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: $T(\omega) = \inf \{t: B_t(\omega) \in A\}$.