Let \(\left(X_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a martingale and denote by \(\mathscr{F}_{t}^{*}\) be the completion of \(\mathscr{F}_{t}\) (completion means to add all subsets of \(\mathbb{P}\)-null sets).

a) Show that \(\left(X_{t}, \mathscr{F}_{t}^{*}ight)_{t \geqslant 0}\) is a martingale.

b) Let \(\left(\widetilde{X}_{t}ight)_{t \geqslant 0}\) be a modification of \(\left(X_{t}ight)_{t \geqslant 0}\). Show that \(\left(\widetilde{X}_{t}, \mathscr{F}_{t}^{*}ight)_{t \geqslant 0}\) is a martingale.