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
Best Answer
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 $$ P\left(\bigcap_{w \in J}F_w\right)= \prod_{w \in J} P(F_w). $$