# 可测空间与概率测度

从有限分组列出事件域，检验函数可测性，并区分几乎处处相等与合法版本。

Entry: zh-qt09 | Node: QT09 | Language: zh | Editorial revision: 2026-09-21

## Teaching instructions
你正在使用一份公开教学包，读者具有高年级本科至研究生的数学背景。先实际读取 required_readings 中本次采用的完整单元，再进入讲解；记录版本、页/节、关键假设与支持范围。仅取得目录、摘要或失败链接不算读完。已在本次会话完整取得的同一版本可以复用。找不到正文时尝试已核公开等价版本并说明版本变化，仍缺失则明确缺少的单元，不编造已读内容。
以下正文、输入和解析为共同来源，不能自行改变数字或把教学概率改称样本频率、现实真值或定价测度。不要访问账户。每次推进一个完整推理任务；已掌握的基础可跳过讲解，但承重条件不能省略。静态说明与交互同义；工具不能运行时直接使用完整静态输入与精确计算，不声称运行过实验。

当前词条：QT09《可测空间与概率测度》。本次只教这个节点及选择的证明分支。
诊断任务：给分组 {1,2}/{3,4} 和函数 (0,1,2,2)，请读者列事件域并找出不可测的具体原像。
通过标准：能证明所有事件恰为原子的并，检查同组常数；在零概率组反例中区分 a.s. 与可测性。
先让读者作解释或计算，再依完整解析反馈；最后更换分组、概率或条件做迁移，不能只询问“懂了吗”。

Before substantive teaching, actually retrieve every required reading unit for the selected scope. Read its complete designated section, including necessary assumptions, tables and footnotes. A working URL or an editorial access date is not a runtime reading receipt. Record the actual version, location, scope and what it supports. If unavailable, use a previously verified equivalent source; if the required unit remains unavailable, identify that gap rather than teach it from memory. Start runtime_reading_log empty. Once reading is complete, use a substantive diagnostic or follow the reader's request for direct explanation. Advance one complete reasoning task at a time; skip mastered basics. Distinguish original facts, supplied teaching assumptions and inference.

## Required readings and runtime protocol
```json
{
  "export_mode": "public",
  "required_readings": [
    {
      "source_id": "QTB-MIT-EVENTS",
      "access": {
        "kind": "pdf_full_text",
        "uri": "https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/c37dc8b61cdf6bde689a627bfa5b4942_MIT6_436JF18_lec01.pdf"
      },
      "required_unit": {
        "locator": "§4–§5 开头，pp.4–8",
        "scope": "sigma-代数、生成、部分观测、概率测度与 a.s. 例",
        "purpose": "明确事件域与概率是两个对象"
      },
      "supports": "支持 sigma-代数、生成事件域、部分观测的信息解释、概率测度与几乎必然。本文的有限分区及可测性判据给出独立完整证明。",
      "title": "Lecture 1: Probabilistic Models and Probability Measures",
      "authors": [
        "MIT 6.436J/15.085J",
        "Yury Polyanskiy（课程教师）"
      ],
      "version": "Fall 2018"
    },
    {
      "source_id": "QTB-DEMBO-2021",
      "access": {
        "kind": "pdf_full_text",
        "uri": "https://adembo.su.domains/stat-310b/lnotes.pdf"
      },
      "required_unit": {
        "locator": "§1.2.1：Definition 1.2.1–1.2.2 p.18；Theorem 1.2.9、Definition 1.2.12 p.20；§4.1.1 pp.153–156",
        "scope": "可测映射/随机变量定义、生成族与半直线原像判据，以及条件期望版本的条件",
        "purpose": "检查可测性和零测集修改的边界"
      },
      "supports": "定义与条件的公开依据。采用 L1 条件期望及 RN 构造、塔式性质和 L2 投影；正文把指示函数到有界函数再到 L2 检验函数的延伸逐步写出。一般 RN 证明单元保留其 Hahn 分解依赖，未声称重新证明全部测度论。",
      "title": "Probability Theory: STAT310/MATH230",
      "authors": [
        "Amir Dembo"
      ],
      "version": "2021-04-15（PDF 标题页；不等同于此前规划的 2019 版）"
    }
  ],
  "optional_readings": [],
  "runtime_reading_log": [],
  "supplied_inputs": {
    "content_version": "2026-09-21-QT-B-review-v3",
    "experiment": {
      "id": "EXP-QTB-MEASURABLE-01",
      "title": "信息能否辨认这个函数",
      "anchor": "qt09-partition",
      "description": "可测性先看函数在每组是否恒定，而不是先看哪个组的概率很小。",
      "inputs": {
        "identity": "教学构造；真实产品条款仅用于支付乘数和结算值的解释，不是历史行情、样本频率或风险中性模型",
        "shared_file": "shared_inputs.json",
        "paths": [
          "shared_case.partitions",
          "defaults.measurable"
        ],
        "values": {
          "shared_case": {
            "id": "EXP-COND-01",
            "identity": "教学构造；真实产品条款仅用于支付乘数和结算值的解释，不是历史行情、样本频率或风险中性模型",
            "state_ids": [
              "w1",
              "w2",
              "w3",
              "w4"
            ],
            "probabilities": [
              0.1,
              0.3,
              0.4,
              0.2
            ],
            "signals": [
              "L",
              "L",
              "H",
              "H"
            ],
            "settlement_points": [
              5900,
              6000,
              6100,
              6200
            ],
            "strike_points": 6000,
            "multiplier_usd_per_point": 100,
            "contracts": 1,
            "payoff_type": "call",
            "probability_measure": "P_model",
            "clocks": {
              "t0": "未观察信号",
              "t1": "模型约定已收到 L/H 信号",
              "t2": "合约规定结算值确定；支付金额可计算",
              "cash_delivery": "与 t2 的金额确定区分，不在本实验建立账户到账/交收模型"
            },
            "partitions": {
              "trivial": [
                [
                  0,
                  1,
                  2,
                  3
                ]
              ],
              "signal": [
                [
                  0,
                  1
                ],
                [
                  2,
                  3
                ]
              ],
              "full": [
                [
                  0
                ],
                [
                  1
                ],
                [
                  2
                ],
                [
                  3
                ]
              ],
              "cross": [
                [
                  0,
                  2
                ],
                [
                  1,
                  3
                ]
              ]
            },
            "partition_labels": {
              "trivial": "尚无信号",
              "signal": "只知 L/H",
              "full": "完整状态",
              "cross": "交叉分组 {1,3}/{2,4}"
            },
            "events": {
              "H": [
                2,
                3
              ],
              "T": [
                3
              ],
              "L": [
                0,
                1
              ],
              "all": [
                0,
                1,
                2,
                3
              ],
              "empty": []
            },
            "case_source_ids": [
              "QTB-CBOE-SPX"
            ]
          },
          "defaults": {
            "events": {
              "event": "T",
              "given": "H"
            },
            "conditional": {
              "partition": "signal"
            },
            "measurable": {
              "partition": "signal",
              "candidate": [
                0,
                1,
                2,
                2
              ]
            },
            "projection": {
              "partition": "signal",
              "candidate_usd": [
                0,
                0,
                10000,
                10000
              ],
              "outer_partition": "trivial"
            },
            "rn": {
              "partition": "signal",
              "shift_usd": 15000
            },
            "integration": {
              "pareto_alpha": 1.5,
              "cap": 100,
              "spike_n": 10,
              "probe_x": 0.02,
              "dyadic_n": 4
            }
          }
        }
      },
      "outputs": {
        "measurable": false,
        "violations": [
          [
            0,
            1
          ]
        ],
        "events": [
          [],
          [
            0,
            1
          ],
          [
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ]
      },
      "algorithm": "枚举各组的所有并；用严格相等检查候选在每组恒定，独立于该组概率；列出违例分组。",
      "units": {
        "probabilities": "dimensionless",
        "settlement_points": "index points (contract-defined settlement value)",
        "payoff": "USD / one contract",
        "expectation": "USD",
        "mse": "USD^2",
        "pareto": "dimensionless pedagogical variable, support [1,infinity)",
        "spike": "dimensionless on (0,1) with Lebesgue probability"
      },
      "boundaries": [
        "零概率组同样检查可测性",
        "可测性使用同组值严格相等，而积分验算另用浮点容差",
        "事件枚举最多12组，当前4状态"
      ],
      "static_equivalent": {
        "body_anchor": "qt09-partition",
        "default_table_html": "<div class=\"table-wrap\"><table><thead><tr><th>信息</th><th>事件数</th><th>(0,1,2,2) 可测？</th></tr></thead><tbody><tr><td>平凡信息</td><td>2</td><td>否</td></tr><tr><td>L/H</td><td>4</td><td>否；L 组中取值不同</td></tr><tr><td>完整状态</td><td>16</td><td>是</td></tr></tbody></table></div><p>零概率不会改变函数是否在每个信息组内恒定的检查。</p>",
        "scope": "默认及正文列明的迁移算例，不依赖点击状态"
      },
      "execution": {
        "status": "executed",
        "runtime": "Chromium 144.0.7559.96 JavaScript in author sandbox",
        "independent_check": "Python Fraction 的共同四状态精确结果；其余边界见验证报告"
      },
      "source_ids": [
        "QTB-MIT-EVENTS",
        "QTB-DEMBO-2021"
      ]
    },
    "shared_inputs": {
      "schema_version": "qt-b-inputs-1",
      "content_version": "2026-09-21-QT-B-review-v3",
      "shared_case": {
        "id": "EXP-COND-01",
        "identity": "教学构造；真实产品条款仅用于支付乘数和结算值的解释，不是历史行情、样本频率或风险中性模型",
        "state_ids": [
          "w1",
          "w2",
          "w3",
          "w4"
        ],
        "probabilities": [
          0.1,
          0.3,
          0.4,
          0.2
        ],
        "signals": [
          "L",
          "L",
          "H",
          "H"
        ],
        "settlement_points": [
          5900,
          6000,
          6100,
          6200
        ],
        "strike_points": 6000,
        "multiplier_usd_per_point": 100,
        "contracts": 1,
        "payoff_type": "call",
        "probability_measure": "P_model",
        "clocks": {
          "t0": "未观察信号",
          "t1": "模型约定已收到 L/H 信号",
          "t2": "合约规定结算值确定；支付金额可计算",
          "cash_delivery": "与 t2 的金额确定区分，不在本实验建立账户到账/交收模型"
        },
        "partitions": {
          "trivial": [
            [
              0,
              1,
              2,
              3
            ]
          ],
          "signal": [
            [
              0,
              1
            ],
            [
              2,
              3
            ]
          ],
          "full": [
            [
              0
            ],
            [
              1
            ],
            [
              2
            ],
            [
              3
            ]
          ],
          "cross": [
            [
              0,
              2
            ],
            [
              1,
              3
            ]
          ]
        },
        "partition_labels": {
          "trivial": "尚无信号",
          "signal": "只知 L/H",
          "full": "完整状态",
          "cross": "交叉分组 {1,3}/{2,4}"
        },
        "events": {
          "H": [
            2,
            3
          ],
          "T": [
            3
          ],
          "L": [
            0,
            1
          ],
          "all": [
            0,
            1,
            2,
            3
          ],
          "empty": []
        },
        "case_source_ids": [
          "QTB-CBOE-SPX"
        ]
      },
      "defaults": {
        "events": {
          "event": "T",
          "given": "H"
        },
        "conditional": {
          "partition": "signal"
        },
        "measurable": {
          "partition": "signal",
          "candidate": [
            0,
            1,
            2,
            2
          ]
        },
        "projection": {
          "partition": "signal",
          "candidate_usd": [
            0,
            0,
            10000,
            10000
          ],
          "outer_partition": "trivial"
        },
        "rn": {
          "partition": "signal",
          "shift_usd": 15000
        },
        "integration": {
          "pareto_alpha": 1.5,
          "cap": 100,
          "spike_n": 10,
          "probe_x": 0.02,
          "dyadic_n": 4
        }
      },
      "named_cases": {
        "zero_group": {
          "probabilities": [
            0.25,
            0.75,
            0,
            0
          ],
          "partition": "signal"
        },
        "zero_atom": {
          "probabilities": [
            0.25,
            0.75,
            0,
            0
          ],
          "partition": "full"
        },
        "bad_future_candidate": {
          "candidate_usd": [
            0,
            0,
            10000,
            20000
          ]
        },
        "cap_grid": [
          10,
          100,
          1000
        ],
        "alpha_grid": [
          0.8,
          1,
          1.5,
          3
        ],
        "integration_spike_grid": [
          2,
          10,
          100
        ],
        "mse_probe_grid_usd": [
          0,
          10000,
          13333.333333333334,
          20000
        ],
        "dyadic_grid": [
          1,
          2,
          4,
          8
        ]
      },
      "units": {
        "probabilities": "dimensionless",
        "settlement_points": "index points (contract-defined settlement value)",
        "payoff": "USD / one contract",
        "expectation": "USD",
        "mse": "USD^2",
        "pareto": "dimensionless pedagogical variable, support [1,infinity)",
        "spike": "dimensionless on (0,1) with Lebesgue probability"
      },
      "numerical_policy": {
        "probability_sum_absolute_tolerance": 1e-12,
        "calculation_relative_tolerance": 1e-10,
        "calculation_absolute_tolerance": 1e-08,
        "null_group": "mass exactly zero; conditional ratios undefined; canonical CE constant 0 on that group; never infer impossibility outside the model",
        "normalization": "invalid probabilities rejected, never silently normalized",
        "finite": "all finite numeric inputs required",
        "identity": "fixed finite model; no stochastic simulation and no financial return denominator used"
      }
    },
    "body_scope": "完整正文、静态核算与题目解析"
  }
}
```

## Supplied entry
“目前知道什么”不必等于“目前知道哪个精确数值”。我们可以只知道结果落在哪一组，却仍能判断许多事件有没有发生。可测空间先记录哪些问题可以被辨认，概率测度随后给这些事件赋予权重；两层不要混在一起。

<a id="qt09-definitions"></a>
## 定义与必要约定

令 $\Omega$ 为集合，则 $\Omega$ 上的 **$\sigma$-代数**是指集合族 $\mathcal F\subseteq\mathcal P(\Omega)$，满足 $\Omega\in\mathcal F$、对补集封闭、对可数并封闭。由 De Morgan 律，它也对可数交封闭。二元组 $(\Omega,\mathcal F)$ 称为**可测空间**；$\mathcal F$ 中的集合称为可测集，在概率语境中称为事件。[^sigma]

令 $(\Omega,\mathcal F)$ 为可测空间，则其上的**概率测度**是函数 $P:\mathcal F\to[0,1]$，满足 $P(\Omega)=1$、$P(\varnothing)=0$，并对两两不交的事件 $A_n$ 满足
$$
P\!\left(\bigcup_{n\ge1}A_n\right)=\sum_{n\ge1}P(A_n).
$$
对一般测度可允许总质量不是 $1$，甚至为无穷。是否可测由集合族决定；概率是多少由测度决定。[^measure]

实线的 Borel $\sigma$-代数 $\mathcal B(\mathbb R)$ 由开集生成。实值函数 $Y$ **对 $\mathcal G$ 可测**，是指对每个 $B\in\mathcal B(\mathbb R)$，都有 $Y^{-1}(B)\in\mathcal G$。检验时可只看半直线原像 $\{Y\le a\}$，因为这些半直线生成 Borel $\sigma$-代数，原像运算又保留补集与可数并。[^measurable]

<a id="qt09-partition"></a>
## 有限信息：究竟包含哪些集合

取 $\Omega=\{\omega_1,\omega_2,\omega_3,\omega_4\}$，只知道信号
$$
L=\{\omega_1,\omega_2\},\qquad H=\{\omega_3,\omega_4\}.
$$
则可以判断 $L$ 或 $H$ 是否发生，也能判断空集与全空间，但不能区分 $H$ 内的两个结果。相应信息就是
$$
\mathcal G=\{\varnothing,L,H,\Omega\}.
$$
用更一般的记号说，它是包含 $L,H$ 的最小 $\sigma$-代数 $\sigma(L,H)$。[^information]

**有限分区命题。** 若非空集合 $A_1,\ldots,A_m$ 两两不交并覆盖 $\Omega$，则 $\sigma(A_1,\ldots,A_m)$ 恰是这些组的所有并集。

**证明。** 所有组的并集组成的集合族包含各 $A_i$，对补集、可数并均封闭，所以是一个 $\sigma$-代数；由最小性，生成的 $\sigma$-代数包含在它里面。反过来，任何包含各 $A_i$ 的 $\sigma$-代数也包含它们的所有并集。因此两者相等。

这一步说明，分组不是随手画的图：每个框就是当前信息不能再拆开的一个最小非空事件，称为原子。全部可辨认事件从这些原子组成。

**可测性判据。** 对上述有限分区，实值 $Y$ 可测，当且仅当它在每个原子上为常数。

**证明。** 若同组中两个函数值不同，选一个夹在它们之间的阈值 $a$，则 $\{Y\le a\}$ 会把这个原子拆开，不能是原子的并，故不可测。反过来，若每组取常数，则任意 Borel 集的原像都是某些组的并，因而可测。

例如 $Y=(0,1,2,2)$ 对 $L/H$ 信息不可测，因为 $L$ 中两个值不同。若完整状态已知，对应事件域是 $\mathcal P(\Omega)$，这个函数就可测。改变的不是函数，而是允许使用的信息。

<div data-experiment-slot="EXP-QTB-MEASURABLE-01"></div>

| 信息分组 | 全部可辨认事件个数 | $Y=(0,1,2,2)$ 是否可测 |
|---|---:|---|
| $\{\Omega\}$ | $2$ | 否 |
| $\{L,H\}$ | $4$ | 否 |
| 四个单点 | $16$ | 是 |

这个判据正是 [有限条件期望](https://ou-liu-red-sugar.github.io/zh/notebook/conditional-expectation/) 要求“同组取同一条件平均”的原因。它不保证预测准确，只保证函数没有要求区分当时尚无法区分的状态。

<a id="qt09-null"></a>
## 零概率、几乎处处与版本

若 $N\in\mathcal F$ 且 $P(N)=0$，称 $N$ 为零概率事件。性质在某个可测零概率事件以外都成立，就称**几乎处处成立**，在概率空间也称几乎必然成立，记作 a.s.。零概率事件不必是空集；例如给四状态中的某个状态赋零权重，并没有把这个状态从集合里删除。[^measure]

但不能由此推断“零概率部分怎样修改都可测”。看一个更小的例子：
$$
\Omega=\{a,b,c\},\quad
\mathcal G=\{\varnothing,\{a,b\},\{c\},\Omega\},\quad
P(\{a,b\})=0,\quad P(\{c\})=1.
$$
函数 $f=(0,0,1)$ 对 $\mathcal G$ 可测；函数 $g=(0,1,1)$ 只在零概率组里改了值，却不再可测，因为它把 $\{a,b\}$ 拆开了。

如果测度空间包含所有可测零测集的子集，就称为完备空间；在未假设完备时，任意修改可能离开原来的可测函数类。即使较大的事件域 $\mathcal F$ 完备，某个较小的信息域 $\mathcal G$ 也不必自动包含这些子集。因此条件期望的“版本”必须先满足所要求的 $\mathcal G$-可测性，再谈只差一个零概率事件。[^completion]

<a id="qt09-tests"></a>
## 自检与解析

**题 1。** 在四状态 $L/H$ 信息下，$Y=(3,3,7,7)$ 是否可测？$\{Y>5\}$ 是哪个事件？若四状态概率改为 $(0.25,0.75,0,0)$，答案是否改变？

**解析。** 它在每组取常数，故可测；$\{Y>5\}=H$。概率改动不改变这个结论，因为可测性只看函数和事件域。但新的概率会改变期望以及“哪些差异可以忽略至 a.s.”。

**题 2。** $\mathcal G_1$ 由 $\{1,2\},\{3,4\}$ 生成，$\mathcal G_2$ 由 $\{1,3\},\{2,4\}$ 生成。是否可以说 $\mathcal G_2$ 比 $\mathcal G_1$ 信息更多？

**解析。** 不能。$\{1,2\}$ 属于 $\mathcal G_1$ 而不属于 $\mathcal G_2$，$\{1,3\}$ 反过来；二者互不包含。这也是使用 [塔式性质](https://ou-liu-red-sugar.github.io/zh/notebook/conditional-expectation-projection/) 前必须检查嵌套条件的原因。

[^sigma]: MIT, *Lecture 1: Probabilistic Models and Probability Measures*，Fall 2018，§4、pp.4–6：$\sigma$-代数、生成和 Borel 例。[公开讲义](https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/c37dc8b61cdf6bde689a627bfa5b4942_MIT6_436JF18_lec01.pdf)。
[^measure]: 同讲义 §5、pp.7–8：测度、概率测度和 a.s. 与处处的区别。
[^information]: 同讲义 §4.1、p.7：较小 $\sigma$-代数与部分观测。本文的有限命题及判据给出了所用情形的完整证明。
[^measurable]: Amir Dembo, *Probability Theory: STAT310/MATH230*，2021-04-15 版，§1.2.1 的 Definition 1.2.1–1.2.2（p.18）给出可测映射/随机变量定义；Theorem 1.2.9 与 Definition 1.2.12（p.20）给出在生成族上检查原像以及实值随机变量的半直线判据。[作者公开讲义](https://adembo.su.domains/stat-310b/lnotes.pdf)。
[^completion]: Dembo，同版，§4.1.1、pp.153–156 对版本和 $\mathcal G$-可测性的要求。本文三点例直接显示：忽略零概率差异不能代替可测性检查。


## Experiment inputs and static equivalents
```json
[
  {
    "id": "EXP-QTB-MEASURABLE-01",
    "title": "信息能否辨认这个函数",
    "anchor": "qt09-partition",
    "description": "可测性先看函数在每组是否恒定，而不是先看哪个组的概率很小。",
    "inputs": {
      "identity": "教学构造；真实产品条款仅用于支付乘数和结算值的解释，不是历史行情、样本频率或风险中性模型",
      "shared_file": "shared_inputs.json",
      "paths": [
        "shared_case.partitions",
        "defaults.measurable"
      ],
      "values": {
        "shared_case": {
          "id": "EXP-COND-01",
          "identity": "教学构造；真实产品条款仅用于支付乘数和结算值的解释，不是历史行情、样本频率或风险中性模型",
          "state_ids": [
            "w1",
            "w2",
            "w3",
            "w4"
          ],
          "probabilities": [
            0.1,
            0.3,
            0.4,
            0.2
          ],
          "signals": [
            "L",
            "L",
            "H",
            "H"
          ],
          "settlement_points": [
            5900,
            6000,
            6100,
            6200
          ],
          "strike_points": 6000,
          "multiplier_usd_per_point": 100,
          "contracts": 1,
          "payoff_type": "call",
          "probability_measure": "P_model",
          "clocks": {
            "t0": "未观察信号",
            "t1": "模型约定已收到 L/H 信号",
            "t2": "合约规定结算值确定；支付金额可计算",
            "cash_delivery": "与 t2 的金额确定区分，不在本实验建立账户到账/交收模型"
          },
          "partitions": {
            "trivial": [
              [
                0,
                1,
                2,
                3
              ]
            ],
            "signal": [
              [
                0,
                1
              ],
              [
                2,
                3
              ]
            ],
            "full": [
              [
                0
              ],
              [
                1
              ],
              [
                2
              ],
              [
                3
              ]
            ],
            "cross": [
              [
                0,
                2
              ],
              [
                1,
                3
              ]
            ]
          },
          "partition_labels": {
            "trivial": "尚无信号",
            "signal": "只知 L/H",
            "full": "完整状态",
            "cross": "交叉分组 {1,3}/{2,4}"
          },
          "events": {
            "H": [
              2,
              3
            ],
            "T": [
              3
            ],
            "L": [
              0,
              1
            ],
            "all": [
              0,
              1,
              2,
              3
            ],
            "empty": []
          },
          "case_source_ids": [
            "QTB-CBOE-SPX"
          ]
        },
        "defaults": {
          "events": {
            "event": "T",
            "given": "H"
          },
          "conditional": {
            "partition": "signal"
          },
          "measurable": {
            "partition": "signal",
            "candidate": [
              0,
              1,
              2,
              2
            ]
          },
          "projection": {
            "partition": "signal",
            "candidate_usd": [
              0,
              0,
              10000,
              10000
            ],
            "outer_partition": "trivial"
          },
          "rn": {
            "partition": "signal",
            "shift_usd": 15000
          },
          "integration": {
            "pareto_alpha": 1.5,
            "cap": 100,
            "spike_n": 10,
            "probe_x": 0.02,
            "dyadic_n": 4
          }
        }
      }
    },
    "outputs": {
      "measurable": false,
      "violations": [
        [
          0,
          1
        ]
      ],
      "events": [
        [],
        [
          0,
          1
        ],
        [
          2,
          3
        ],
        [
          0,
          1,
          2,
          3
        ]
      ]
    },
    "algorithm": "枚举各组的所有并；用严格相等检查候选在每组恒定，独立于该组概率；列出违例分组。",
    "units": {
      "probabilities": "dimensionless",
      "settlement_points": "index points (contract-defined settlement value)",
      "payoff": "USD / one contract",
      "expectation": "USD",
      "mse": "USD^2",
      "pareto": "dimensionless pedagogical variable, support [1,infinity)",
      "spike": "dimensionless on (0,1) with Lebesgue probability"
    },
    "boundaries": [
      "零概率组同样检查可测性",
      "可测性使用同组值严格相等，而积分验算另用浮点容差",
      "事件枚举最多12组，当前4状态"
    ],
    "static_equivalent": {
      "body_anchor": "qt09-partition",
      "default_table_html": "<div class=\"table-wrap\"><table><thead><tr><th>信息</th><th>事件数</th><th>(0,1,2,2) 可测？</th></tr></thead><tbody><tr><td>平凡信息</td><td>2</td><td>否</td></tr><tr><td>L/H</td><td>4</td><td>否；L 组中取值不同</td></tr><tr><td>完整状态</td><td>16</td><td>是</td></tr></tbody></table></div><p>零概率不会改变函数是否在每个信息组内恒定的检查。</p>",
      "scope": "默认及正文列明的迁移算例，不依赖点击状态"
    },
    "execution": {
      "status": "executed",
      "runtime": "Chromium 144.0.7559.96 JavaScript in author sandbox",
      "independent_check": "Python Fraction 的共同四状态精确结果；其余边界见验证报告"
    },
    "source_ids": [
      "QTB-MIT-EVENTS",
      "QTB-DEMBO-2021"
    ]
  }
]
```

## Sources
- [Probability Theory: STAT310/MATH230](https://adembo.su.domains/stat-310b/lnotes.pdf): 定义与条件的公开依据。采用 L1 条件期望及 RN 构造、塔式性质和 L2 投影；正文把指示函数到有界函数再到 L2 检验函数的延伸逐步写出。一般 RN 证明单元保留其 Hahn 分解依赖，未声称重新证明全部测度论。
- [Lecture 1: Probabilistic Models and Probability Measures](https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/c37dc8b61cdf6bde689a627bfa5b4942_MIT6_436JF18_lec01.pdf): 支持 sigma-代数、生成事件域、部分观测的信息解释、概率测度与几乎必然。本文的有限分区及可测性判据给出独立完整证明。

## Content relations
```json
[
  {
    "from": "zh-qt09",
    "relation": "part_of",
    "to": "quant-foundations",
    "reason": "主要 topic 归属"
  },
  {
    "from": "zh-qt09",
    "relation": "supported_by",
    "to": "QTB-MIT-EVENTS",
    "reason": "支持本篇采用的定义、条件或真实产品条款",
    "locator": "§4–§5 开头，pp.4–8",
    "scope": "支持 sigma-代数、生成事件域、部分观测的信息解释、概率测度与几乎必然。本文的有限分区及可测性判据给出独立完整证明。"
  },
  {
    "from": "zh-qt09",
    "relation": "supported_by",
    "to": "QTB-DEMBO-2021",
    "reason": "支持本篇采用的定义、条件或真实产品条款",
    "locator": "§1.2.1：Definition 1.2.1–1.2.2 p.18；Theorem 1.2.9、Definition 1.2.12 p.20；§4.1.1 pp.153–156",
    "scope": "定义与条件的公开依据。采用 L1 条件期望及 RN 构造、塔式性质和 L2 投影；正文把指示函数到有界函数再到 L2 检验函数的延伸逐步写出。一般 RN 证明单元保留其 Hahn 分解依赖，未声称重新证明全部测度论。"
  },
  {
    "from": "qt09-partition",
    "relation": "illustrated_by",
    "to": "EXP-QTB-MEASURABLE-01",
    "reason": "可测性先看函数在每组是否恒定，而不是先看哪个组的概率很小。"
  },
  {
    "from": "zh-qt09",
    "relation": "informs",
    "to": "zh-qt11",
    "reason": "提供哪些候选可由当前信息辨认的定义"
  }
]
```

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