Definition:Independent Sigma-Algebras/Binary Case

Definition
Let $\left({\Omega, \mathcal E, P}\right)$ be a probability space.

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

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


 * $\forall E \in \Sigma, E' \in \Sigma': P \left({E \cap E'}\right) = P \left({E}\right) P \left({E'}\right)$