Edited
I am working on the following problem
Let $\{B_t\}$ is Brownian motion. Here, $\tau = \sup\{t\in [0,5]: B_t=0\}$. Show that $$\{\tau \leq T\}=\bigcap_{t\in(T,5]}\{|B_t|>0 \}\not\in \mathcal{F}_T$$ where $\mathcal{F}_t$ is canonical sigma algebra of $B_t$ i.e., is generated by $B_t$.
I think it is true because the intersection is about 'future event' which should not be determined by observing the current event at time $T$. But I cannot see why it should not be there. Is there any good reason that the intersection is not $\mathcal{F}_T-$measurable?
This problem has been arisen from proving that last zero random time is not a stopping time.