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



Brownian motion and Durret book
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.


