Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space, let \(u\) be a further measure on \(\mathscr{A}\) and let \(\left(A_{n, i}ight)_{i \in \mathbb{N}} \subset \mathscr{A}\) be for each \(n \in \mathbb{N}\) a sequence of mutually disjoint sets such that \(X=\biguplus_{i \in \mathbb{N}} A_{n, i}\). Assume, moreover, that each set \(A_{n, i}\) is the union of finitely many sets from the sequence \(\left(A_{n+1, k}ight)_{k \in \mathbb{N}}\). Show that

(i) the \(\sigma\)-algebras \(\mathscr{A}_{n}:=\sigma\left(A_{n, i}: i \in \mathbb{N}ight)\) form a filtration;

(ii) if \(\mu\left(A_{n, i}ight)>0\) for all \(n, i \in \mathbb{N}\), then

\[u_{n}:=\sum_{i=1}^{\infty} \frac{u\left(A_{n, i}ight)}{\mu\left(A_{n, i}ight)} \mathbb{1}_{A_{n, i}}\]

is a martingale w.r.t. \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\).