I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to me that $T$ depends on random variable $\omega$ in the measure space $\Omega$, so, my question is the previous definition indeed: $T(\omega) = \inf \{t: B_t(\omega) \in A\}$.
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Yes: assuming $B_t$ is a random variable, $T$ is also a random variable, and the inf is done pointwise with respect to the sample space.
answered Jul 22, 2019 at 13:38
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$\begingroup$ Indeed, $B_t$ is a Brohnian motion. $\endgroup$– LiraCommented Jul 22, 2019 at 14:07