Martingale difference sequence. Let \(\left(d_{i}ight)_{i \in \mathbb{N}}\) be a sequence in \(\mathcal{L}^{2}(\mathscr{A}) \cap \mathcal{L}^{1}(\mathscr{A})\). Define \(\mathscr{A}_{0}:=\{\emptyset, X\}\) and \(\mathscr{A}_{n}:=\sigma\left(d_{1}, d_{2}, \ldots, d_{n}ight)\). Suppose that for each \(n \in \mathbb{N}\)

\[\int_{A} d_{n} d \mu=0 \quad \forall A \in \mathscr{A}_{n-1}\]

Show that \(\left(u_{n}^{2}ight)_{n \in \mathbb{N}}, u_{n}:=d_{1}+\cdots+d_{n}\), is a submartingale which satisfies

\[\int u_{n}^{2} d \mu=\sum_{i=1}^{n} \int d_{i}^{2} d \mu\]

Show that on \(\left(\mathbb{R}, \mathscr{B}(\mathbb{R}), \lambda^{1}ight)\) the sequence \(d_{i}(x):=\operatorname{sgn} \sin \left(2^{i} \pi xight), x \in \mathbb{R}, i \in \mathbb{N}\), is a martingale difference sequence. (See Chapter 28, in particular pages 389 and 391 for further details.)