I am working with the script of Günter Last and Mathew Penrose Lectures on the Poisson Process (available online).
Definition 2.4 We shall refer to a point process $\eta$ on $\mathbb{X}$ as a proper point if there exist random variables $X_1, X_2, \dots$ in $\mathbb{X}$ and a $\overline{\mathbb{N}_0}$-valued random variable $\kappa$ such that almost surely $$\eta = \sum_{n=1}^\kappa \delta_{X_n} $$
At the end of the section is struggle with
Exercise 2.4: Let $\eta_1, \eta_2, \dots$ be a sequence of proper point processes. Show that $\eta:= \eta_1 + \eta_2 + \dots$ is a proper point process
My approach: My main problem is that I can't exactly reproduce the Definition 2.4, evidently I have for every $i \in \overline{\mathbb{N}}$ that $$\eta_i = \sum_{n=1}^{\kappa_i} \delta_{X_{n,i}} $$ Where the $\kappa_i$ are $\overline{\mathbb{N}_0}$-valued RV and the $X_{n,i}$ are the random variables for every $i \in \overline{\mathbb{N}}$, therefore $$ \eta = \sum_{i=1}^\infty \eta_i= \sum_{i=1}^\infty\sum_{n=1}^{\kappa_i} \delta_{X_{n,i}}\overset{?}=\sum_{n=1}^\gamma \delta_{X_n} $$ Where the Questionmark refers to Definition 2.4, can I define such a suitable RV $\gamma$ such that my double sum collapses into a (single) sum? My naive approach seems to lead me nowhere so far.
Update: I believe one approach that could get me out of my dilemma would be to take the random variables $K_1, K_2, \dots$ with values in $\overline{\mathbb{N}_0}$ and enumerate them anew such that $$K_1\leq K_2 \leq \dots \leq K_\infty:= \max_{i \in \overline{\mathbb{N}_0}} \{ K_i \} $$ This should allow me to naturally redefine $$ \lbrace \underbrace{X_{1,1}}_{=Y_1}, \underbrace{X_{2,1}}_{=Y_2}, \dots, \underbrace{X_{K_1,1}}_{=Y_{K_1}}, \underbrace{X_{1,2}}_{=Y_{K_1+1}}, \dots , \underbrace{X_{K_2,2}}_{=Y_{K_2}}, \dots ,\underbrace{X_{K_n,n}}_{Y_{K_n}}, \dots \rbrace $$
All of the $Y_i$ are indeed RV in $\mathbb{X}$ and $K_\infty$ is a RV with values in $\overline{\mathbb{N}_0}$, thus I can write $$\eta = \sum_{i=1}^{K_\infty} Y_i $$