Let (M in mathcal{M}_{T, text { loc }}^{c}) and (f(t, omega)) be an adapted cdlg (right continuous,

Let \(M \in \mathcal{M}_{T, \text { loc }}^{c}\) and \(f(t, \omega)\) be an adapted càdlàg (right continuous, finite left limits) process and \(t \in[0, T]\).

a) Show that \(f \in \mathcal{L}_{T}^{0}(M)\), i.e. \(\int_{0}^{T} f(t, \omega) d\langle Mangle_{t}<\infty\) a.s.

b) Show that for any sequence of partitions \(\Pi\) of \([0, T]\) with \(|\Pi| \downarrow 0\) we have \(f^{\Pi} \bullet M ightarrow f \bullet M\) in ucp-sense. Here

\[f^{\Pi}(t, \omega):=\sum_{\Pi} f\left(t_{i-1}, \omegaight) \mathbb{1}_{\left[t_{i-1}, t_{i}ight)}(t) .\]