Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a (sub, super)martingale and let \(\left(\mathscr{B}_{n}ight)_{n \in \mathbb{N}}\) and \(\left(\mathscr{C}_{n}ight)_{n \in \mathbb{N}}\) be filtrations in \(\mathscr{A}\) such that \(\mathscr{B}_{n} \subset \mathscr{A}_{n} \subset \mathscr{C}_{n}\).

(i) Show that \(\left(u_{n}, \mathscr{B}_{n}ight)_{n \in \mathbb{N}}\) is again a (sub, super)martingale.

(ii) Show that \(\left(u_{n}, \mathscr{C}_{n}ight)_{n \in \mathbb{N}}\) is, in general, no longer a (sub, super)martingale.