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Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space and $\mathcal{A_1}, \mathcal{A_2}\subseteq \mathscr{F}$. Suppose that:

1) $\mathcal{A}_1$ and $\mathcal{A}_2$ contain $\Omega$ and are stable under pairwise intersections

2) $\mathcal{A}_1$ is independent of $\sigma(\mathcal{A}_2)$ and $\mathcal{A_2}$ is independent of $\sigma(\mathcal{A}_1)$.

How can I conclude that $\sigma(\mathcal{A}_1)$ is independent of $\sigma(\mathcal{A}_2)$?

The motivation of this question comes from trying to understand the very last step of the argument in Independent $\sigma$-algebras using $\pi$-$\lambda$-theorem, where the author simply just says "repeat" the above argument. I don't understand how repeating the argument will allow us to understand complete the proof and conclude that the $\sigma$-algebra's generated by the generating sets are independent of one another.

BCLC
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User086688
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    ...the very last step of which argument? – Did Aug 23 '17 at 07:51
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    Fix some random variable $X$ which is $\sigma(\mathcal A_2)$-measurable. Let $\mathcal D:= {A \in \mathscr F : E[1_A X] = P(A)E[X]}$. Show that this is a Dynkin system containing $\mathcal A_1$. Then apply the $\pi$-$\lambda$ theorem in order to deduce that $\sigma(\mathcal A_1) \subset \mathcal D$. Conclude. – shalin Aug 23 '17 at 12:11
  • Woops sorry I completely forgot to attach a link for the argument! This is the argument I was meant to link https://math.stackexchange.com/questions/1665533/independent-sigma-algebras-using-pi-lambda-theorem – User086688 Aug 23 '17 at 15:03
  • Thanks Shalop for your reply - I will follow your suggestion and tackle it now! – User086688 Aug 23 '17 at 15:04
  • @Shalop Do you really have to use random variables? It seems kind of a cheat like proving Borel-Cantelli Lemma Part 1 using indicator random variables, expectations and Fubini's Theorem. – BCLC Apr 20 '18 at 23:15

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