Let \(\left(X, \mathscr{A}, \mathscr{A}_{n}, \muight)\) be a \(\sigma\)-finite filtered measure space and denote by \(\langle u, \phiangle\) the canonical dual pairing between \(u \in L^{p}\) and \(\phi \in L^{q}\), where \(p^{-1}+q^{-1}=1\), namely \(\langle u, \phiangle:=\int u \phi d \mu\). A sequence \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset L^{p}\) is weakly relatively compact if there exists a subsequence \(\left(u_{n_{k}}ight)_{k \in \mathbb{N}}\) such that

\[\left\langle u_{n_{k}}-u, \phiightangle \underset{k ightarrow \infty}{ } 0\]

holds for all \(\phi \in L^{q}\) and some \(u \in L^{p}\). Show that for a martingale \(\left(u_{n}ight)_{n \in \mathbb{N}}\) and every \(p \in(1, \infty)\) the following assertions are equivalent:

(i) there exists some \(u \in L^{p}(\mathscr{A})\) such that \(\lim _{n ightarrow \infty}\left\|u_{n}-uight\|_{p}=0\);

(ii) there exists some \(u_{\infty} \in L^{p}\left(\mathscr{A}_{\infty}ight)\) such that \(u_{n}=\mathbb{E}^{\mathscr{A}_{n}} u_{\infty}\);

(iii) the sequence \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is weakly relatively compact.