Let $(E,\mathcal E,\alpha)$ be a measure space. A Poisson random measure on $(E,\mathcal E)$ with intensity $\alpha$ is a transition kernel $\zeta$ from a probability space $(\Omega,\mathcal A,\operatorname P)$ to $(E,\mathcal E)$ such that
- if $B\in\mathcal E$, then $$\zeta(B):=\zeta(\;\cdot\;,B)\sim\operatorname{Poi}(\alpha(B))\tag1;$$
- if $n\in\mathbb N$ with $n\ge2$ and $B_1,\ldots,B_n$ are disjoint, then $\zeta(B_1),\ldots,\zeta(B_n)$ are independent.
If $(E,\mathcal E)=([0,\infty),\mathcal B([0,\infty))$ and $\alpha$ is a positive constant, we can consider the "event times" $$\tau_n:=\inf\{t\ge0:\zeta([0,t])=n\}$$ for $n\in\mathbb N$. These are the times at which the process $t\mapsto\zeta([0,t])$ jumps (with a jump size of 1).
Now I wonder how this generalizes and the only thing I found was a highly confusing description in Non-Uniform Random Variate Generation. Clearly, we can think of $\zeta(B)$ as the number of points placed in the set $B\in\mathcal E$. For simplicity, we can assume $\alpha(B)\in(0,\infty)$, but that might be unnecessary. Now the (obscure) claim is that the points $X_1,\ldots,X_{\zeta(B)}$ in $B$ "placed by $\zeta$" are conditionally independent given $\zeta(B)$ and identically conditionally distributed given $\zeta(B)$ with distribution $$A\mapsto\frac{\alpha(A\cap B)}{\alpha(B)}\tag2.$$ My questions are: (a): How can we make this claim rigorous and also prove it rigorously? I mean, what is $X$ at all? Do we assume it is an $(E,\mathcal E)$-valued process $(X_n)_{n\in\mathbb N}$? Cause simply saying "let $X_1,\ldots,X_{\zeta(B)}$ denote the points in $B$ placed" seems to be handwavy to me (among other things cause the length of that sequence of points is random).
(b): Once we got (a), how are these "event locations" related to the jump times $\tau_n$ in the special case consider above?
