Sigma-Algebras with Independent Generators are Independent

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

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

Suppose that $\mathcal G, \mathcal H$ are $\cap$-stable generators for $\mathcal A, \mathcal B$, respectively.

Suppose that, for all $G \in \mathcal G, H \in \mathcal H$:


 * $P \left({G \cap H}\right) = P \left({G}\right) P \left({H}\right)$

Then $\mathcal A$ and $\mathcal B$ are $P$-independent.