{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-almost-sure-convergence",
  "semantic_family_id": "sf-almost-sure-convergence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "幾乎必然收斂"
  },
  "summary": {
    "zh-Hant": "同一機率空間上的隨機變數序列若在零機率例外集外逐點收斂，則幾乎必然收斂。"
  },
  "statement": {
    "zh-Hant": "本輪用完整有限原子機率空間與每個原子的symbolic eventual certificate。所有正質量原子都須收斂；零質量原子失敗可忽略。有限prefix與單獨依機率收斂都不足。"
  },
  "formula": {
    "tex": "A(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "dde799663ba218245a9b35d2d1baa486043c6b4b87197f3f6293eaa6ac210367"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "所有X_n與X綁定同一完整有限樣本空間及精確有理質量。"
    },
    {
      "id": "a2",
      "zh-Hant": "eventual certificate明示每個正質量原子的settling index或全稱尾端公式。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "almost_sure_convergence",
      "role_zh": "幾乎必然收斂紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "random_variable_sequence",
      "role_zh": "隨機變數序列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "共同樣本空間上的可測函數。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "時間索引。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "零機率例外集合。"
    },
    {
      "id": "mko-sigma-algebra-measure-space",
      "title_zh": "σ代數與測度空間",
      "reason_zh": "共同有限機率空間。"
    },
    {
      "id": "mko-convergence-in-probability",
      "title_zh": "依機率收斂",
      "reason_zh": "較弱模式的區分。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "逐正質量原子核尾端收斂。",
      "零質量原子只影響點態，不影響a.s.結論。",
      "有限prefix與依機率收斂不能反推a.s.。"
    ],
    "common_misconception_zh": "從有限n或依機率收斂直接判幾乎必然收斂。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限原子與明示全稱尾端證書。",
    "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Lp收斂與收斂模式關係。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-almost-sure-convergence.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-01 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}