Why is $\{(\Delta_1,...,\Delta_k) \in B_1 : k\geq 1, B\in \mathcal{B}(\mathbb{R}^k)\}$ a $\cap$-stable generator of $\mathcal{F}_u$?
Suppose $(W_t)_t$ is a standard Brownian motion and $0=s_0<s_1<\cdots s_k\leq u$. Define $\Delta_1=W_{s_1}-W_{s_0}$, $\Delta_2=W_{s_2}-W_{s_1}$ and so on up to $\Delta_k=W_{s_k}-W_{s_{k-1}}$.
If I remember correctly, $$\mathcal{F}_u=\sigma(W_t : t\leq u)=\sigma\left(\bigcup \limits_{t\leq u}\sigma(W_t)\right)=\sigma\left(\bigcup\limits_{t\leq u} \{W_t^{-1}(A) : A\in \mathcal{B}(\mathbb{R})\}\right)$$
I think $\{(\Delta_1,...,\Delta_k) \in B_1 : k\geq 1, B\in \mathcal{B}(\mathbb{R}^k)\}$ is a generator since choosing $k=1$ and $\Delta_1=W_{s_1}-W_{s_0}=W_{s_1}-W_0=W_{s_1}$, we can range $s_1$ over all $[0,u]$ and thus "know everything" up to $u$. But this is just an informal thought. I don't really know how to approach it formally, especially showing the stability under intersection. Can someone help?