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Let $(\Omega, \mathcal{A}, P)$ be a probability space and $(A_i)_{i\in I}\subset \mathcal{A}$ a family of events in $\mathcal{A}$.

Is there any way to prove the following statement?

$(\sigma(A_i))_{i\in I}$ independent $\Rightarrow$ $(A_i)_{i\in I}$ independent

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This is by definition. A family of sigma-algebras is independent iff for every finite sequence of events in distinct algebras of the family, these events are independent. But also by definition, $A_i$ belongs to $\sigma(A_i)$ ( indeed, $\sigma(A_i)$ is just $\{\emptyset, A_i, A_i^c, \Omega\}$)

You asked only for one implication, the converse is a bit more interesting, but basically boils down to using the following lemma : $(A_1,\ldots,A_k)$ independent iff $(A_1^c,\ldots,A_k)$ independent

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