Assume that \(\left(B_{t}, \mathscr{G}_{t}ight)_{t \geqslant 0}\) and \(\left(M_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) are independent martingales. Show that \(\left(M_{t}ight)_{t \leqslant T}\) and \(\left(B_{t}ight)_{t \leqslant T}\) are martingales for the enlarged filtration \(\mathscr{H}_{t}:=\sigma\left(\mathscr{F}_{t}, \mathscr{G}_{t}ight)\).