# De Finetti-style theorem for Point Processes

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

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

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?