Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{R}$ be the set of \reftext{def:real-numbers-c54-2026c}{real numbers}, and let $\mathcal{B}(\mathbb{R})$ be the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra}.

\textbf{Generated $\sigma$-algebra of a family of random variables.} Let $J$ be a nonempty set and let $(X_j)_{j\in J}$ be a \reftext{def:family-subfamily-subsets-set-2026a}{family} of random variables on $(\Omega,\mathcal{F},P)$. The \textbf{$\sigma$-algebra generated by $(X_j)_{j\in J}$}, written $\sigma(X_j: j\in J)$, is the \reftext{def:generated-sigma-algebra-2026a}{generated $\sigma$-algebra} $\sigma(\mathcal{C})$ on $\Omega$ of the family

Since each $X_j$ is a random variable, every member of $\mathcal{C}$ lies in $\mathcal{F}$, and $\mathcal{F}$ is a \reftext{def:sigma-algebra-measurable-space-2026a}{$\sigma$-algebra} containing $\mathcal{C}$; hence $\sigma(X_j:j\in J)\subseteq\mathcal{F}$, i.e., it is a \textbf{sub-$\sigma$-algebra} of $\mathcal{F}$.

For a single random variable $X$ we write $\sigma(X)=\sigma(X:j\in\{1\})$. In this case the family $\{X^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$ is itself a $\sigma$-algebra: $X^{-1}(\mathbb{R})=\Omega$, $\Omega\setminus X^{-1}(B)=X^{-1}(\mathbb{R}\setminus B)$, and $\bigcup_m X^{-1}(B_m)=X^{-1}(\bigcup_m B_m)$, with $\mathbb{R}\setminus B$ and $\bigcup_m B_m$ Borel because $\mathcal{B}(\mathbb{R})$ is a $\sigma$-algebra. Hence $\sigma(X)=\{X^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$.

\textbf{Independence of $\sigma$-algebras.} Let $B$ be a nonempty set and for each $b\in B$ let $\mathcal{G}_b\subseteq\mathcal{F}$ be a sub-$\sigma$-algebra. The family $(\mathcal{G}_b)_{b\in B}$ is \textbf{independent} if for every finite nonempty set of distinct indices $b_1,\dots,b_p\in B$ and every choice of events $A_l\in\mathcal{G}_{b_l}$ ($1\le l\le p$),

with the \reftext{def:finite-product-notation-2026a}{finite product notation}.

Because $\Omega\in\mathcal{G}_b$ for every $b$ and $P(\Omega)=1$, the family $(\mathcal{G}_b)_{b\in B}$ is independent if and only if every finite subfamily is independent, and events $A_1,\dots,A_r$ are \reftext{def:independence-events-rvs-2026a}{independent} if and only if the product identity above holds for every choice of indices after inserting $\Omega$ for omitted factors; in particular, random variables $X_1,\dots,X_r$ are independent in the sense of \ref{def:independence-events-rvs-2026a} if and only if the $\sigma$-algebras $\sigma(X_1),\dots,\sigma(X_r)$ are independent, since $\sigma(X_i)=\{X_i^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$ as shown above.