Let $(W_t)_{t\in [0,\infty)}$ be a Wiener process, and let $\mathcal F$ be a filtration adapted to $(W_t)$. The book 'Applied Stochastic Analysis' claims the following:
Let $t_1< t_2<t_3<t_4$. Let $e_1, e_2$ be a function on the sample space which is measurable in $\mathcal F_{t_1}, \mathcal F_{t_3}$, respectively. Define $\delta W_1 = W_{t_2}-W_{t_1}$ and $\delta W_2= W_{t_4}-W_{t_3}$. Then, the book asserts that $e_1e_2\delta W_1$ and $\delta W_2$ are independent. Why is it?
$\delta W_2$ is independent of $\mathcal F_{t_3}$ and $e_1e_2\delta W_1$ is measurable w.r.t $\mathcal F_{t_3}$. Hence they are independent.