What is missing from the following statement?
If $W$ is a Wiener process, then it has independent increments, which means for arbitrary $$0=t_0\leq t_1\leq ...\leq t_n\leq s\leq t,$$ $$ \left\{ W_{t_{i}}-W_{t_{i-1}}\;(i=1,...,n),W_{t}-W_{s}\right\} $$ are independent random variables. In this case, the $$X=\left(W_{u}\right)_{u\leq s}$$ process is independent from $$ W_{t}-W_{s},$$ which is the same if we write $$\mathcal{F}_{s}=\sigma\left(W_{u}:u\leq s\right)$$ is independent from $$W_{t}-W_{s}.$$