The following is of Philip Protter at page 26 of the book *Stochastic integration and Differential* equations that I have not been able to proved yet.

Let $X$ a Levy process, and $\Lambda$ a borel set in $\mathbb{R}$ away from $0$ (that is, $0 \notin \bar{\Lambda}$), then $\nu(\Lambda) = E[N_{1}^{\Lambda}] < \infty$. Where $N_{t}^{\Lambda} = \sum_{0 < s \leq t} 1_{\Lambda}(\Delta X_{s})$. Protter says that it follows from Theorem 34, but for me it means I have to prove first that the jumping times have independent and stationary increments something that I am not sure.

**My attempt**
I assume that the jumping times $T_{\Lambda}^{1} := \lbrace t >0 : \Delta X_t \in \Lambda \rbrace, \cdots, T_{\Lambda}^{n}:= \lbrace t > T_{\Lambda}^{n-1}: \Delta X_{t} \in \Lambda \rbrace,...$ have independent and stationary increments. Therefore, taking into account that $ T_{\Lambda}^{n} = T_{\Lambda}^{1} + \sum_{k=1}^{n} T_{\Lambda}^{k} - T_{\Lambda}^{k-1} $ we get

\begin{align} \nu(\Lambda) &= E[N_{1}^{\Lambda}] \\ &=\sum_{n=1}^{\infty}P[T_{\Lambda}^{n} \leq 1] \\ &\leq \sum_{n=1}^{\infty}P[T_{\Lambda}^{1} \leq 1, ..., T_{\Lambda}^{n} - T_{\Lambda}^{n-1} \leq 1 ] \\ &\leq \sum_{n=1}^{\infty}(P[T_{\Lambda}^{1} \leq 1])^{n} < \infty, \end{align} and the last part depends on the fact $P[T_{\Lambda}^{1} \leq 1] < 1$ something that I am not quite sure. Apparently, this is not at all an intuitive fact.

Any hint will be welcome.