有有趣的例子都介于这两个极端之间。\n>\n显然，任意两个西格玛代数 $\\mathcal { M } \\cap \\mathcal { M } ^ { \\prime }$ 的交集仍是西格玛代数——只需验证两个西格玛代数共有的集合满足性质 (Meas1)–(Meas3)。这一结论可推广到任意一族西格玛代数的交集 $\\cap _ { i \\in I } { \\mathcal { M } } _ { i }$。因此，给定任意子集族 $\\mathcal { A } \\subset 2 ^ { X }$，存在唯一的“最小”西格玛代数 $\\mathcal S\\supseteq\\mathcal A$。它是所有包含 ${ \\mathcal { A } } .$ 的西格玛代数的交集。它称为由 ${ \\mathcal { A } } .$ 生成的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:6923d159-9120-4de8-aad5-aaf9baa64b37\" data-concept-route=\"/concept-identities#concept-6923d159-9120-4de8-aad5-aaf9baa64b37\" id=\"first-5139e4371bda3b6512b3-zh\">生成的西格玛代数(generated sigma-algebra)</u>。对于拓扑空间 $X ,$，由开集生成的西格玛代数称为 $X$ 上的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:e41d2f97-2b81-41a5-a89e-a1df482c0028\" data-concept-route=\"/concept-identities#concept-e41d2f97-2b81-41a5-a89e-a1df482c0028\" id=\"first-cdc12b6ea25268c0b522-zh\">博雷尔集(Borel set)</u>。它们包含所有开集和所有闭集，并且一般还包含许多既非开也非闭的集合。",
        "quote_sha256": "0d017051cd6d9da3bc977386f98ac028730416d002bcd5769170f1093a318a23",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "en/chapter-11/01-measurable-spaces-and-functions",
          "lang": "en",
          "source_category": "original_transcription",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-af2b5f79db67dda9b07eaea2"
    }
  ],
  "registered_positions": [
    {
      "route": "zh/chapter-11/01-measurable-spaces-and-functions",
      "lang": "zh-CN",
      "section": "11.1",
      "signal": "term-mark",
      "label": "生成的西格玛代数",
      "english": "generated sigma-algebra",
      "offset": 1927
    }
  ],
  "statements": [],
  "resolved_references": [
    {
      "raw": "href=\"#example-11-1\"",
      "href": "#example-11-1",
      "range": [
        6642,
        6662
      ],
      "relation": "references",
      "target_route": "zh/chapter-11/01-measurable-spaces-and-functions",
      "target_anchor": "example-11-1",
      "resolved": true,
      "reason": null,
      "source_revision": "0e5801479b5859d6fa0081d22792f6e5063856edace04ac2dfc1dab208a32811",
      "source_route": "zh/chapter-11/01-measurable-spaces-and-functions",
      "occurrence_id": "occ-af2b5f79db67dda9b07eaea2"
    }
  ],
  "reading_order": [
    "occ-af2b5f79db67dda9b07eaea2"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 1,
    "located_occurrences": 1,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "f9a273f23c854ef8b0b2a4d97684dea4f338f33fe3fcd31d71a039faaf0a94ad",
  "optional_enrichments": []
}