If $B=\{B_t\}_{t\ge{0}}$ is a Brownian motion adapted to $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t\ge{0}},P)$, let's define a Poisson random measure (or jump measure) as follows:
Definition. Let $\textbf{B}_0$ be the family of Borel sets $U\in{\mathbb{R}}$ whose closure $\bar{U}$ does not contain 0. For $U\in{\textbf{B}_0}$ we define the Poisson random measure as:$$N(t,U)=N(t,U,\omega)=\displaystyle\sum_{s:0<s\le{t}}{\chi_{U}(B_s-B_{s-})},$$ where $\chi_U$ is the indicator funcion of $U$.
Since $N(t,U)$ counts the number of jumps of the Brownian motion whose size is in $U$ in the period $(0,t)$, and since the Brownian motion does an infinite number of jumps of intinitesimal size in every interval $(0,t)$, I'm trying to understand if $N(t,U)=\infty$ or $N(t,U)=0.$
My attempt: If I'm not wrong, a Brownian motion is a continuous process, so for all $t\ge{0}$ we have $B_t-B_{t-}=0$ a.s., but $0\notin{\bar{U}}$ which means that $N(t,U)=0$ a.s. for every $t\ge{0}$ and $U\in\mathbf{B_0}.$
