In order to prove that $(B_t^4)_{t \in \mathbb{R+}}$ is a continuous semi-martingale I need to compute. $E(\int_s^t B_u^2 du |F_s)$ I was first thinking that $\int_s^t B_u^2 du$ independent of $ F_s$ but this can't be true but I can't argue why ? Could someone help me out here. I say that it can't be true because I am given the Doob-decomposition of $(B_t^4)_{t \in \mathbb{R+}}$ as
$B_t^4=(B_t^4-6\int_0^t B_u^2 du) +6\int_0^t B_u^2 du$
and when I try to show that $(B_t^4-6\int_0^t B_u^2 du)_{t \in \mathbb{R_+}}$ is a martinagale , for the proof to work , $\int_s^t B_u^2 du$ can't be independent of $ F_s$ because if it were , then $(B_t^4-6\int_0^t B_u^2 du)_{t \in \mathbb{R_+}}$ won't be a martingale