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}