> [!definition] > > Let $\bracs{X_i}_{i \in I}$ be an indexed collection of non-empty [[Set|sets]], $X = \prod_{i \in I}X_i$ be their [[Cartesian Product|Cartesian product]], and $\pi_i: X \to X_i$ be the coordinate maps. If $\cm_i$ is a [[Sigma Algebra|sigma algebra]] on $X_i$ for each $i$, then the **product $\sigma$-algebra** on $X$ is the $\sigma$-algebra generated by > $ > \ce = \bracs{\pi_i^{-1}(E_i): E_i \in \cm_i, i \in I} > $ > Denoted as $\bigotimes_{i \in I}\cm_i$. > [!theorem] > > If the index set $I$ is [[Cardinality|countable]], then $\bigotimes_{i \in I}\cm_i$ is the $\sigma$-algebra generated by > $ > \ce = \bracs{\prod_{i \in I} E_i: E_i \in \cm_i} > $ > *Proof*. If $E_i \in \cm_i$, then $\pi^{-1}_i(E_i) = \prod_{j \in I}E_j$ where $E_j = X_j$ for $j \ne i$. On the other hand, $\prod_{i \in I}E_i = \bigcap_{i \in I}\pi^{-1}_i(E_i)$. Since they generate each other, the $\sigma$-algebras are equal. > [!theorem] > > Let $\cm_i$ be a $\sigma$-algebra generated by $\ce_i$ for each $i \in I$. Then $\bigotimes_{i \in I}\cm_i$ is generated by $\cf_1 = \bracs{\pi_i^{-1}(E_i): E_i \in \ce_i, i \in I}$. > > If $A$ is countable and $X_i \in \ce_i$ for all $i$, then $\bigotimes_{i \in I}M_i$ is generated by $\cf_2 = \bracs{\prod_{i \in I}E_i: E_i \in \ce_i}$. > > *Proof*. Obviously $\agb{\cf_1} \subseteq \bigotimes_{i \in I}\cm_i$. On the other hand, for each $i \in I$, the collection $\bracs{E \subseteq X_i: \pi^{-1}_i(E) \in \agb{\cf_1}}$ is a $\sigma$-algebra on $X_i$ that contains $\ce_i$ and $\cm_i$. Therefore $\pi^{-1}_i(E) \in \agb{\cf_1}$ for all $E \in M_i$, $i \in I$, and $\bigotimes_{i \in I}\cm_i \subseteq \agb{\cf_1}$. > > The second statement follows using the previous theorem.