Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$\mathcal{G}_j \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ for each $j \in J$
(ii)	$\mathcal{H}_j \subseteq \mathcal{G}$ is a D470: Subsigma-algebra of $\mathcal{G}_j$ on $\Omega$ for each $j \in J$
(iii)	$\mathcal{G} = \{ \mathcal{G}_j \}_{j \in J}$ is an D471: Independent collection of sigma-algebras on $P$

Then $\mathcal{H} = \{ \mathcal{H}_j \}_{j \in J}$ is an D471: Independent collection of sigma-algebras on $P$.