The following comes from some remarks of Philip Protter at page 26 of the book Stochastic integration and Differential equations that I have not been able to prove yet.

Let $X$ a Levy process, under a filtration satisfying the usual conditions. If $\Lambda$ is a Borel set in $\mathbb{R}$ bounded away from zero (that is $0 \notin \bar{\Lambda}$), then the jumping times

\begin{align} &T_{\Lambda}^{1} = \lbrace t \geq : \Delta X_{t} \in \Lambda \rbrace \\ &\vdots \\ &T_{\Lambda}^{n} = \lbrace t > T_{\Lambda}^{n-1} : \Delta X_{t} \in \Lambda \rbrace \end{align} are stopping times.

My attempt Since the filtration satisfies the usual conditions, we only need to prove that $\lbrace T_{\Lambda} < t \rbrace \in \mathcal{F}_{t}$.

Let $\epsilon := d(0, \Lambda) >0$ and $M:= ( - \infty, - \epsilon] \cup [\epsilon, \infty)$, I am trying to prove

\begin{align} \lbrace T_{\Lambda} < t \rbrace = \left( \bigcup_{r \in [0, t) \cap \mathbb{Q}} \lbrace \Delta X_{r} \in \Lambda \rbrace \right) \cap \lbrace T_{M} < t\rbrace \end{align} If we can prove this equation, we are done. This is due to the fact that $\lbrace T_{M} \leq t\rbrace \in \mathcal{F}_{t}$ since \begin{align} \lbrace T_{M} < t\rbrace = \bigcap_{n} \bigcup_{r,s \in [0, t+1/n)\\ \vert r-s \vert < 1/n} \lbrace \vert X_{s} - X_{r} \vert > \epsilon \rbrace \end{align}

We know that the "$\supset$" is the easy part, but the "$\subset$" part is the only part that I need to prove. I was trying to prove this by contradiction, and seems that it is the best way.

If $w \in \lbrace T_{M} < t\rbrace $ and $w \notin \lbrace T_{M} < t\rbrace $ is a contradiction. This can be done using lemmas of discontinuities and the fact that $d(0, \Lambda) >0$. However the part $w \in \lbrace T_{M} < t\rbrace $ and $w \notin \left( \bigcup_{r \in [0, t) \cap \mathbb{Q}} \lbrace \Delta X_{r} \in \Lambda \rbrace \right)$ is the difficult one.

Any hint will be welcome.


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