# Two Questions about Brownian Motion

How do you show $B_T\in\mathcal{F}_T$ for T is a stopping time?

Note the filtration is generated by the Brownian motion (and not necessarily completed, in particular, $\mathcal{F}_T\neq\mathcal{F}_{T+}$)

and a much harder question:

Are all Brownian Motion stopping times previsible? (Please point me to a proof or reference)

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Since you assert that the first question is much easier than the second, what did you try to solve it? –  Did Feb 26 '13 at 19:52
@Did someone has shown me a proof, which I vaguely remember, you take a map from the sample path to $I_{(t\leq T)}, B_t$ and something else and use some composition of maps. He thinks there is an easier method and I did not quite follow the proof. If we wish to prove it is measurable with respect to $\mathcal{F}_{T+}$, then we can take a sequence of stopping time $T_n\downarrow T$, but the assertion above is a bit harder. –  Lost1 Feb 26 '13 at 20:00

Answer to the first question is an application Proposition 2.18 on Karatzas and Shreve, where we take $X$ to be the Brownian motion in question.