Let \(\tau\) be a stopping time for the filtration \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\). Show that

a) \(F \in \mathscr{F}_{\tau+} \Longleftrightarrow \forall t \geqslant 0: F \cap\{\tau

b) \(\tau \wedge t\) is again a stopping time;

c) \(\{\tau \leqslant t\} \in \mathscr{F}_{\tau \wedge t}\) for all \(t \geqslant 0\).