Let $\mathcal{E}_1, …,\mathcal{E}_n$ be collections of measurable sets on $(\Omega,\mathcal{F},P)$, each closed under intersection. Suppose
\begin{align*}
P(A_1\cap…\cap\ A_n)=P(A_1)\cdot … \cdot P(A_n),
\end{align*}
for all $A_i \in \mathcal{E}_i$ for $1 \leq i \leq n$.
Now I want to show that the $\sigma$-algebras $\sigma(\mathcal{E}_i)$ for $1 \leq i \leq n$ are independent, using an application of the $\pi$-$\lambda$-theorem.
Since $\mathcal{E}_i$ for $1 \leq i \leq n$ are closed under intersection, each $\mathcal{E}_i$ is a $\pi$-system. Now, for me it is unclear how to define a $\lambda$-system and how to apply the $\pi$-$\lambda$-theorem.
Best Answer
Fix $A_i,\forall i=2,\cdots,n$, define $G=\{B\in \sigma(\mathcal{E}_1)|P(B\cap\cdots\cap\ A_n)=P(B)\cdot \cdots \cdot P(A_n)\}$,
By assumption $\mathcal{E}_1\subset G$, it suffices to show $G$ is a $\lambda$-system by definition.
Then by $\pi-\lambda$ theorem, we have $\sigma(\mathcal{E}_1)\subset G$, this shows given $\mathcal{E}_1, \cdots,\mathcal{E}_n$ independent, we have $\sigma(\mathcal{E}_1), \cdots,\mathcal{E}_n$ independent, hence $\mathcal{E}_2, \cdots,\mathcal{E}_n,\sigma(\mathcal{E}_1)$ independent. THen repeart