23. Measures
本章主题：测度。 测度把一个非负实数赋给每个合适的子集，直观上是该子集的"大小"，是长度/面积/体积的推广。§23.1 定义：定义 23.1 测度 \(\mu:\Sigma\to\mathbb R\)（非负性、空集为零、可数可加性 \(\mu(\bigcup E_k)=\sum\mu(E_k)\)）；定义 23.2 \(\sigma\) 代数（含 \(\varnothing\)、对补与可数并封闭；最小 \(\{\varnothing,X\}\)、最大幂集 \(2^X\)）；定义 23.3 Borel \(\sigma\) 代数（含 \(\mathbb R^n\) 所有开集）；定义 23.4 可测空间 \((X,\Sigma)\)；定义 23.5 测度空间 \((X,\Sigma,\mu)\)；定义 23.6 概率测度（\(\mu(X)=1\)）与概率空间。§23.2 例：掷两骰子（\(\mathcal F_0\subseteq\mathcal F_1\subseteq\mathcal F_2\) 表示各时点已知信息、\(\mu(A)=\frac{\text{card}(A)}{36}\)）；计数测度；勒贝格测度（外测度 \(\lambda^\star(E)=\inf\sum\ell(I_k)\)、Carathéodory 准则定义勒贝格 \(\sigma\) 代数、存在不可测集故严格小于幂集）。
Chapter theme: measures. A measure assigns a non-negative real number to each suitable subset, intuitively the "size" of that subset, generalizing length / area / volume. §23.1 Definitions: Definition 23.1 measure \(\mu:\Sigma\to\mathbb R\) (non-negativity, null empty set, countable additivity \(\mu(\bigcup E_k)=\sum\mu(E_k)\)); Definition 23.2 \(\sigma\)-algebra (contains \(\varnothing\), closed under complement and countable union; smallest \(\{\varnothing,X\}\), biggest power set \(2^X\)); Definition 23.3 Borel \(\sigma\)-algebra (contains all open sets in \(\mathbb R^n\)); Definition 23.4 measurable space \((X,\Sigma)\); Definition 23.5 measure space \((X,\Sigma,\mu)\); Definition 23.6 probability measure (\(\mu(X)=1\)) and probability space. §23.2 Examples: rolling two dice (\(\mathcal F_0\subseteq\mathcal F_1\subseteq\mathcal F_2\) as information known at each time, \(\mu(A)=\frac{\text{card}(A)}{36}\)); counting measure; Lebesgue measure (outer measure \(\lambda^\star(E)=\inf\sum\ell(I_k)\), the Carathéodory criterion defining the Lebesgue \(\sigma\)-algebra, the existence of non-measurable sets so it is strictly smaller than the power set).
23.1 Definitions
测度定义在一个集合上，用以给每个合适的子集赋一个非负实数。直观上，这个非负数可解释为该子集的大小。因此，测度是长度、面积与体积的推广概念。
下面正式给出测度、可测空间、测度空间、概率测度与概率空间的定义。
定义 23.1（测度 Measure） 设 \(X\) 为一个集合，\(\Sigma\) 为 \(X\) 上的一个 \(\sigma\) 代数。函数 \(\mu:\Sigma\to\mathbb R\) 称为测度，若它具有如下性质： 1. 非负性：对 \(\Sigma\) 中所有 \(E\)，\(\mu(E)\ge0\)。 2. 空集为零：\(\mu(\varnothing)=0\)。 3. 可数可加性（或 \(\sigma\) 可加性）：对 \(\Sigma\) 中所有由两两不交集合构成的可数族 \(\{E_i\}_{i=1}^\infty\)， $$\mu\!\left(\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty\mu(E_k)$$
定义 23.2（\(\sigma\) 代数 \(\sigma\)-algebra） \(\Sigma\) 是 \(X\) 上的一个 \(\sigma\) 代数，若： - \(\varnothing\in\Sigma\) 或 \(X\in\Sigma\)； - \(A\in\Sigma\) 蕴含 \(A^c=X\setminus A\in\Sigma\)； - \(A_i\in\Sigma\)（\(i=1,2,3,\dots\)）蕴含 \(\bigcup_{i=1}^\infty A_i\in\Sigma\)。
注意 \(X\) 上最小的 \(\sigma\) 代数是 \(\{\varnothing,X\}\)、最大的 \(\sigma\) 代数是 \(X\) 的幂集 \(2^X\)（它包含 \(X\) 的每个子集）。二者都说成是 \(X\) 上的平凡 \(\sigma\) 代数。
定义 23.3（Borel \(\sigma\) 代数 Borel \(\sigma\)-algebra） \(\mathcal B(\mathbb R^n)\) 是 \(\mathbb R^n\) 上的一个 Borel \(\sigma\) 代数，若它包含 \(\mathbb R^n\) 中所有开集作为元素。
A measure is defined on a set to assign a non-negative real number to each suitable subset. Intuitively, the non-negative number can be interpreted as the size of that subset. Therefore, measure is a generalized concept of length, area and volume.
Formally, below are the definitions of measure, measurable space, measure space, probability measure and probability space.
Definition 23.1 (Measure) Let \(X\) be a set and let \(\Sigma\) be a \(\sigma\)-algebra over \(X\). A function \(\mu:\Sigma\to\mathbb R\) is called a measure if it has the following properties: 1. Non-negativity: for all \(E\) in \(\Sigma\), \(\mu(E)\ge0\). 2. Null empty set: \(\mu(\varnothing)=0\). 3. Countable additivity (or \(\sigma\)-additivity): for all countable collections \(\{E_i\}_{i=1}^\infty\) of pairwise disjoint sets in \(\Sigma\), $$\mu\!\left(\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty\mu(E_k)$$
Definition 23.2 (\(\sigma\)-algebra) \(\Sigma\) is a \(\sigma\)-algebra over \(X\) if: - \(\varnothing\in\Sigma\) or \(X\in\Sigma\); - \(A\in\Sigma\) implies \(A^c=X\setminus A\in\Sigma\); - \(A_i\in\Sigma\) (\(i=1,2,3,\dots\)) implies \(\bigcup_{i=1}^\infty A_i\in\Sigma\).
Note that the smallest \(\sigma\)-algebra over \(X\) is \(\{\varnothing,X\}\) and the biggest \(\sigma\)-algebra over \(X\) is the power set of \(X\), i.e. \(2^X\) which includes every subset of \(X\). Both of them are said to be trivial \(\sigma\)-algebra over \(X\).
Definition 23.3 (Borel \(\sigma\)-algebra) \(\mathcal B(\mathbb R^n)\) is a Borel \(\sigma\)-algebra over \(\mathbb R^n\) if it contains all the open sets in \(\mathbb R^n\) as elements.
定义 23.4（可测空间 Measurable Space） 一对 \((X,\Sigma)\) 称为可测空间，\(\Sigma\) 的成员称为可测集。
定义 23.5（测度空间 Measure Space） 一个三元组 \((X,\Sigma,\mu)\) 称为测度空间。
定义 23.6（概率测度与概率空间 Probability Measure and Probability Space） 若总测度（即 \(X\) 上的测度）为 1，即 \(\mu(X)=1\)，则称该测度为概率测度。带有概率测度的测度空间称为概率空间。
23.2 Examples
下面是关于 \(\sigma\) 代数的一个例子。
例 23.1（掷两枚骰子） 考虑如下测度空间 \((\Omega,\mathcal F,\mu)\)： - \(\Omega=\{\omega=(x,y):x,y\in\{1,2,\dots,6\}\}\)。 - \(\mathcal F\)： - \(\mathcal F_0=\{\varnothing,\Omega\}\)； - \(\mathcal F_1=\{A=A_x\times\{1,2,\dots,6\}:A_x\subseteq\{1,2,\dots,6\}\}\)； - \(\mathcal F_2=\{A:A\subseteq\Omega\}\)。
事实上，\(\mathcal F_0\)、\(\mathcal F_1\)、\(\mathcal F_2\) 都是 \(\Omega\) 上的 \(\sigma\) 代数，这由 \(\sigma\) 代数的定义不难验证。
此外，注意 \(\mathcal F_0\subseteq\mathcal F_1\subseteq\mathcal F_2\)。
而 \(\mathcal F_t\)（\(t=0,1,2\)）可解释为时点 \(t\) 已知事件的信息：我们在 \(t=1\) 掷第一枚骰子、在 \(t=2\) 掷第二枚骰子。
\(\mu(A)=\frac{\text{card}(A)}{36}\)，这是一个概率测度。
Definition 23.4 (Measurable Space) A pair \((X,\Sigma)\) is called a measurable space, and the members of \(\Sigma\) are called measurable sets.
Definition 23.5 (Measure Space) A triple \((X,\Sigma,\mu)\) is called a measure space.
Definition 23.6 (Probability Measure and Probability Space) If the total measure (the measure on set \(X\)) is one, i.e. \(\mu(X)=1\), then we call the measure a probability measure. A measure space with a probability measure is called a probability space.
23.2 Examples
Below is an example for \(\sigma\)-algebra.
Example 23.1 (Rolling two dice) Consider the following measure space \((\Omega,\mathcal F,\mu)\): - \(\Omega=\{\omega=(x,y):x,y\in\{1,2,\dots,6\}\}\). - \(\mathcal F\): - \(\mathcal F_0=\{\varnothing,\Omega\}\); - \(\mathcal F_1=\{A=A_x\times\{1,2,\dots,6\}:A_x\subseteq\{1,2,\dots,6\}\}\); - \(\mathcal F_2=\{A:A\subseteq\Omega\}\).
In fact, \(\mathcal F_0\), \(\mathcal F_1\), \(\mathcal F_2\) are all \(\sigma\)-algebra over \(\Omega\), which is not hard to prove by the definition of \(\sigma\)-algebra.
Moreover, notice that \(\mathcal F_0\subseteq\mathcal F_1\subseteq\mathcal F_2\).
And \(\mathcal F_t\) (\(t=0,1,2\)) can be interpreted as the information for the events known at \(t\): we roll the first dice at \(t=1\) and roll the second dice at \(t=2\).
\(\mu(A)=\frac{\text{card}(A)}{36}\), which is a probability measure.
下面是几个常用测度。
例 23.2（计数测度 Counting measure） 计数测度定义为 $$\mu(A)=\begin{cases}\text{card}(A)&\text{if }A\text{ is finite}\\+\infty&\text{if }A\text{ is infinite}\end{cases}$$
例 23.3（勒贝格测度 Lebesgue measure） \(\mathbb R\) 上的勒贝格测度：给定子集 \(E\subseteq\mathbb R\)，区间 \(I=[a,b]\)（或 \(I=(a,b)\)）的长度由 \(\ell(I)=b-a\) 给出，则勒贝格外测度 \(\lambda^\star(E)\) 定义为 $$\lambda^\star(E)=\inf\left\{\sum_{k=1}^\infty\ell(I_k):(I_k)_{k\in\mathbb N}\text{ is a sequence of open intervals that satisfies }E\subseteq\bigcup_{k=1}^\infty I_k\right\}$$ 勒贝格测度定义在勒贝格 \(\sigma\) 代数上，它是满足"Carathéodory 准则"的所有集合 \(E\) 之集合：对每个 \(A\subseteq\mathbb R\)， $$\lambda^\star(A)=\lambda^\star\!\left(A\cap E\right)+\lambda^\star\!\left(A\cap E^c\right)$$ 对勒贝格 \(\sigma\) 代数中的任一集合，其勒贝格测度由其勒贝格外测度给出，即 \(\lambda(E)=\lambda^\star(E)\)。
（Birkhoff 大数定律）注意：不在勒贝格 \(\sigma\) 代数中的集合不是勒贝格可测的。这样的集合确实存在，即勒贝格 \(\sigma\) 代数严格包含于 \(\mathbb R\) 的幂集之中。
Below are some frequently used measures.
Example 23.2 (Counting measure) Counting measure is defined as $$\mu(A)=\begin{cases}\text{card}(A)&\text{if }A\text{ is finite}\\+\infty&\text{if }A\text{ is infinite}\end{cases}$$
Example 23.3 (Lebesgue measure) Lebesgue measure on \(\mathbb R\): Given a subset \(E\subseteq\mathbb R\), with the length of interval \(I=[a,b]\) (or \(I=(a,b)\)) given by \(\ell(I)=b-a\), the Lebesgue outer measure \(\lambda^\star(E)\) is defined as $$\lambda^\star(E)=\inf\left\{\sum_{k=1}^\infty\ell(I_k):(I_k)_{k\in\mathbb N}\text{ is a sequence of open intervals that satisfies }E\subseteq\bigcup_{k=1}^\infty I_k\right\}$$ The Lebesgue measure is defined on the Lebesgue \(\sigma\)-algebra, which is the collection of all sets \(E\) which satisfy the "Carathéodory criterion": for every \(A\subseteq\mathbb R\), $$\lambda^\star(A)=\lambda^\star\!\left(A\cap E\right)+\lambda^\star\!\left(A\cap E^c\right)$$ For any set in the Lebesgue \(\sigma\)-algebra, its Lebesgue measure is given by its Lebesgue outer measure, i.e. \(\lambda(E)=\lambda^\star(E)\).
(Birkhoff Law of Large Numbers) Note that sets that are not included in the Lebesgue \(\sigma\)-algebra are not Lebesgue-measurable. Such sets do exist, i.e. the Lebesgue \(\sigma\)-algebra is strictly contained in the power set of \(\mathbb R\).