Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a
D1159: Probability space.
Let $X_j$ be a
D202: Random variable on $P$ for each $j \in J$ such that
Then $X = \{ X_j \}_{j \in J}$ is an
independent random collection on $P$ 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 \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} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right]
\end{equation}