Is Ito's isometry true for conditional expectations too?
I mean, is it true that:$\mathbb{E}\left[\left(\int_0^tX_sdB_s\right)^2\ |\ \mathcal{F}_t^B\right]=\mathbb{E}\left[\int_0^tX^2_sds\ |\ \mathcal{F}_t^B\right]$ where $B_t$ is a Brownian motion and $\mathcal{F}_t^B$ the natural filtration?