 Solution

The pi-systems criterion says that if ${\mathcal{I}}_1,{\mathcal{I}}_2$ are pi-systems and $\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B)$ for all $A\in {\mathcal{I}}_1$, $B\in {\mathcal{I}}_2$, then their generated sigma-algebras are independent. Necessity is immediate. We may add $\Omega$ to either pi-system, since it does not alter the generated sigma-algebra or the product identities.

For sufficiency, fix $A\in {\mathcal{I}}_1$ and defineIt contains $\Omega$, is closed under complements by subtraction from $\mathbb{P}(A)$, and is closed under countable disjoint unions by countable additivity. Thus it is a lambda-system. It contains ${\mathcal{I}}_2$, so the pi-lambda theorem gives ${\mathcal{D}}_A={\mathcal{G}}_2$.

Now fix any $B\in {\mathcal{G}}_2$ and apply the same lambda-system argument to the sets $A\in {\mathcal{G}}_1$ satisfying the identity. The first step shows that this lambda-system contains ${\mathcal{I}}_1$, so it contains all ${\mathcal{G}}_1$. Thus the product identity holds on both entire sigma-algebras, proving their independence.