I am new to point processes. I know there are a number of theorems along the lines that if a point process $\eta$ satisfies:


1. Complete independence (the random variables $\eta(B\_1), \ldots, \eta(B\_n)$ are independent for pairwise disjoint bounded measurable $B\_1, \ldots, B\_n$) and
2. Some regularity conditions like being simple and uniform $\sigma$-finite on a Borel subset of a complete separable metric space,


then $\eta$ is a Poisson process.


It seems that something like the following must be known. I think (I haven't worked through all the details) that if $\eta$ satisfies:


1. (a) A condition one might call "complete exchangeability": for any disjoint bounded measurable $B\_1, \ldots, B\_n$, there are $A\_i \subseteq B\_i$, with equality for at least one $i$, such that $\eta(A\_1), \ldots, \eta(A\_n)$ are exchangeable random variables;


or equivalently


1. (b) For any disjoint bounded measurable $B\_1, \ldots, B\_n$ with $\mathbb{E}\eta(B\_1) = \ldots = \mathbb{E}\eta(B\_n)$, $\eta(B\_1),\ldots, \eta(B\_n)$ are exchangeable random variables;


as well as


2. Similar regularity conditions including $\mathbb{E}\eta(B)<\infty$ for all bounded measurable $B$; and
3. $\mathbb{E}\eta(\text{whole space}) = \infty$


then there exists a nonnegative-valued scalar random variable $G$ with $\mathbb{E}G = 1$ such that conditioned on $G$, $\eta$ is a Poisson process with intensity measure $G\mathbb{E}\eta$.


Without (3) there are simple counterexamples, e.g. $\eta = \delta\_x$ where $x$ is distributed according to a nonatomic probability distribution.


Could anyone provide a reference for such a point process de Finetti theorem?



De Finetti-style theorem for Point Processes
This is Theorem 3.34 in 


*Kallenberg, Olav*, Random measures, Berlin: Akademie-Verlag. London - New York - San Francisco: Academic Press. 104 p. M 28.00 (1976). [ZBL0345.60032](https://zbmath.org/?q=an:0345.60032).


