I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I am trying to understand:
$$ \mathbb{P}(T_a \lt \infty ) = 1 $$
$$ \mathbb{E}[T_a]=\infty $$
These are from pg 6 of http://www.pstat.ucsb.edu/faculty/ludkovski/bmNotes.pdf, for example.
My question is what these mean and how they are not in conflict. I interpret the first one to say that with probability one the hitting time for any $a$ is finite. I interpret the second one to say that the expectation of the stopping time for any $a$ is infinite. This seems in conflict to me (i.e. saying that something is finite with probability one yet has infinite expectation?). Please help me gain some intuition here, and correct me if I am misunderstanding one or both of these properties.