# Sequence of sub-$\sigma$-algebras independent

I'm studying for a stochastic analysis exam and I want to solve this task:

Let $$(F_n)_{n\in\mathbb N}$$ be a sequence of sub-$$\sigma$$-algebras on a given probability space $$(\Omega,\mathcal F,\mathbb P)$$. Prove that $$(\mathcal F_n)_{n\in\mathbb N}$$ are independent if and only if $$\mathcal F_{n+1}$$ is independent of $$\sigma(\mathcal F_1\cup\cdots\cup\mathcal F_n)$$ for every $$n \in \mathbb N$$.

Update: I was able to show the $$\Rightarrow$$ direction. How does the $$\Leftarrow$$ direction work? I appreciate any help

Suppose $$F_{n+1}$$ is independent of $$\sigma(\mathcal F_1\cup\cdots\cup\mathcal F_n)$$ for every $$n \in \mathbb N$$.
Let $$J$$ be a finite non-empty subset of $$\Bbb N$$. Let $$F_i \in \mathcal F_i$$ (with $$i \in J$$). Let $$m=\max J\ge1$$. By our hypothesis, $$\mathcal F_m$$ is independent of $$\sigma(\mathcal F_1\cup\cdots\cup\mathcal F_{m-1})$$. Since $$\bigcap_{w \in J-\{m\}}F_w \in \sigma(\mathcal F_1\cup\cdots\cup\mathcal F_{m-1})$$ (this is because $$\forall w \in J -\{m\}, \mathcal F_w \subset \mathcal F_1\cup\cdots\cup\mathcal F_{m-1}$$ so $$\mathcal F_w \subset \sigma(\mathcal F_1\cup\cdots\cup\mathcal F_{m-1})$$ and by usual properties of $$\sigma$$-algebras.)
and $$F_m \in \mathcal F_m$$ we have $$P\left(\bigcap_{w \in J}F_w\right)= P(F_m)P\left(\bigcap_{w \in J-\{m\}}F_w\right).$$ $$J$$ is a finite subset so by finite induction we have $$P\left(\bigcap_{w \in J}F_w\right)= \prod_{w \in J} P(F_w).$$
• By $J-\{m\}$ you mean $J\setminus \{m\}$, right? Oct 25, 2020 at 12:50