Assume that \(\left(X_{t}ight)_{t \geqslant 0}\) is a uniformly bounded stochastic process with exclusively continuous sample paths, \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\) is some filtration, and \(\sigma\) is an everywhere finite stopping time.

a) Show that \(\sigma \wedge n+t\) are again stopping times.

b) Show that \(\mathscr{F}_{\sigma}=\sigma\left(\bigcup_{n \geqslant 1} \mathscr{F}_{\sigma \wedge n}ight)\). This remains true for \(\sigma<\infty\) almost surely if \(\mathscr{F}_{0}\) contains all measurable null sets.

c) Use Lévy's upwards martingale convergence theorem and the (conditional) dominated convergence theorem to show that \(\mathbb{E}\left(X_{\sigma \wedge n} \mid \mathscr{F}_{\sigma \wedge n}ight) ightarrow \mathbb{E}\left(X_{\sigma} \mid \mathscr{F}_{\sigma}ight)\) a.s. and in \(L^{1}\).