I'm reading through the proof of Kolmogorovs 0-1 law and one statement is made but I can't find the proof to this statement anywhere. Could anyone help me out?
The statement is given by:
Let $A_1, A_2, \ldots$ be independent events. If $I \cap J =\emptyset$, then $\sigma(A_k : k \in I)$ and $\sigma(A_k : k \in J)$ are independent.
I'm really stuck on this, so help is much appreciated.
It's by definition when $I$ and $J$ are finite, so the problem is when these sets are infinite. Here are the main ideas:
Let $\mathcal A$ and $\mathcal B$ be algebras generating respectively $\sigma(A_i,i\in I)$ and $\sigma(B_j,j\in J)$. According to this thread, it's enough to show that for each $A\in \mathcal A$ and $B\in \mathcal B$, $P(A\cap B)=P(A)P(B)$.
We can write each element of $A$ as a finite disjoint union of elements of the form $\bigcap_{i\in I'}C_i$, where $C_i$ is $A_i$ or $A_i^c$ and $I'\subset I$ is finite, and the same for elements of $\mathcal B$.
