Conditionally independent random collection

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ (ii) $X_j$ be a D202: Random variable on $P$ for each $j \in J$ (iii) $\sigma_{\text{pullback}} \langle X_j \rangle$ is the D1730: Pullback sigma-algebra of $X_j$ on $P$ for each $j \in J$
Then $X = \{ X_j \}_{j \in J}$ is a conditionally independent random collection on $P$ given $\mathcal{G}$ if and only if $$\forall \, N \in 1, 2, 3, \ldots : \forall \, j_1, \ldots, j_N \in J \left[ E_{j_1} \in \sigma_{\text{pullback}} \langle X_{j_1} \rangle, \ldots, E_{j_N} \in \sigma_{\text{pullback}} \langle X_{j_N} \rangle \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \mid \mathcal{G} \right) \overset{a.s.}{=} \prod_{n = 1}^N \mathbb{P}(E_{j_n} \mid \mathcal{G}) \right]$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ (ii) $X_j$ is a D202: Random variable on $P$ for each $j \in J$
Then $X = \{ X_j \}_{j \in J}$ is a conditionally independent random collection on $P$ given $\mathcal{G}$ if and only if $$\forall \, N \in 1, 2, 3, \ldots : \forall \, j_1, \ldots, j_N \in J \left[ \{ X_{j_1} \in E_{j_1} \}, \ldots, \{ X_{j_N} \in E_{j_N} \} \in \mathcal{F} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N \{ X_{j_n} \in E_{j_n} \} \mid \mathcal{G} \right) \overset{a.s.}{=} \prod_{n = 1}^N \mathbb{P}( X_{j_n} \in E_{j_n} \mid \mathcal{G}) \right]$$
Also known as
Conditionally independent collection of random variables