Given is a pure jump Lévy process $L$ with Lévy measure $\nu$. Let $m>0$ and set $J = \{x: ||x|| > m\}$. Moreover let $t_i$ be a strictly increasing sequence with $t_0=0$, $t_i \rightarrow \infty$ and define $\tau_i= \inf\{t: t_{i-1} < t \leq t_i, ||\Delta L_t|| \in J\}$, so $\tau_i$ is the first time in $(t_{i-1},t_i]$ where $L$ has a jump bigger than $m$, otherwise, if this jump doesn't exist, set $\tau_i = \infty$.
I did some calculations and ended up with needing the following property:
$$ E[1_{(0,\infty)}(\tau_i) \Delta L_{\tau_i}] = \frac{1-e^{-\nu(J)(t_i-t_{i-1})}}{\nu(J)} \int_J x \nu(dx) $$
How do I estblish this equality (if it even holds)?