{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-measurable-function",
  "semantic_family_id": "sf-measurable-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "可測函數"
  },
  "summary": {
    "zh-Hant": "來源與目標各固定sigma代數時，函數可測表示每個目標可測集的逆像皆在來源sigma代數。"
  },
  "statement": {
    "zh-Hant": "給有限可測空間(X,Sigma_X)與(Y,Sigma_Y)，總函數f:X→Y可測，若每個B∈Sigma_Y的逆像f^{-1}(B)都屬Sigma_X。本輪用完整有限列舉逐一核逆像。"
  },
  "formula": {
    "tex": "M(f)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7792820a01412d22988891a18d3ac9f830b61d36cf3b053b0563a3b6b2d9e5c4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "載體以0起算有限標籤、sigma代數以canonical完整子集族、函數以完整總映射表表示。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限完整列舉只證指定有限實例；可測性不推出連續、可積或保測。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "measurability_record",
      "role_zh": "可測性紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "total_mapping",
      "role_zh": "總映射",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sigma-algebra-measure-space",
      "title_zh": "sigma代數與測度空間",
      "reason_zh": "來源與目標可測集合族必須先固定。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "逆像由完整總映射導出。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "逆像、補集與聯集使用集合運算。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "可測責任的方向是目標可測集取逆像。",
      "indicator函數可測恰由其事件及補集在來源sigma代數中控制。",
      "兩個可測映射的合成可由逆像合成逐一讀回。"
    ],
    "common_misconception_zh": "檢查像集而非逆像，或把可測誤當連續、可積或保測。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限列舉不外推任意無限可測空間。",
    "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": "加入簡單函數、積分與測度收斂。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-measurable-function.md 整理；2026-10-13 脈衝 R02（使用者手動接續 2026-10-24 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"
  }
}