\(\mathcal{L}^{2}\)-bounded martingales. A martingale \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) is called \(\mathcal{L}^{2}\)-bounded, if the \(\mathcal{L}^{2}\)-norms are bounded: \(\sup _{n \in \mathbb{N}} \int u_{n}^{2} d \mu<\infty\). For ease of notation set \(u_{0}:=0\).

(i) Show that \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is \(\mathcal{L}^{2}\)-bounded if, and only if,

\[\sum_{n=1}^{\infty} \int\left(u_{n}-u_{n-1}ight)^{2} d \mu<\infty\]

[ use Problem 23.6 .]

Assume from now on that \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is \(\mathcal{L}^{2}\)-bounded.

(ii) Show that \(\lim _{n ightarrow \infty} u_{n}=u\) exists a.e.

[ \(\left(\mathbb{1}_{K} u_{i}ight)_{i \in \mathbb{N}}\) is an \(L^{1}\)-bounded martingale for any \(K \in \mathscr{A}_{0}\) such that \(\mu(K)<\infty\).]

(iii) Show that

\[\lim _{n ightarrow \infty} \int\left(u-u_{n}ight)^{2} d \mu=0\]

[ check that \(\int\left(u_{n+k}-u_{n}ight)^{2} d \mu=\sum_{\ell=n+1}^{n+k} \int\left(u_{\ell}-u_{\ell-1}ight)^{2} d \mu\) and apply Fatou's lemma.]

(iv) Assume now that \(\mu(X)<\infty\). Show that \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is uniformly integrable, that \(u_{n} ightarrow u\) in \(\mathcal{L}^{1}\) and that \(u_{\infty}:=u\) closes the martingale to the right, i.e. that \(\left(u_{n}ight)_{n \in \mathbb{N} \cup\{\infty\}}\) is again a martingale.

Data from problem 23.6

Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a martingale with \(u_{n} \in \mathcal{L}^{2}\left(\mathscr{A}_{n}ight)\). Show that

\[
\int u_{n} u_{k} d \mu=\int u_{n \wedge k}^{2} d \mu
\]

[assume that \(n