Let (left(B_{t}, mathscr{F}_{t} ight)_{t geqslant 0}) be a one-dimensional Brownian motioin and (tau) a stopping time. Show

Let \(\left(B_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}\) be a one-dimensional Brownian motioin and \(\tau\) a stopping time. Show that \(f(s, \omega):=\mathbb{1}_{[0, T \wedge \tau(\omega))}(s), 0 \leqslant s \leqslant T

Data From 15.15 Theorem 

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15.15 Theorem. Let (B)to be a BM, (F)to be an admissible filtration and f = 2. Then we have for all t