Brownian closed-set hitting times are stopping times

If $C=\mathrm{\varnothing }$, then ${\tau }_C=\mathrm{\infty }$ and all its finite-time test events are empty; this case is settled. Now assume $C\mathrm{\ne }\mathrm{\varnothing }$. Fix $t\ge 0$ and put $G_t:={\bigcap }_{n\ge 1}{\bigcup }_{q\in \mathbb{Q}\cap [0,t]}\{\mathrm{dist}(B_q,C)<1\mathrm{/}n\}$. For a point of $G_t$, use the declared AC to select $q_n\in \mathbb{Q}\cap [0,t]$ with $\mathrm{dist}(B_{q_n},C)<1\mathrm{/}n$ for every $n\ge 1$. Apply [F4] to the bounded sequence $(q_{j+1}{)}_{j\in \mathbb{N}}$, obtaining a subsequence $q_{n_k}$ converges to some $q^{\ast }\in [0,t]$; continuity of the path and of the distance give $0\le \mathrm{dist}(B_{q^{\ast }},C)\le {\mathrm{lim\hspace{0.17em}inf}}_k(\mathrm{dist}(B_{q^{\ast }},C)-\mathrm{dist}(B_{q_{n_k}},C))+{\mathrm{lim\hspace{0.17em}sup}}_{k}1\mathrm{/}{n}_k=0$, hence $\mathrm{dist}(B_{q^{\ast }},C)=0$ and $B_{q^{\ast }}\in C$ by [F3], so ${\tau }_C\le q^{\ast }\le t$. Thus $G_t\subseteq \{{\tau }_C\le t\}$.

Conversely suppose ${\tau }_C\le t$. The hitting set is nonempty and bounded below. By its infimum property choose, using AC, a hit $s_n\in [{\tau }_C,{\tau }_C+1\mathrm{/}n)$ for each $n\ge 1$. Then $s_n\to {\tau }_C$, so continuity and [F3] imply $d_C(B_{{\tau }_C})={\lim }_n{d}_C(B_{s_n})=0$; hence $B_{{\tau }_C}\in C$. Given $n\ge 1$, continuity at $s={\tau }_C$ and rational density supply $q\in \mathbb{Q}\cap [0,t]$ with $\mathrm{\mid }{B}_q-B_s\mathrm{\mid }<1\mathrm{/}n$: use an interior rational sufficiently near $s$ when $s>0$, and $q=0$ when $s=0$. Thus $d_C(B_q)\le \mathrm{\mid }{B}_q-B_s\mathrm{\mid }<1\mathrm{/}n$. This proves $\{{\tau }_C\le t\}\subseteq G_t$. Together with step 1.1 it yields equality for every $t\ge 0$. At $t=0$, $G_0={\bigcap }_{n\ge 1}\{d_C(B_0)<1\mathrm{/}n\}=\{B_0\in C\}$.

Each set $\{\mathrm{dist}(B_q,C)<1\mathrm{/}n\}$ is in ${\mathcal{F}}_q^0\subseteq {\mathcal{F}}_t^0$ for $q\le t$, because $\mathrm{dist}(\cdot ,C)$ is continuous hence Borel and $B_q$ is ${\mathcal{F}}_q^0$-measurable; the union over the countable set $\mathbb{Q}\cap [0,t]$ and the intersection over $n$ are therefore in ${\mathcal{F}}_t^0$. By step 2.1, $\{{\tau }_C\le t\}\in {\mathcal{F}}_t^0$ for every $t\ge 0$, so ${\tau }_C$ is a stopping time for $({\mathcal{F}}_t^0)$ by [F1], and hence for $({\mathcal{F}}_t)$ by [F2].

The result concerns the given everywhere-continuous process; [F5] supplies the canonical realization as an example. No transfer to another raw natural filtration by null modification is used. The usual-augmentation conclusion follows solely by inclusion in step 3.1. For $C=\mathbb{R}$, ${\tau }_C=0$ and $G_t=\mathrm{\Omega }$; the empty-set case was settled in step 1.1. AC supplies the countable selections of approximating rational times and hit times made in steps 1.1 and 2.1, as well as the ambient construction [F6].