Let $(\Omega ,\mathcal F,P)$ a proba space and $(\mathcal F_t)_{t\geq 0}$ a filtration. Let $(B_t)_t$ a Brownian motion adapted to $\mathcal F_t$. We know that $(B_{t+s}-B_t)_{s}$ is independent of $\mathcal F_t$.
My question are the following one:
1) I know that $B_{t+s}-B_t$ is independent of $B_t$ for all $s$. Does it has sense to say that $(B_{t+s}-B_t)_s$ is independent of $(B_t)_t$ ?
2) What does it mean that $(B_{t+s}-B_t)_s$ is independent of $\mathcal F_t$ ? We it mean that for all $s$ $B_{t+s}-B_t$ is independent of $B_t$ for all $t$ ?