Independence (1). Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\mathscr{B}, \mathscr{C} \subset \mathscr{A}\) be two sub- \(\sigma\) algebras of \(\mathscr{A}\). We call \(\mathscr{B}\) and \(\mathscr{C}\) independent, if

\[\mathbb{P}(B \cap C)=\mathbb{P}(B) \mathbb{P}(C) \quad \forall B \in \mathscr{B}, C \in \mathscr{C} .\]

Assume now that \(\mathscr{B}=\sigma(\mathscr{G})\) and \(\mathscr{C}=\sigma(\mathcal{H})\), where \(\mathscr{G}, \mathcal{H}\) are \(\cap\)-stable collections of sets. Prove that \(\mathscr{B}\) and \(\mathscr{C}\) are independent if, and only if,

\[\mathbb{P}(G \cap H)=\mathbb{P}(G) \mathbb{P}(H) \quad \forall G \in \mathscr{G}, H \in \mathcal{H}\]