$
\def\P{\mathsf{\sf P}}
\def\E{\mathsf{\sf E}}
\def\Var{\mathsf{\sf Var}}
\def\Cov{\mathsf{\sf Cov}}
\def\std{\mathsf{\sf std}}
\def\Cor{\mathsf{\sf Cor}}
\def\R{\mathbb{R}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\mathsf{\sf diag}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
$

Measurable rectangles are products of sets in the corresponding $\sigma$-algebras $\mathcal{F}_t$, $t\in T$. Some examples of measurable rectangles in $\Omega=\R^3=\R\times \R\times \R$ (assuming the the Borel $\sigma$-algebra on each copy of $\R$) are \begin{align*} &(1,2)\times \R \times [2,3)\\ &([1,2]\cup [3,4]) \times[2,3]\times [4,5]\\ &\R\times \R \times \cup_{n\in\mathbb{N}} [n,n+1/2). \end{align*} For example, the set $\{\bb x: \sum_i x_i=1\}$ is not a measurable rectangle. Figure 4.2 illustrates some sets in $\R^2$ that are measurable rectangles and some sets that are not.

Measurable cylinders $\{A_s \times (\times_{t\neq s} \Omega_t):A_s\in\mathcal{F}_s, s\in T\}$ are special cases of measurable rectangles, in which only one of the products is allowed to differ from $\Omega_t$. A general characterization of the measurable cylinders in $\R^3$ is \begin{align*} &A \times \R \times \R, \qquad A\in\mathcal{B}(\R)\\ &\R \times A \times \R, \qquad A\in\mathcal{B}(\R)\\ &\R \times \R \times A, \qquad A\in\mathcal{B}(\R). \end{align*}

The following definitions constructs a new measurable space from several measurable spaces. The new measurable space, called the product measurable space, leads to the construction of the product measure, which is instrumental in studying probability over Euclidean spaces $\R^n$.

Since $\sigma($all open measurable cylinders) includes all measurable cylinders and therefore \begin{align*} \otimes_{t\in T} \mathcal{F}_t = \sigma\left( \left\{ A: A \text{ is an open measurable cylinder}\right\} \right). \end{align*}

In the proposition above and elsewhere, whenever we refer to the set $\R^{\infty}$, we assume the metric structure defined in Example B.4.4.

A set in $\mathcal{L}_2$ is an open measurable cylinder and is therefore an open set in $\R^d$ or in $\R^{\infty}$ (see Example B.4.4).

In the case of a finite number of products, let $A$ be an open set in $\R^d$. Then for each $a\in A$ there exists an open ball $B_r(a')$ such that $a\in B_r(a')\subset A$. For each $a\in A$ and each open ball $B_r(a')$ containing $a$ there exists a rectangle $\times_{i=1}^d (a_i,b_i)$ with rational endpoints such that $a\in \times_{i=1}^d (a_i,b_i) \subset B_r(a')\subset A$ and the union of all such open rectangles equals $A$. Since open rectangles are finite intersections of cylinder sets, and there are a countable number of rectangles with rational end-points, $A$ is a countable union of finite intersections of cylinders, implying that $A\in\sigma(\mathcal{B}(\R) \otimes \cdots \otimes \mathcal{B}(\R))$.

The case of an infinite number of products proceeds similarly. By second countability of $\R^{\infty}$ (see Proposition B.4.5), an open set $A\subset R^{\infty}$ is a countable union of finite intersections of cylinders. It follows that $A=\sigma(\mathcal{L}_2)$.

In the following, we concentrate on product of two spaces. This is done for the sake of simplicity, and the definitions and propositions carry over to a product of a finite number of spaces (for example, by induction).

We denote the set of sets $E$ for which $f_{E}$ is measurable as $\mathcal{L}$, and show that it is a $\lambda$-system on $X\times Y$. We have $X\times Y \in\mathcal{L}$ since $f_{E}(x)= \P_Y(\{y: (x,y)\in X\times Y\})= \P_Y(Y)$. If $A\in\mathcal{L}$, then $f_{E^c}(x)$ is measurable and therefore $f_{E}(x)= \P_Y(\{y: (x,y)\in A^c\})= \P_Y(Y)-f_{E^c}(x)$ is measurable as well. If $A_n, n\in\mathbb{N}$ is a sequence of disjoint sets in $\mathcal{L}$ then $f_{\cup A_n}(x)= \P_Y(\{y: (x,y)\in \cup_n A_n\})=\sum_{n\in\mathbb{N}}f_{A_n}(x)$ implying that $\cup A_n$ is in $\mathcal{L}$ as well. We have thus shows that $\mathcal{L}$ is a $\lambda$ system. This derivation also shows that $f_E(x)$ and $g_E(x)$ are measure functions for all $x$.

The function $f_E$ is also measurable for all measurable rectangles $E=A\times B$, since $f_{A\times B}(x)=I_A(x) \P_X(B)$. $\mathcal{L}$ is a $\lambda$-system containing the $\pi$-system of measurable rectangles, which generates the $\sigma$ algebra $\mathcal{X}\otimes\mathcal{Y}$. It follows from Dynkin's Theorem (Proposition E.3.4) that $\mathcal{X}\otimes\mathcal{Y}\subset \mathcal{L}$ implying that $f_E$ is measurable. The proof of measurability of $g_E$ is similar.

Since for all $E\in\mathcal{X}\otimes\mathcal{Y}$, the functions $f_E$ and $g_E$ are measurable, we can define the following functions that assign real values to sets in $\mathcal{X}\otimes\mathcal{Y}$ \begin{align} \P_{X\times Y}^{(1)}(E)&=\int_X \P_X(\{y: (x,y)\in E\})\, d \P_X,\\ \P_{X\times Y}^{(2)}(E)&=\int_Y \P_Y(\{y: (x,y)\in E\})\, d \P_Y. \end{align} For measurable rectangles, $f_{A\times B}(x)=I_A(x) \P_X(B)$ and $g_{A\times B}(x)=I_B(x) \P_Y(A)$, implying that ${\P_{X\times Y}}^{(1)}(A\times B)= \P_{X\times Y}^{(2)}(A\times B)= \P_X(A)\cdot \P_Y(B)$. Since measurable rectangles generate $\mathcal{X}\otimes\mathcal{Y}$, it follows from Corollary E.3.1 that $\P_{X\times Y}^{(1)}$ agrees with $\P_{X\times Y}^{(2)}$ on $\mathcal{X}\otimes\mathcal{Y}$. It also follows that if there is any other measure $\P_{X\times Y}^{(3)}$ which assigns the value $\P_{X\times Y}^{(3)}=(A\times B)=\P_X(A)\cdot \P_Y(B)$ on measurable rectangles, then $\P_{X\times Y}^{(3)}$ agrees with $\P_{X\times Y}^{(1)}$ and $\P_{X\times Y}^{(2)}$ on $\mathcal{X}\otimes\mathcal{Y}$.

- We usually denote the product measure of $\P_X$ and $\P_Y$ as $\P_X\times \P_Y$.
- The product measure corresponds to independence between the random variables $X$ and $Y$. Other probability measures are also available on the product space $(X\times Y,\mathcal{X}\otimes\mathcal{Y})$ (see Chapters 4 and 5).
- The product measure theorem can be generalized to measures that are not probability functions. See (Billingsley, 1995) for a more general version.
- The product measure theorem can be generalized for products of more than two spaces. The generalization is straightforward since the definitions above are associative: \begin{align*} (X\times Y)\times Z &= X\times (Y\times Z)\\ (\mathcal{X}\otimes \mathcal{Y})\otimes \mathcal{Z} &= \mathcal{X}\otimes (\mathcal{Y}\otimes \mathcal{Z})\\ (\P_X\times \P_Y)\times\P_Z &= \P_X\times( \P_Y\times\P_Z). \end{align*} Further extensions exist for a product of an infinite number of measure spaces (both countably infinite and uncountably infinite).