Definition:Independent Sigma-Algebras/Binary Case

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

Let $\mathcal A, \mathcal B$ be sub-$\sigma$-algebras of $\mathcal E$.

Then $\mathcal A$ and $\mathcal B$ are said to be ($P$-)independent iff:


 * $\forall A \in \mathcal A, B \in \mathcal B: P \left({A \cap B}\right) = P \left({A}\right) P \left({B}\right)$