Let $(\Sigma, \mathbb{A}, \mathbb{P})$ be a probability space and $\mathbb{A}_1 \subset \mathbb{A_2} \subset \mathbb{A}$ subalgebras of $\mathbb{A}$. Prove that $\mathbb{A}_1$ and $\mathbb{A}_2$ are independent iff $P(A) \in \{0,1\}$ for every $A \subset \mathbb{A}_1$.
I would be grateful for your help or any hints.
Thanks