{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-sigma-algebra-measure-space",
  "semantic_family_id": "sf-sigma-algebra-measure-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "σ-代數與測度空間"
  },
  "summary": {
    "zh-Hant": "σ-代數含空集並對補集與可數聯集封閉；測度非負、空集為零且可數可加。"
  },
  "statement": {
    "zh-Hant": "在集合X上，σ-代數含空集，對補集及可數聯集封閉。測度對可測集給非負值、空集為零，並對兩兩互斥可數族可數可加。本輪完整有限載體可把可數責任歸約為有限distinct sets。"
  },
  "formula": {
    "tex": "M(S)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "S",
          "glyph": "S"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7955d94067e6d42aebae1f6764a62d6596ef7e768dc6752820fc49b6d9ba35e6"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用有限載體、canonical完整集合族與精確非負有理測度值。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限歸約只證指定有限測度空間，不外推任意無限σ-代數或extended-real行為。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "finite_measure_space_record",
      "role_zh": "有限測度空間紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "S",
      "role": "sigma_family",
      "role_zh": "可測集合族",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "載體與可測集為集合。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "補集與聯集封閉。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "測度為可測集合到非負值的函數。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "可數族以序列索引。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "可數封閉與可加性含全稱責任。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "有限題的測度值為非負實數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "有限載體上的canonical集合族可完整核補集與二元聯集；有限distinct集合使可數聯集歸約成立。",
      "測度核對空集零與全部互斥對的有限可加性；有限載體中互斥非空族最多有限個。",
      "總量為1時才是機率測度。"
    ],
    "common_misconception_zh": "只含空集與全體就叫任意σ-代數，或把有限加法無條件外推無限載體。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "可數語義只在指定有限載體由有限歸約封閉。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入可測函數、Lebesgue積分與收斂定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-sigma-algebra-measure-space.md 整理；2026-10-13 脈衝 R05（使用者手動接續 2026-10-23 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}