[proofplan] The forward implication is the definition of independence applied to sets in the generating classes. For the converse, fix all but one coordinate and use the pi-lambda theorem successively to extend the rectangle identity from the generators to the full sigma-algebras. [/proofplan] [step:Prove the forward implication] If $X_1,\ldots,X_n$ are independent, then for any $A_i\in\mathcal A_i\subset\mathcal E_i$, \begin{align*} \mathbb P\left(\bigcap_{i=1}^n X_i^{-1}(A_i)\right) =\prod_{i=1}^n\mathbb P(X_i^{-1}(A_i)). \end{align*} This is the displayed identity. [/step] [step:Extend one coordinate] Assume the displayed identity holds for every $A_i\in\mathcal A_i$. More generally, suppose that for some $m\in\{1,\ldots,n\}$ the identity is already known whenever the first $m-1$ coordinates are chosen from $\mathcal E_1,\ldots,\mathcal E_{m-1}$ and the remaining coordinates are chosen from their generating classes. Fix $B_1\in\mathcal E_1,\ldots,B_{m-1}\in\mathcal E_{m-1}$ and fix $A_{m+1}\in\mathcal A_{m+1},\ldots,A_n\in\mathcal A_n$. Let $\mathcal D_m$ be the class of all $B\in\mathcal E_m$ for which \begin{align*} \mathbb P\left(\bigcap_{i=1}^{m-1}X_i^{-1}(B_i)\cap X_m^{-1}(B)\cap\bigcap_{i=m+1}^{n}X_i^{-1}(A_i)\right) =\left(\prod_{i=1}^{m-1}\mathbb P(X_i^{-1}(B_i))\right)\mathbb P(X_m^{-1}(B)) \left(\prod_{i=m+1}^{n}\mathbb P(X_i^{-1}(A_i))\right). \end{align*} The class $\mathcal D_m$ contains $E_m$: this is exactly the induction hypothesis with the $m$th coordinate chosen as $E_m\in\mathcal A_m$. It contains $\mathcal A_m$ by the same induction hypothesis. It is a lambda-system: complements relative to $E_m$ follow by subtracting both sides from the identity for $E_m$, and countable disjoint unions follow by countable additivity on both sides. Since $\mathcal A_m$ is a pi-system that generates $\mathcal E_m$, the [Dynkin Pi System Lemma](/theorems/505) gives $\mathcal D_m=\mathcal E_m$. [/step] [step:Iterate over coordinates] Start with $m=1$, where the induction hypothesis is just the assumed identity on $\mathcal A_1\times\cdots\times\mathcal A_n$. The previous step promotes the first coordinate from $\mathcal A_1$ to $\mathcal E_1$. Repeating the argument for $m=2,\ldots,n$ promotes one coordinate at a time while keeping the already-promoted coordinates in their full sigma-algebras. After the last step, the rectangle identity holds for every choice of sets $B_i\in\mathcal E_i$: \begin{align*} \mathbb P\left(\bigcap_{i=1}^n X_i^{-1}(B_i)\right) =\prod_{i=1}^n\mathbb P(X_i^{-1}(B_i)). \end{align*} This is exactly the independence of $X_1,\ldots,X_n$. [/step]