Let \((X, \mathscr{A}, \mu)\) be a measure space and assume that \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) is a sequence of pairwise disjoint sets such that \(\bigcup_{n \in \mathbb{N}} A_{n}=X\) and \(0<\mu\left(A_{n}ight)<\infty\). Then denote \(\mathscr{A}_{n}:=\sigma\left(A_{1}, A_{2}, \ldots, A_{n}ight)\) and \(\mathscr{A}_{\infty}:=\sigma\left(A_{n}: n \in \mathbb{N}ight)\).

(i) Show that \(L^{2}\left(\mathscr{A}_{n}ight) \subset L^{2}(\mathscr{A})\) and that \(L^{2}\left(\mathscr{A}_{n}ight)\) is a closed subspace.

(ii) Find an explicit formula for \(E^{\mathscr{A}_{n}} u\), where \(E^{\mathscr{A}_{n}}\) is the orthogonal projection \(E^{\mathscr{A}_{n}}: L^{2}(\mathscr{A}) ightarrow L^{2}\left(\mathscr{A}_{n}ight)\)

(iii) Determine the orthogonal complement of \(L^{2}\left(\mathscr{A}_{n}ight)\).

(iv) Show that \(\left(E^{\mathscr{A}_{n}} uight)_{n \in \mathbb{N} \cup\{\infty\}}, u \in L^{1}(\mathscr{A}) \cap L^{2}(\mathscr{A})\), is a martingale.

(v) Show that \(\lim _{n ightarrow \infty} E^{\mathscr{A}_{n}} u=E^{\mathscr{A} \infty} u\) a.e. and in \(L^{2}\) for all \(u \in L^{1}(\mathscr{A}) \cap L^{2}(\mathscr{A})\).

(vi) Conclude that \(L^{2}\left(\mathscr{A}_{\infty}ight)\) is separable.