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Let $(\Omega, M, P)$ be a probability space and let $O_1, O_2$ be independent events. Does the sigma algebras generated by $F_1 := \sigma(\{O_1\})$ and $F_2 := \sigma(\{O_1,O_2\})$ are independent? Here $O_1,O_2$ are not disjoint and $1 > P(O_1),P(O_2) > 0$.

The sigma algebra generated by $O_1$ is $\{\emptyset, O_1, O_1^\complement, \Omega\}$ where $O_1^\complement$ is taken as the complement relative to $\Omega$. We have,

$$F_1 = \{\emptyset, O_1, O_1^\complement, \Omega\}.$$


On the other hand, the sigma algebra generated by $F_2$ is given by $$F_2=\{\emptyset, \Omega, O_1,O_2,O_1^\complement,O_2^\complement,O_1\cup O_2, O_1 \cap O_2,O_1^\complement \cap O_2^\complement, O_1^\complement\cup O_2^\complement\}.$$

Now, on the other hand, I know that $O_1^\complement, O_2$, are independent and that $O_1^\complement, O_2^\complement$ are also independent. I think they are not independent, because for instance, $$P(O_1\cap(O_1^\complement \cap O_2^\complement)) = P((O_1\cap O_1^\complement) \cap O_2^\complement) = P(\emptyset) = 0 \neq P(O_1)P(O_1^\complement) P(O_2^\complement).$$

Does this shows that the sigma algebras generated by $F_1, F_2$ are not independent?

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Any two sigma algebras that have an event $E$ in common with $0<P(E)<1$ are not independent. Why? Let $E\in\mathcal F_1$ and $E\in\mathcal F_2.$ Then for $\mathcal F_1$ and $\mathcal F_2$ to be independent we would need $$ P(E\cap E) = P(E)P(E)$$ but $$ P(E\cap E) = P(E)$$ so we have $$ P(E) = P(E)^2$$ which means $P(E) = 0$ or $1.$

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  • $\begingroup$ Thank you very much, I was looking for a simple proof like this one. $\endgroup$ – okie Feb 20 '17 at 5:11
  • $\begingroup$ @richitesenpai no problem. Check my edit though, there is one corner case where they can still be independent in a strict reading of your question. $\endgroup$ – spaceisdarkgreen Feb 20 '17 at 5:16
  • $\begingroup$ Yes, actually, I know the probability of both sets, which are less than 1. I wanted to see a general case. Will edit the post and thanks again. $\endgroup$ – okie Feb 20 '17 at 5:19

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