Independent collection of sigma-algebras

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 $j \in J$
Let $\mathcal{P}_{\mathsf{finite}}(J)$ be the D2337: Set of finite subsets in $J$.
Then $\mathcal{G} = \{ \mathcal{G}_j \}_{j \in J}$ is an independent collection of sigma-algebras in $P$ if and only if $$\forall \, I \in \mathcal{P}_{\mathsf{finite}}(J) \left[ \forall \, i \in I : E_i \in \mathcal{G}_i \quad \implies \quad \mathbb{P} \left( \bigcap_{i \in I} E_i \right) = \prod_{i \in I} \mathbb{P}(E_i) \right]$$

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 $$\forall \, N \in 1, 2, 3, \ldots : \forall \text{ distinct } j_1, \ldots, j_N \in J \left[ E_{j_1} \in \mathcal{G}_{j_1}, \ldots, E_{j_N} \in \mathcal{G}_{j_N} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right]$$
Also known as
Mutually independent collection of sigma-algebras
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