Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $\mathcal{G}_j \subseteq \mathcal{F}$ be a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ for each $j \in J$ such that

(i)	$\mathcal{G} = \{ \mathcal{G}_j \}_{j \in J}$ is an D471: Independent collection of sigma-algebras on $P$

Let $I \subseteq J$ be a D78: Subset.

Then $\{ \mathcal{G}_i \}_{i \in I}$ is an D471: Independent collection of sigma-algebras on $P$.