Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space $C([0,\infty),\mathbb{R})$ .
I'm trying to prove that the set $\left((W_{t})_{t\geq0}\;\mathrm{hits}\; A\right)$ is $\mathcal{F}$ -measurable if $A$ is a Borel measure set in $\mathbb{R}^{2}$.
If $A$ is open, then it is clear to me that only need to consider rationals, since $W$ will have to enter $A$. Similarly, if it closed then it hits iff it get arbitrarily close, so rationals again suffice. But I'm not sure how to proceed for a general borel set.
Also, what if we do not know anything about $\mathcal{F}$ . Can we then still get a conclusion?
And finally, is the event $\left((W_{t})_{t\geq0}\;\mathrm{hits}\; A\right)$ measurable for more general sets than Borel measurable ones?
EDIT: I am aware of the so-called Debut Theorem about hitting times being stopping times (implying measurability), but I am looking for a more elementary proof of just the fact that the event is measurable.