我们接下来探讨的问题是: 独立随机变量是否是存在的? 为了严格地说明这一点, 钟开莱书上引入了乘积空间的概念, 而为了更方便理解, 以下使用的是程士宏书上的定义.

定义1 (乘积空间) 设\(n\geq 2\), \((\Omega_i,\mathscr{F}_i)(1\le i\le n)\)是可测空间, 记 \[ \prod_{i=1}^{n}\Omega_i:=\{(\omega_1,\omega_2,\cdots,\omega_n):\omega_1\in\Omega_1,\omega_2\in\Omega_2,\cdots,\omega_n\in\Omega_n\} \] 则\(\displaystyle\prod_{i=1}^{n}\Omega_i\)称为\(\Omega_1,\Omega_2,\cdots,\Omega_n\)的乘积空间; 再记 \[ \mathscr{L}:=\{A_1\times A_2\times\cdots\times A_n:A_1\in\mathscr{F}_1,A_2\in\mathscr{F}_2,\cdots,A_n\in\mathscr{F}_n\},\quad\prod_{i=1}^{n}\mathscr{F}_i:=\sigma(\mathscr{L}), \] 则\(\displaystyle\prod_{i=1}^{n}\mathscr{F}_i\)称为\(\mathscr{F}_1,\mathscr{F}_2,\cdots,\mathscr{F}_n\)的乘积, \(\left(\displaystyle\prod_{i=1}^{n}\Omega_i,\prod_{i=1}^{n}\mathscr{F}_i\right)\)称为乘积可测空间.

定义2 (投影) 设\(\displaystyle\prod_{i=1}^{n}\Omega_i\)是\(\Omega_1,\Omega_2,\cdots,\Omega_n\)的乘积空间, \(\omega_i\in\Omega_i\), 映射 \[ \pi_j(\omega_1,\omega_2,\cdots,\omega_n)=\omega_j \] 称为从\(\displaystyle\prod_{i=1}^{n}\Omega_i\)到\(\Omega_j\)的.

在上述定义的基础上, 可以在\((\Omega_i,\mathscr{F}_i)\)上赋予概率测度\(\mathbb{P}_i\). 如果我们考虑概率空间\((\Omega_i,\mathscr{F}_i,\mathbb{P}_i)(1\le i\le n)\)所生成的乘积可测空间\(\left(\displaystyle\prod_{i=1}^{n}\Omega_i,\prod_{i=1}^{n}\mathscr{F}_i\right)\), 自然会想在其上赋予概率测度. 然而, 这一点是比较麻烦的. 为了方便, 我们首先考虑两个概率空间\((\Omega_1,\mathscr{F}_1,\mathbb{P}_1)\)和\((\Omega_2,\mathscr{F}_2,\mathbb{P}_2)\)的情形, 并记它们生成的乘积空间为\((\Omega_1\times\Omega_2,\mathscr{F}_1\times\mathscr{F}_2)\). 我们不加证明地给出Fubini定理的内容, 以及有限维乘积概率空间的存在性.

定理3 (Fubini定理) 设\((\Omega_1,\mathscr{F}_1,\mathbb{P}_1)\)和\((\Omega_2,\mathscr{F}_2,\mathbb{P}_2)\)是概率空间.

  1. 在\((\Omega_1\times\Omega_2,\mathscr{F}_1\times\mathscr{F}_2)\)上存在唯一的概率\(\mathbb{P}_1\times\mathbb{P}_2\), 使得对任意的\(A_1\in\mathscr{F}_1\), \(A_2\in\mathscr{F}_2\), 都有 \[ (\mathbb{P}_1\times\mathbb{P}_2)(A_1\times A_2)=\mathbb{P}_1(A_1)\cdot\mathbb{P}_2(A_2). \] 这里的\(\mathbb{P}_1\times\mathbb{P}_2\)称为\(\mathbb{P}_1\)和\(\mathbb{P}_2\)的乘积, \((\Omega_1\times\Omega_2,\mathscr{F}_1\times\mathscr{F}_2,\mathbb{P}_1\times\mathbb{P}_2)\)称为乘积概率空间.

  2. 对\((\Omega_1\times\Omega_2,\mathscr{F}_1\times\mathscr{F}_2,\mathbb{P}_1\times\mathbb{P}_2)\)上的可积随机变量\(X(\omega_1,\omega_2)\), 有 \[ \begin{aligned} \int_{X_1\times X_2}X\mathrm{d}(\mathbb{P}_1\times\mathbb{P}_2) &=\int_{X_1}\mathbb{P}_1(\mathrm{d}\omega_1)\int_{X_2}X(\omega_1,\omega_2)\mathbb{P}_2(\mathrm{d}\omega_2)\\ &=\int_{X_2}\mathbb{P}_2(\mathrm{d}\omega_2)\int_{X_1}X(\omega_1,\omega_2)\mathbb{P}_1(\mathrm{d}\omega_1). \end{aligned} \]

推论 设\(n\geq 2\), \((\Omega_i,\mathscr{F}_i,\mathbb{P}_i)(1\le i\le n)\)是概率空间, 则在\(\left(\displaystyle\prod_{i=1}^{n}\Omega_i,\prod_{i=1}^{n}\mathscr{F}_i\right)\)上存在唯一的概率测度\(\displaystyle\prod_{i=1}^{n}\mathbb{P}_i\), 使得对任意的\(A_1\in\mathscr{F}_1, A_2\in\mathscr{F}_2, \cdots, A_n\in\mathscr{F}_n\), 都有 \[ \left(\prod_{i=1}^{n}\mathbb{P}_i\right)\left(\prod_{i=1}^{n}A_i\right)=\prod_{i=1}^{n}\mathbb{P}_i(A_i). \] 这里的\(\displaystyle\prod_{i=1}^{n}\mathbb{P}_i\)称为\(\mathbb{P}_1, \mathbb{P}_2, \cdots, \mathbb{P}_n\)的乘积, \(\left(\displaystyle\prod_{i=1}^{n}\Omega_i, \prod_{i=1}^{n}\mathscr{F}_i, \prod_{i=1}^{n}\mathbb{P}_i\right)\)称为乘积概率空间.

在乘积概率空间的基础上, 我们来说明独立随机变量的存在性.

例4 设\(n\geq 2\), \((\Omega_i,\mathscr{F}_i,\mathbb{P}_i)(1\le i\le n)\)是概率空间. 对\(1\le i\le n\), 设\(X_i\)是\((\Omega_i,\mathscr{F}_i,\mathbb{P}_i)\)上的随机变量, \(B_i\in\mathscr{F}_i\), 则有 \[ \left(\prod_{i=1}^{n}\mathbb{P}_i\right)\left(\prod_{i=1}^{n}(X_i\in B_i)\right)=\prod_{i=1}^{n}\mathbb{P}_i(X_i\in B_i). \] 设\(\omega_i\in\Omega_i\), 记\(\omega=(\omega_1,\omega_2,\cdots,\omega_n)\in\displaystyle\prod_{i=1}^{n}\Omega_i\), 考虑\(\left(\displaystyle\prod_{i=1}^{n}\Omega_i, \prod_{i=1}^{n}\mathscr{F}_i, \prod_{i=1}^{n}\mathbb{P}_i\right)\)上的随机变量 \[ \tilde{X}_i(\omega):=X_i(\omega_i),\quad i=1,2,\cdots,n, \] 再任取\(\tilde{B}_1, \tilde{B}_2, \cdots, \tilde{B}_n\in\displaystyle\prod_{i=1}^{n}\mathscr{F}_i\), 并设对任意的\(1\le i\le n\), \(\tilde{B}_i\)在\(\Omega_i\)上的投影为\(B_i\), 则有 \[ \begin{aligned} &\left(\prod_{i=1}^{n}\mathbb{P}_i\right)\left(\tilde{X}_1\in\tilde{B}_1,\tilde{X}_2\in\tilde{B}_2,\cdots,\tilde{X}_n\in\tilde{B}_n\right)\\ &=\left(\prod_{i=1}^{n}\mathbb{P}_i\right)\left(\prod_{j=1}^{n}(X_j\in B_j)\right)\\ &=\prod_{j=1}^{n}\mathbb{P}_j(X_j\in B_j)\\ &=\prod_{j=1}^{n}\left(\prod_{i=1}^{n}\mathbb{P}_i\right)\left(\tilde{X}_j\in\tilde{B}_j\right), \end{aligned} \] 因此\(\tilde{X}_1,\tilde{X}_2,\cdots,\tilde{X}_n\)是\(\left(\displaystyle\prod_{i=1}^{n}\Omega_i, \prod_{i=1}^{n}\mathscr{F}_i, \prod_{i=1}^{n}\mathbb{P}_i\right)\)上的独立随机变量.