User:Caliburn/s/prob/Definition:Independent Sigma-Algebras/Binary Case

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 :


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