I am reading the "Construction of Diffusion processes with Wentzell's Boundary conditions by means of poisson point processes of Browninan excursions" of S. Watanabe and he says:
I understand that to say that $N_p((0,t]\times U)$ is $\mathcal{F}_t$ adapted means $$\forall a >0\qquad\{\omega\mid N_{p(\omega)}((0,t]\times U))>a\} \in \mathcal{F}_t $$ But what does it mean to say that $\{N_p((0,t]\times U)\}_{U \in \mathbb{B}(X)}$ is $\mathcal{F}_t$ adapted?
The only way I make sense of this is that for all $U \in \mathcal{B}(X)$ and $t$ it is required to be true that the random variables $f_{U, t}$ given by $$f_{U, t}(\omega) = N_{p(\omega)}((0, t] \times U)$$ are $\mathcal{F}_t$ measurable. I have placed $U$ and $t$ in the subscript of $f$ to highlight the dependence of its definition on those two parameters.