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 $j \in J$

Then $\mathcal{G} = \{ \mathcal{G}_j \}_{j \in J}$ is an independent collection of sigma-algebras in $P$ if and only if $$\begin{equation} \forall \, N \in 1, 2, 3, \ldots : \forall \text{ distinct } j_1, \ldots, j_N \in J \left[ E_{j_1} \in \mathcal{G}_{j_1}, \ldots, E_{j_N} \in \mathcal{G}_{j_N} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right] \end{equation}$$