Independent subalgebras 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

Two events of probability in $\{0,1\}$ are independent, which gives the 'if' part. 
For the 'only if', fix $A\in\cal A_1$. Then $A\in \cal A_2$. As $\cal A_1$ and $\cal A_2$ are independent, we have $P(A\cap A)=P(A)\cdot P(A)$, that is, $P(A)\in\{0,1\}$.
Note that we didn't use the fact that $\cal A_1$ and $\cal A_2$ are algebras. 
