2 problems

Let $f\in L^2_{\operatorname{imp}}$, let $(A_k)$ be pairwise disjoint Borel subsets of $\mathbb R_+$, and define $f_k=f\mathbf 1_{A_k}$. Let $(S_n)$ be the arrival times of a Poiss…

Let $N$ be a Poisson random measure and let $L^2_{\operatorname{imp}}$ be the space of bounded, square-integrable functions for which the relevant improper integrals exist. For a f…