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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

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    Thank you very much, I was looking for a simple proof like this one. – 2017-02-20
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    @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. – 2017-02-20
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    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. – 2017-02-20