Continuous adapted processes are measurable at bounded stopping times

Fix $t\in (0,T]$ and, for each integer $n\ge 1$, form the deterministic grid $G_{n,t}=\{kt{2}^{-n}:0\le k\le 2^n\}$. Let ${\tau }_{n,t}$ be the least point of $G_{n,t}$ greater than or equal to $\tau \wedge t$. For $s<t$, $\{{\tau }_{n,t}\le s\}={\bigcup }_{q\in G_{n,t},q\le s}\{\tau \le q\}\in {\mathcal{F}}_s$ by [F2] and [F3]; for $s\ge t$ the event is all of $\mathrm{\Omega }$. Thus ${\tau }_{n,t}$ is a finite-valued stopping time, ${\tau }_{n,t}\le t$, and ${\tau }_{n,t}\downarrow \tau \wedge t$. Each event $\{{\tau }_{n,t}=q\}$ belongs to ${\mathcal{F}}_q$ by subtracting consecutive sublevel events, so the finite-valued variable $X_{{\tau }_{n,t}}$ is ${\mathcal{F}}_t$-measurable by [F3].

Define $Z_t$ as the limit of $X_{{\tau }_{n,t}}$ where that finite vector limit exists, and as $0$ otherwise. Since each approximant is ${\mathcal{F}}_t$-measurable, $Z_t$ is ${\mathcal{F}}_t$-measurable. On $\{\tau \le t\}$ and on the full-probability continuity event, ${\tau }_{n,t}\downarrow \tau$ and continuity gives $Z_t=X_{\tau }$. The case $t=0$ follows directly from $\tau \wedge 0=0$ and $Z_0=X_0$.

For any Borel set $C\subseteq {\mathbb{R}}^d$, the events $\{X_{\tau }\in C\}\cap \{\tau \le t\}$ and $\{Z_t\in C\}\cap \{\tau \le t\}$ differ only on the measurable null set where path continuity may fail. The second event is in ${\mathcal{F}}_t$ by [F2] and step 2.1; [F1] places their symmetric difference in ${\mathcal{F}}_t$, so the first event is in ${\mathcal{F}}_t$. At $t=T$ this also gives $\{X_{\tau }\in C\}\in {\mathcal{F}}_T$. All tests in [F2] hold, proving $X_{\tau }\in {\mathcal{F}}_{\tau }$. AC enters only through [A1].