Definition:Independent Sigma-Algebras/Countable Case

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {\GG_n}_{n \mathop \in \N}$ be a sequence of sub-$\sigma$-algebras of $\Omega$.

We say that $\sequence {\GG_n}_{n \mathop \in \N}$ is a sequence of ($\Pr$-)independent $\sigma$-algebras :


 * for each $n \in \N$ and distinct natural numbers $i_1, i_2, \ldots, i_n$, we have:


 * $\ds \map \Pr {\bigcap_{k \mathop = 1}^n G_{i_k} } = \prod_{k \mathop = 1}^n \map \Pr {G_{i_k} }$


 * for all $G_{i_1}, G_{i_2}, \ldots, G_{i_n}$ with $G_{i_k} \in \GG_{i_k}$ for each $k$.