Definition:Independent Sigma-Algebras/Binary Case

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

Let $\Sigma$ and $\Sigma'$ be sub-$\sigma$-algebras of $\EE$.

Then $\Sigma$ and $\Sigma'$ are said to be ($\Pr$-)independent :


 * $\forall E \in \Sigma, E' \in \Sigma': \map \Pr {E \cap E'} = \map \Pr E \map \Pr {E'}$