Independent Sigma Algebras

Definition

Given a probability space $(\Omega , \mathcal{F} , \mathbfsf{P} )$, a sequence of $\mathcal{G} _1, \dots , \mathcal{G} _n$ of sub-$\sigma $-algebras of $\mathcal{F} $, are independent if

\[ \textstyle \mathbfsf{P} (A_1 \cap A_2 \cap \cdots \cap A_n) = \prod_{i = 1}^{n} \mathbfsf{P} (A_i) \]

for any $A_1 \in \mathcal{G} _1$, $A_2 \in \mathcal{G} _2$, $\dots $, $A_n \in \mathcal{G} _n$. A family $\set{\mathcal{G} _i}_{i \in I}$ is a family of sub-$\sigma $-algebras of $\mathcal{F} $ is independent if any finitely many of them are independent.