the probability measure $\mathbf{P}$. For two sets $A, B\in\mathcal{F}$ we shall write $A\subseteq B \ \; \mathbf{P}$-a.s. if $\mathbf{P}(B\setminus A)=0$. If $A\subseteq B \ \; \mathbf{P}$-a.s. and $B\subseteq A \ \; \mathbf{P}$-a.s., then we write $A=B \ \; \mathbf{P}$-a.s.

\[lem:For-all-,\]For all $t>0$, $\{ \beta_{t}=0\} =\{ \tau\leq t\},\ \mathbf{P}$-a.s. In particular, $\tau$ is a stopping time with respect to the filtration $\mathbb{F}^{P}$.

Using the formula of total probability and Corollary \[cor:LEM if–is\], we have that $$\begin{aligned}
\mathbf{P}\left(\beta_{t}=0,\,\tau>t\right) &=\int_{\left(t,+\infty\right)}\mathbf{P}\left(\beta_{t}=0|\tau=r\right)dF(r)\\
&=\int_{\left(t,+\infty\right)}\mathbf{P}\left(\beta_{t}^{r}=0\right)dF\left(r\right)=0\,,\end{aligned}$$ where the latter equality holds because the random variable $\beta_{t}^{r}$ is nondegenerate and normally distributed for $0<t<r$ and, therefore, we obtain $\mathbf{P}(\beta_{t}^{r}=0)=0,\,0<t<r$. Hence, for all $t>0$, $\{ \beta_{t}=0\} \cap\{ \tau>t\} \in\mathcal{N}_{P}$ and, consequently, $\{ \beta_{t}=0\} \subseteq\{ \tau\leq t\} ,\,\mathbf{P}$-a.s. In view of $\{ \tau\leq t\} \subseteq\{ \beta_{t}=0\}$, this yields the first part of the statement: $\{ \tau\leq t\} =\{\beta_{t}=0\} ,\,\mathbf{P}$-a.s. Since $\{ \beta_{t}=0\} \in\mathcal{F}_{t}^{0}$, the event $\{ \tau\leq t\}$ belongs to $\mathcal{F}_{t}^{0}\vee\mathcal{N}_{P}$, for all $t\geq 0$. Hence $\tau$ is a stopping time with respect to $\mathbb{F}^{P}$.

$\tau$ is a stopping time with respect to the filtration $\mathbb{F}^{\beta}$.

The above proposition states that the process $(\mathbb{I}_{\{ \tau\leq t\} },\, t>0)$ is a modification, under the probability measure $\mathbf{P}$, of the process $(\mathbb{I}_{\{ \beta_{t}=0\} 