Let us define the following two stopping time $\tau\_B=\inf\{t\geq 0: X\_t\in B\}, \tau'\_B=\inf\{t> 0: X\_t\in B\}$, where $\tau\_B$ is entrance time and $\tau'\_B$ is hitting time. It is clear $\tau\_B=\tau'\_B, \mathbb{P}-a.s.$. By the book [Levy process](http://www.maphysto.dk/oldpages/events/LevyBranch2000/) by Bertoin P22, when $B$ is closed $\tau\_B=\tau'\_B$ may fail on $\partial B$. These point is called irregular point of B. When our starting point $x$ is irregular point of $B$, using Bulmenthal 0-1 law, $\mathbb{P}(\tau'\_B>0)=1$.


I don't understand how to use Bulmenthal 0-1 law? Is there someone to explain a little bit?



Blumenthal 0-1 law
$\newcommand\F{\mathcal F}\newcommand\N{\mathbb N}$First, some preliminary remarks:


1. Your link to Bertoin's book is not very good. Here is a link with a [better reference to the book](https://www.ams.org/bull/1998-35-04/S0273-0979-98-00761-7/S0273-0979-98-00761-7.pdf).
2. It is Blumenthal, not Bulmenthal.
3. It is not true that "when $B$ is closed $\tau\_B=\tau'\_B$ is failed on $\partial B$". This kind of failure may occur only at some (not all) points on $\partial B$, which are then called irregular points for $B$.
4. In Bertoin's book, $(X\_t)\_{t\ge0}$ is a right-continuous Lévy process, and $\F\_t$ is the completion of the sigma-field generated by $(X\_s\colon0\le s\le t)$. According to the book, let $T\_B:=\tau\_B$ and $T'\_B:=\tau'\_B$.


Now, to your question, "how to use Bulmenthal 0-1 law":


Let $B$ be a closed set. By part (iii) of Corollary 8 on p. 22 in Bertoin's book, $T'\_B$ is a stopping time and hence $[T'\_B=0]=[T'\_B\le0]\in\F\_0$, where $[\cdots]$ means the event $\cdots$. So, by the definition of $\F\_t$ cited above in Remark 4, we see that $P\_x(T'\_B=0)$ is $0$ or $1$ for any $x$. Suppose now that a point $x\in\partial B$ is irregular for $B$. Then $P\_x(T\_B=T'\_B)\ne1$, while $P\_x(T\_B=0)=1$. So, $P\_x(T'\_B=0)\ne1$. Thus, $P\_x(T'\_B=0)=0$.


In the above proof we did not explicitly use the Blumenthal 0-1 law, which states that the filtration $(\F\_t)$ is right-continuous at $0$, that is, $\F\_{0+}=\F\_0$. However, in the proof in Bertoin's book of the fact that $T'\_B$ is a stopping time he used (not quite explicitly) Proposition 4 on p. 18 in that book, which states that the filtration $(\F\_t)$ is right-continuous at all real $t\ge0$, and the proof of Proposition 4 on p. 18 in that book is based on Kolmogorov's 0-1 law. So, this reference in Bertoin's book to the Blumenthal 0-1 law is indeed imprecise and possibly confusing.


