If \((\Omega_1, \mathcal{F}_1)\) and \((\Omega_2, \mathcal{F}_2)\) are measurable spaces, then any set of the form
\(A_1\times A_2=\{(\omega_1,\omega_2): \omega_1\in A_1, \omega_2\in A_2\}\), with \(A_1\in \mathcal{F}_1\) and \(A_2\in \mathcal{F}_2\), is called a measurable rectangle.
The product \(\sigma\)-field, denoted \(\mathcal{F}_1\times \mathcal{F}_2\) is defined to be the \(\sigma\)-field on \(\Omega_1\times \Omega_2\) generated by the class of measurable rectangles. Thus if \[ \text{Rect}(\mathcal{F_1},\mathcal{F_2})=\{A_1\in \mathcal{F}_1, A_2\in \mathcal{F}_2\} \] represents the class of measurable rectangles, then \[ \mathcal{F_1}\times \mathcal{F_2}:=\sigma(\text{Rect}(\mathcal{F_1},\mathcal{F_2})). \] More generally, if \((\Omega_1,\mathcal{F}_1),\ldots,(\Omega_k,\mathcal{F}_k)\) are measurable spaces, then \[ \times_{i=1}^k\mathcal{F}_i:=\sigma(\text{Rect}(\mathcal{F_1},\ldots\mathcal{F_k})) \] where \[ \text{Rect}(\mathcal{F_1},\ldots,\mathcal{F_k})=\{A_1\times\cdots A_k: A_i\in \mathcal{F}_i,i=1,\ldots,k\}. \]
Suppose that \((\Omega_1,\mathcal{F}_1)\) and \((\Omega_2\mathcal{F}_2)\) are measurable spaces, and that \(\mathcal{A}_1\) and \(\mathcal{A}_2\) are generating classes for \(\mathcal{F}_1\) and \(\mathcal{F}_2\), respectively, with \(\Omega_1\in \mathcal{A}_1\), \(\Omega_2\in \mathcal{A}_2\). Then, \[ \mathcal{F_1}\times \mathcal{F_2}:=\sigma(\text{Rect}(\mathcal{A_1},\mathcal{A_2})). \] More generally, if \((\Omega_1,\mathcal{F}_1),\ldots,(\Omega_k,\mathcal{F}_k)\) are measurable spaces, with \(\mathcal{F}_i=\sigma(\mathcal{A}_i)\), and \(\Omega_i\in \mathcal{A}_i\), \(i=1,\ldots,n\), then \[ \times_{i=1}^k\mathcal{F}_i:=\sigma(\text{Rect}(\mathcal{A_1},\ldots\mathcal{A_k})) \]
If \((\Omega_1,\mathcal{F}_1),\ldots,(\Omega_{m+n},\mathcal{F}_{m+n})\) are measurable spaces, for some \(m,n\ge 1\), then \[ \times_{i=1}^{m+n}\mathcal{F}_i=\left(\times_{i=1}^{m}\mathcal{F}_i\right)\times \left(\times_{j=m+1}^{m+n}\mathcal{F}_i\right). \]
If \((\Omega,\mathcal{F}),(\Omega_1,\mathcal{F}_1)\) and \((\Omega_2,\mathcal{F}_2)\) are measurable spaces, and \(T:\Omega\rightarrow \Omega_1\times \Omega_2\), with \(T(\omega)=(T_1(\omega), T_2(\omega))\), then \(T\) is measurable-\(\mathcal{F}/(\mathcal{F}_1\times\mathcal{F}_2)\) if and only if each \(T_i\) is measurable-\(\mathcal{F}/(\mathcal{F}_i\), \(i=1,2\).
More generally, if \((\Omega,\mathcal{F}),(\Omega_1,\mathcal{F}_1)\ldots(\Omega_k,\mathcal{F}_k)\) are measurable spaces, then \(T\) is measurable-\(\mathcal{F}/(\mathcal{F}_1\times\cdots\times\mathcal{F}_k)\) if and only if each \(T_i\) is measurable-\(\mathcal{F}/(\mathcal{F}_i\), \(i=1,2,\ldots,k\).
If \((\Omega_1, \mathcal{F}_1, \mu_1)\) and \((\Omega_2, \mathcal{F}_2, \mu_2)\) are \(\sigma\)-finite measure spaces, then there exists a unique measure \(\mu\) on \(\mathcal{F}_1\times\mathcal{F}_2\) satisfying \[ \mu(A_1\times A_2)=\mu_1(A_1)\mu_2(A_2)\mbox{ }\mbox{ }\mbox{ }\mbox{ for all } A_1\in \mathcal{F}_1, A_2\in \mathcal{F_2}. \] Moreover, \(\mu\) is \(\sigma\)-finite on \(\mathcal{F}_1\times\mathcal{F}_2\).
This measure \(\mu\) in the theorem is called a product measure, and is usually denoted \(\mu_1\times \mu_2\). The measurable space \((\Omega_1\times\Omega_2, \mathcal{F}_1\times \mathcal{F}_2, \mu_1\times\mu_2 )\) is called a product measure space.
If \(\mathcal{A}_1\) is a semiring on \(\Omega_1\) and \(\mathcal{A}_2\) is a semiring on \(\Omega_2\), then the class \(\text{Rect}(\mathcal{A}_1, \mathcal{A}_2)\) is a semiring on \(\Omega_1\times\Omega_2\).
\(\phi=\phi\times \phi\in \text{Rect}(\mathcal{A}_1\times \mathcal{A}_2).\)
If \(A_1\times A_2,B_1 \times B_2\in \text{Rect}(\mathcal{A}_1\times \mathcal{A}_2)\), then \[ (A_1\times A_2)\cap (B_1\times B_2) = (A_1\times B_1)\cap (A_2\times B_2)\in \text{Rect}(\mathcal{A}_1\times \mathcal{A}_2). \]
\(A_1\times A_2,B_1\times B_2 \in\text{Rect}(\mathcal{A}_1\times \mathcal{A}_2)\), with \(A_1\times A_2\subset B_1\times B_2\) 라고 가정하자. 그렇다면 \(A_1\subset B_1\)이고 \(A_2\subset B_2\)이다. 따라서, \[ B_1\times B_2-A_1\times A_2= [(B_1-A_1)\times B_2]\uplus [(B_2-A_2)\times A_1]=\left[\uplus_{i=1}^m (C_{1i}\times B_2)\right]\uplus \left[\uplus_{j=1}^n (C_{2j}\times A_1)\right]. \] (큰사각형에서 작은 사각형을 뺀다고 생각하면 된다.)
전에 Uniqueness Theorem과 Caratheodory Extension Theorem을 배웠었다. Uniqueness Theorem은 두 measure가 주어진 field \(\mathcal{P}\)에서 agree한다면 \(\sigma(\mathcal{P})\)에서도 agree한다는 내용이다. Caratheodory Extension Theorem은 주어진 field \(\mathcal{P}\)에서 정의된 measure \(\mu\)가 \(\sigma(\mathcal{P})\)에서도 unique하게 정의된다는 것을 의미한다.
이 내용을 똑같이 Product measure에서도 증명할 수 있다 (너무 복잡하여 생략).