Working through the book "Brownian Motion & Stochastic Processes" by Karatzas and Shreve, 
I found the following problem (page 6, Problem 2.2):

Let $ X $ be a stochastic process and $ T $ a stopping time of $ \{ \mathcal{F}\_t^X \}$, where $ \mathcal{F}\_t^X := \sigma (X\_s, 0 \leq s \leq t) $.
Suppose that for any $ \omega, \omega ' \in \Omega $, we have $ X\_t (\omega) = X\_t ( \omega '),
 $ for all $ t \in [0, T ( \omega )] \cap [ 0, \infty ) $ .

Show that $ T ( \omega ) = T ( \omega ') $.

Does anybody know how to prove this? Thanks a lot for your efforts!
Regards, Si