Let $E$ be a normed $\mathbb{R}$-vector space, $({X}_{t}{)}_{t\ge 0}$ be an $E$-valued càdlàg Lévy process on a filtered probability space $(\mathrm{\Omega},\mathcal{A},({\mathcal{F}}_{t}{)}_{t\ge 0},\mathrm{P})$, $B\in \mathcal{B}(E)$ with $0\notin \overline{B}$ and

${N}_{t}(\omega ):=\left|\{s\in (0,t]:\mathrm{\Delta}{X}_{s}(\omega )\in B\}\right|=\sum _{{\scriptstyle \begin{array}{c}s\in [0,\phantom{\rule{mediummathspace}{0ex}}t]\\ \mathrm{\Delta}{X}_{s}(\omega )\end{array}}}{1}_{B}(\mathrm{\Delta}{X}_{s}(\omega ))$

for $\omega \in \mathrm{\Omega}$ and $t\ge 0$.

How do we see that $t\mapsto {N}_{t}(\omega )$ is càdlàg?

${N}_{t}(\omega ):=\left|\{s\in (0,t]:\mathrm{\Delta}{X}_{s}(\omega )\in B\}\right|=\sum _{{\scriptstyle \begin{array}{c}s\in [0,\phantom{\rule{mediummathspace}{0ex}}t]\\ \mathrm{\Delta}{X}_{s}(\omega )\end{array}}}{1}_{B}(\mathrm{\Delta}{X}_{s}(\omega ))$

for $\omega \in \mathrm{\Omega}$ and $t\ge 0$.

How do we see that $t\mapsto {N}_{t}(\omega )$ is càdlàg?