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Assume that $F_1$ and $F_2$ are two independent sigma fields. We know that union of $F_1$ and $F_2$ is not necessarily a sigma-field. Suppose we define $ \mathcal{A} = \{A \cap B: A\in F_1, B\in F_2\} $. How to show that:

$$\sigma(\mathcal{A}) = \sigma(F_1 \cup F_2) $$

Thanks,

Here is how I thought about it. I divide the proof into two parts:

1) $\sigma(\mathcal{A}) \subset \sigma(F_1 \cup F_2)$

2) $\sigma(F_1 \cup F_2) \subset \sigma(\mathcal{A})$

I guess, part 1 is easy since $\mathcal{A} \subset (F_1 \cup F_2)$ . Right? How about part 2?

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    The question in the title does not match the question in the body (as a matter of fact, the answer to the question in the title is NO). Please consider editing.2012-11-25
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    Hi @GautamShenoy, you are right, I didn't know how to write a title. I wanted to have something correct but brief and this is what I came up with...2012-11-25
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    You are using $A$ too much.2012-11-25
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    The set is actually script A ! Please see the edits!2012-11-25
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    Dude.. are you sure? If $F_1$ and $F_2$ are disjoint(except for empty and full set), then A will be the trivial sigma algebra (note: intersection of sigma alg is a sigma alg) but the union may be a non-trivial one. Are you sure this is indeed the right question?2012-11-25
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    Well $F_1$ and $F_2$ are not disjoint necessarily. I added one more thing to the question and that is $F_1$ and $F_2$ are independent then for sure they are not disjoint !2012-11-25
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    Ok, that will be helpful.2012-11-25
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    In fact, $F_1\cup F_2\subset\mathcal A$, not (in general) the other way.2012-11-25
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    Hi @did, I just changed the title. The question is why $F_1 \cup F_2 \subset \mathcal{A}$ ?2012-11-25
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    You want to show that $F_1\subseteq\mathcal A$. Thus, let $A$ in $F_1$. To write $A$ as $A=A_1\cap A_2$ with $A_1$ in $F_1$ and $A_2$ in $F_2$, consider...2012-11-25
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    Sam: Did you manage to complete the proof?2012-11-28
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    Hi @did, thank you for following up with me. Actually I've not been able to show $F_1 \cup F_2 \subset A$2012-11-28
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    Hi @did, I accepted since it was the only answer and I wanted to thank you that answered the question.2012-11-29
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    Did you follow the hint I gave you in a comment?2012-11-29

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Consider the sigma-algebras generated by each class of subsets involved in the double inclusion $$ F_1\cup F_2\subseteq\mathcal A\subseteq\sigma(F_1\cup F_2).$$

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    The relevance of the *independence* of $F_1$ and $F_2$ is mysterious to me in the context of this question.2012-11-25