Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers}, let $J\subseteq\mathbb{N}$ be nonempty, and let $(X_m)_{m\in J}$ be an \reftext{def:independence-events-rvs-2026a}{independent} \reftext{def:family-subfamily-subsets-set-2026a}{family} of random variables on it (for $J=\mathbb{N}$ this is an independent \reftext{def:sequence-in-set-2026a}{sequence}). Let $B$ be a nonempty set and let $(I_b)_{b\in B}$ be a family of pairwise disjoint nonempty subsets of $J$.

Then the family of \reftext{def:independence-sigma-algebras-2026a}{generated $\sigma$-algebras} $\mathcal{G}_b=\sigma(X_m: m\in I_b)$, $b\in B$, is independent in the sense of \ref{def:independence-sigma-algebras-2026a}.

Consequently, if for each $b\in B$ a random variable $Y_b$ on $(\Omega,\mathcal{F},P)$ is $\mathcal{G}_b$-measurable, meaning $Y_b^{-1}(B')\in\mathcal{G}_b$ for every \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} $B'$, then the family $(Y_b)_{b\in B}$ is independent in the sense of \ref{def:independence-events-rvs-2026a}.