\begin{definition}[Definition:Independent Sigma-Algebras/Countable Case]
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''' {{iff}}:
: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 $k$.
\end{definition}