I am in the process of learning about the point process (and in particular the Poisson Process). I would like to see that the following notion of a locally finite measure is not a trivial defintion.
Setup:
Let $\mathcal{F},\mathcal{C}$ denote all closed sets and all compact sets of $\mathbb{R}^n$ respectively. Then we define $\mathcal{F}_{l c}:= [ F \in \mathcal{F} \colon \vert F \cap C \vert < \infty \ \forall C \in \mathcal{C} ]$.
A point process $X$ with intensity measure $\mu (A) := \mathbb{E}[\vert X \cap A \vert]$ then satisfies $P(X \in \mathcal{F}_{l c}) = 1$.
We say $\mu$ is locally finite if $\mu(C) < \infty$ for all $C \in \mathcal{C}$.
Question:
What is an example of a point process which has an intensity measure $\mu$ which is not locally finite?
Thoughts:
I find it a bit hard to visualise such a point process as by definition is must be finite a.s. in any "compact window" of $\mathbb{R}^n$, yet we want the expected number of points to be infinite.
My idea was to exploit the Cauchy distribution which is not in $\mathcal{L}^1$ i.e. doesn't have expectation. The first naive attempt inspired by the Poisson process was to define $X$ by $\vert X \cap A \vert \sim \text{Cauchy}(0,1) $ but I think there some problems with the distribution not depending on $A$ (fx when considering $A = \emptyset$). Next I tried $\vert X \cap A \vert \sim \text{Cauchy}(0,\lambda (A)) $ where $\lambda$ is the Lebesgue measure but this need not be finite on non-compact sets and hence is not well-defined. No matter how I try to fiddle with the Cauchy distribution, there occurs some problem...
Is it possible to do this in terms of the Cauchy distribution or is there another (perhaps also more elementary) example?