Let $(\Omega,\Sigma,P)$ be a probability space.

What I know is that if $\{g_n: n \geq 1\}$ is a sequence of $\pi$ systems where $g_n \subset \Sigma$, then $\{g_n: n \geq 1\} $ is independent if and only if $\{\sigma(g_n): n \geq 1\}$ is independent where $\sigma(\cdot)$ denotes the smallest sigma algebra.

Then, I want to know why the following claim is true:

Let $\{E_n\}$ be a sequence of independent events. Then, for every $n \geq 1$, $\sigma(E_1),\sigma(E_2),\dots,\sigma(E_n),\sigma(E_{n+1},E_{n+2},\dots)$ are independent.

Clearly, the singleton set $\{E_n\}$ is a $\pi$ system, but the set $\{E_{n+1},E_{n+2},\dots \}$ is not a $\pi$ system, so I cannot apply the result that I know to verify the claim.

Why is the above claim true?


You can actually show a bit more general claim:

Given a probability space $(\Omega,\Sigma,\mathbb{P})$ and an array of independent events $(A_{i,j})_{i\in I,j\in J_i}$, $\mathcal{F}_i=\sigma(\{A_{i,j}:j\in J_i\})$ are independent.

To show this consider sets $\mathcal{P}_i$ of all finite intersections of sets in row $i$ of $(A_{i,j})$ and note that $\mathcal{F}_i=\sigma(\mathcal{P_i})$. In view of the theorem presented in the question it remains to show that $\mathcal{P}_i$ are independent...


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