Definition:Independent Sigma-Algebras

Definition

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

Let $\GG_1$ and $\GG_2$ be sub-$\sigma$-algebras of $\EE$.

Then $\GG_1$ and $\GG_2$ are said to be ($\Pr$-)independent if and only if:

$\forall E_1 \in \GG_1, E_2 \in \GG_2: \map \Pr {E_1 \cap E_2} = \map \Pr {E_1} \map \Pr {E_2}$

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 if and only if:

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 $k$.