If $E_{j_1} \in \mathcal{H}_{j_1}, \ldots, E_{j_N} \in \mathcal{H}_{j_N}$, then $E_{j_1} \in \mathcal{G}_{j_1}, \ldots, E_{j_N} \in \mathcal{G}_{j_N}$ which implies that $E_{j_1}, \ldots, E_{j_N}$ are independent events in $P$ since $\mathcal{G}_{j_1}, \ldots, \mathcal{G}_{j_N}$ are independent sigma-algebras. $\square$