Independent event collection

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $E_j \in \mathcal{F}$ is an D1716: Event in $P$ for each $j \in J$
Let $\mathcal{P}_{\mathsf{finite}}(J)$ be the D2337: Set of finite subsets of $J$.
Then $E = \{ E_j \}_{j \in J}$ is an independent event collection in $P$ if and only if $$\forall \, I \in \mathcal{P}_{\mathsf{finite}}(J) : \mathbb{P} \left( \bigcap_{i \in I} E_i \right) = \prod_{i \in I} \mathbb{P}(E_i)$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $E_j \in \mathcal{F}$ is an D1716: Event in $P$ for each $j \in J$
Then $E = \{ E_j \}_{j \in J}$ is an independent event collection in $P$ if and only if $$\forall \, N \in 1, 2, 3, \ldots : \forall \, j_1, \ldots, j_N \in J \left[ \forall \, n \neq m : j_n \neq j_m \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right]$$
Also known as
Mutually independent collection of events, Independent collection of events
Child definitions
Results