{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r02-measurable-function-0057",
  "pilot_number": "R0021-11432",
  "number": 11432,
  "family_id": "qf-mod24-measurable-indicator-audit",
  "title": {
    "zh-Hant": "第 11432 題：有限可測函數",
    "en": "Question 11432: An Indicator’s Measurability Depends on Its Event"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r02-measurable-function-0057",
    "batch_id": "batch-20261013-advance-20261024-r02-measurable-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-measurable-function"
    ],
    "title_zh": "有限可測函數",
    "stem_zh": "完整有限來源與目標sigma代數。操作=measurable-indicator-audit；資料={\"source_n\":6,\"source_sigma\":[[],[0,1,2],[3,4,5],[0,1,2,3,4,5]],\"event\":[0]}。請依 [source_n,event,indicator_mapping,measurable] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(6, [0], [1, 0, 0, 0, 0, 0], 否)",
      "explanation_zh": "依賴結構核對=(6, [0], [1, 0, 0, 0, 0, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；逐一核目標可測集的逆像；可測不推出連續可積或保測。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限可測函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-measurable-indicator-audit",
      "seed": 202610242057,
      "parameters": {
        "source_n": 6,
        "source_sigma": [
          [],
          [
            0,
            1,
            2
          ],
          [
            3,
            4,
            5
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5
          ]
        ],
        "event": [
          0
        ],
        "domain": "Complete_finite_measurable_spaces",
        "operation": "measurable-indicator-audit",
        "semantic_template": "qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261024-r02-measurable-function-0057",
    "batch_id": "batch-20261013-advance-20261024-r02-measurable-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7de1ab21d3b8070967093b037cdafb9ea032b4957044d45058deffa5174c407b",
      "family_id": "qf-mod24-measurable-indicator-audit",
      "seed": 202610242057,
      "semantic_template": "qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"
    },
    "fields": {
      "title": "An Indicator’s Measurability Depends on Its Event",
      "stem": "Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2], [3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).",
      "answer": {
        "kind": "expression",
        "value": "(6, [0], [1, 0, 0, 0, 0, 0], No)",
        "explanation": "The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."
      },
      "skills": [
        "An Indicator’s Measurability Depends on Its Event"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "完整有限來源與目標sigma代數。操作=measurable-indicator-audit；資料={\"source_n\":6,\"source_sigma\":[[],[0,1,2],[3,4,5],[0,1,2,3,4,5]],\"event\":[0]}。請依 [source_n,event,indicator_mapping,measurable] 欄序回答。",
        "en": "Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2], [3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "indicator source:carrier Fin(6);complete canonical sigma family=[[], [0, 1, 2], [3, 4, 5], [0, 1, 2, 3, 4, 5]]",
      "after": "empty member=是;whole member=是;missing complements=[];missing pair unions=[]",
      "relation": "implies",
      "rule_id": "qualify_complete_finite_sigma_algebra",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核完整且按位元遮罩排序的集合族：空集、全集、補集與所有兩兩聯集。有限族的可數聯集只含有限種不同成員，可化為有限聯集，故這些封閉條件足夠。",
        "en": "Qualify the complete bit-mask-ordered family:empty set,universe,complements and all pair unions.In a finite family a countable union has only finitely many distinct members,so finite-union closure suffices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source label=0;normalized event=[0]",
      "after": "indicator value=1",
      "relation": "implies",
      "rule_id": "indicator_one_on_event_zero_elsewhere",
      "conditions": [],
      "explanation": {
        "zh-Hant": "事件成員取1，其他來源點取0；指示函數的值域固定{0,1}而非來源載體。",
        "en": "Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source label=1;normalized event=[0]",
      "after": "indicator value=0",
      "relation": "implies",
      "rule_id": "indicator_one_on_event_zero_elsewhere",
      "conditions": [],
      "explanation": {
        "zh-Hant": "事件成員取1，其他來源點取0；指示函數的值域固定{0,1}而非來源載體。",
        "en": "Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source label=2;normalized event=[0]",
      "after": "indicator value=0",
      "relation": "implies",
      "rule_id": "indicator_one_on_event_zero_elsewhere",
      "conditions": [],
      "explanation": {
        "zh-Hant": "事件成員取1，其他來源點取0；指示函數的值域固定{0,1}而非來源載體。",
        "en": "Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source label=3;normalized event=[0]",
      "after": "indicator value=0",
      "relation": "implies",
      "rule_id": "indicator_one_on_event_zero_elsewhere",
      "conditions": [],
      "explanation": {
        "zh-Hant": "事件成員取1，其他來源點取0；指示函數的值域固定{0,1}而非來源載體。",
        "en": "Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source label=4;normalized event=[0]",
      "after": "indicator value=0",
      "relation": "implies",
      "rule_id": "indicator_one_on_event_zero_elsewhere",
      "conditions": [],
      "explanation": {
        "zh-Hant": "事件成員取1，其他來源點取0；指示函數的值域固定{0,1}而非來源載體。",
        "en": "Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source label=5;normalized event=[0]",
      "after": "indicator value=0",
      "relation": "implies",
      "rule_id": "indicator_one_on_event_zero_elsewhere",
      "conditions": [],
      "explanation": {
        "zh-Hant": "事件成員取1，其他來源點取0；指示函數的值域固定{0,1}而非來源載體。",
        "en": "Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "event=[0];indicator mapping=[1, 0, 0, 0, 0, 0];target full sigma=[[],[0],[1],[0,1]]",
      "after": "preimages=[[],[1, 2, 3, 4, 5],[0],[0, 1, 2, 3, 4, 5]];event in source sigma=否",
      "relation": "implies",
      "rule_id": "indicator_all_four_preimages_equivalence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "四個目標可測集的逆像分別是空集、事件補集、事件與全集。已合格的來源sigma含空與全，且對補集封閉，所以指示函數可測當且僅當事件可測。",
        "en": "The four target preimages are empty,event complement,event,and universe.The qualified source sigma contains empty/universe and is complement-closed,so indicator measurability is equivalent to event measurability."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
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      [
        0
      ],
      [
        1,
        0,
        0,
        0,
        0,
        0
      ],
      false
    ]
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    "canonical_value": "(6, [0], [1, 0, 0, 0, 0, 0], 否)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
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      "formalization": "not_formalized",
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    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
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    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
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}