Let \(B=\left(B_{t}ight)_{t \geqslant 0}\) be a canonical \(\operatorname{BM}^{1}\) on Wiener space \((\Omega, \mathscr{A}, \mathbb{P})=\left(\mathcal{C}_{(\mathrm{o})}, \mathscr{B}\left(\mathcal{C}_{(0)}ight), \muight)\) and \(\mathscr{F}_{t}:=\sigma\left(B_{s}, s \leqslant tight), \mathscr{F}_{\infty}=\) \(\sigma\left(\bigcup_{t \geqslant 0} \mathscr{F}_{t}ight)\), the filtration generated by \(B\). Show that the following assertions are equivalent:

a) \(\tau\) is a stopping time for \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\), i.e. \(\{\tau \leqslant t\} \in \mathscr{F}_{t}\) for all \(t \geqslant 0\);

b) If \(\omega, \omega^{\prime} \in \Omega\) satisfy \(\tau(\omega)=t\) and \(B_{s}(\omega)=B_{s}\left(\omega^{\prime}ight)\) for all \(s \leqslant t\), then \(\tau\left(\omega^{\prime}ight)=t\).