I'm reading a book about Brownian Motion and the write the following lemma:
Let $(B_t)_{t \geq 0}$ be a d-dimensional Brownian Motion and $D \subset \mathbb{R}^n$ be an open bounded set. Then $E^x[\tau_D] < \infty$ ($\tau_D$ is the first hitting time of $D$)
They proceed to use it in the following propostion:
If $u$ is continous and $u(B_{t \wedge \tau_G})$ is a martinagle with repect to the law $P^x$ of a Brownian Motion $(B_t)$ started at any $x\in G$ then $u$ has the spherical mean value property. (Here $G$ is a bounded open set in $D$)
The proof starts by considering a small ball $B \subset \bar{G} \subset D$ and defining $\tau_B$ as the exit time. They then assert $\tau_B < \infty$ by the first lemma I wrote.
I don't see why this is true - the lemma only says the expectation is finite. There could be some measure zero sets that blow up - violating $\tau_B < \infty$. What did I miss?
(For those curious: this is Lemma 8.8 and the first part of the proof of Thm 8.10 here.)