Grouping Lemma for Independent Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{N}$ be the set of natural numbers, let $J\subseteq\mathbb{N}$ be nonempty, and let $(X_m)_{m\in J}$ be an independent family of random variables on it (for $J=\mathbb{N}$ this is an independent 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 generated $\sigma$-algebras $\mathcal{G}_b=\sigma(X_m: m\in I_b)$, $b\in B$, is independent in the sense of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras.

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 Borel set $B'$, then the family $(Y_b)_{b\in B}$ is independent in the sense of Independence of Events and of Random Variables.