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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

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    I guess you mean $A\in \mathbb A_1$, right?2012-11-26

1 Answers 1

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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.