I've read two times in different lecture notes that for a Brownian Motion $(B_s)_{s\ge 0}$ and $t<u$ the random variable $B^t_{u}:=B_{t+u}-B_t$ is independent from ${\cal F}_t:=\sigma\{B_s:s\le t\}$. Now the definition of Brownian Motion tells us that $B^t_u$ is independent from the individual $B_s$ but how do I climb up to the whole $\sigma$-field?
If I'm not mistaken it would suffice to prove independence for a $\cap$-closed generator of the $\sigma$-field, using the $\lambda$-$\pi$-theorem and such. But it's still puzzling me how to prove that $B^t_u$ is independent from events of the form $\{B_{s(i)}\in{A_i}:0\le i<n\}$ where $s(i)<t$ and $A_i\in{\cal B}({\bf R})$.