= Solution
{c}
Independence of the full sigma-algebras immediately gives independence of their generating <pi-systems>. For the converse, fix $H\in\mathcal H_0$. On $\mathcal G$, define finite <measures>
$$
\nu_H(G)=\mathbb P(G\cap H),\qquad\lambda_H(G)=\mathbb P(G)\mathbb P(H).
$$
They have the same total mass $\mathbb P(H)$ and agree on $\mathcal G_0$ by assumption. The supplied measure-uniqueness theorem makes them equal on $\mathcal G$. Thus the independence formula holds for every $G\in\mathcal G$ and every $H\in\mathcal H_0$.
Now fix such an arbitrary $G$. The measures $H\mapsto\mathbb P(G\cap H)$ and $H\mapsto\mathbb P(G)\mathbb P(H)$ on $\mathcal H$ have equal total mass and agree on $\mathcal H_0$. A second application gives equality on all of $\mathcal H$. This proves <independence extended from generating pi-systems>.
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