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$X$ is a stochastic process and $T$ a stopping time of $\{F_t^X\}$. If $X_t(\omega) = X_t(\omega')$ for every $t\in[0,T(\omega)]\cap[0,\infty)$ and $\omega,\omega'\in \Omega$, then $T(\omega) = T(\omega')$.

Anyone could please give me a hint? Thank you very much.

R__
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  • Maybe for every $t \in ...$? – Botnakov N. Dec 24 '20 at 12:31
  • oh, it's true. thanks – R__ Dec 24 '20 at 13:32
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    https://math.stackexchange.com/questions/3921393/how-can-i-show-that-t-omega-t-overline-omega-when-x-t-omega-x-t/3923986#3923986 – John Dawkins Dec 24 '20 at 15:58
  • Thank you very much. Do you know how to prove this saturation property or where can I find it? – R__ Dec 24 '20 at 20:16
  • Okay, I think I know how to prove it: Firstly suppose $\omega^\prime\not\in A$ and get that $X_s(\omega) \not = X_s(\omega^\prime)$. To do so, it is enough to show that $X_s(A) \cap X_s(\bar{A}) = \emptyset$. Am I right? – R__ Dec 24 '20 at 20:32

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