I am stomped by the following exam preparation question
Problem: Let $\eta = \sum_{i=1}^\kappa \delta_{X_i}$ be a proper point process on some measurable space $( \mathbb{X}, \mathcal{X})$ with intensity measure $\lambda$ and $\eta' = \sum_{i=1}^{\kappa'} \delta_{Y_i}$ another proper point process (on the same space) with intensity measure $\lambda'$
Define $\xi:= \sum_{i=1}^{\kappa'} \sum_{j=1}^\kappa \delta_{X_i + Y_j}$ and find its intensity measure.
My approach: Nothing much to add here, I can say what I cannot do. I cannot use Marking, since there is no note about either of the marks $X_i, Y_i$ being independent or arise naturally from a $K$-Marking.
I also cannot use conditioning, because I don't know how the variables $X_i, Y_i, \kappa, \kappa'$ are distributed.
I most-likely shouldn't use the Laplace Functional, because the expressions would become very complicated. Hence I might use Campbell's Formula which states that for all $f: \mathbb{X} \to \mathbb{R}_+$ measurable we have $$ \mathbb{E}\left( \int_\mathbb{X} f d \eta \right) = \int_\mathbb{X} f d \lambda $$
I probably want to use this result for $f: \mathbb{X} \times \mathbb{Y} \to \mathbb{R}_+$ given by $f(x,y)=x+y$ and a point process of the form $$ \zeta = \sum_{i=1}^{\kappa'} \sum_{j=1}^\kappa \delta_{(X_i,Y_j)}$$ then $f( \zeta)= \xi$. However this brings me nowhere near a solution.