Let $(X_t)_{t\geq 0}$ be a stochastic process and let $\mathcal F_t=\sigma(X_s:0\leq s\leq t)$ be the generated filtration. If $Y$ is a random variable independent of $X_s$ for all $0\leq s\leq t$, must $Y$ be independent of $\mathcal F_t$?
Edit: Answer is no. Suppose that $Y$ is a random variable independent of $(X_{t_1},\dots,X_{t_n})$ for all $n>0$ and $0\leq X_{t_1}<\dots<X_{t_n}\leq t$, must $Y$ be independent of $\mathcal F_t$?