I have been reading ElSawy et al's paper "Characterizing Random CSMA Wireless Networks: A Stochastic Geometry Approach" and am unsure about a seemingly straightforward equation that appears in the paper.
Given an underlying homogeneous Poisson Point Process (PPP) of uniform intensity $\lambda$, equation (1) requires knowledge of the distribution of the number of points in a random lunar region. The exact situation is as follows.
Fix node $x_o$ at the origin, condition upon it having degree $n\ge 1$ under the $r_e$-disk connection model. That is, there exist $n$ points aside from $x_o$ in $B(x_o,r_e)$. For our purposes, we do not really need to know $n$. We just need to know that $n\ge 1$. Consider a uniformly randomly chosen point $x_L$ amongst these $n$ nodes and define the Euclidean distance between $x_o$ and $x_L$, i.e. $d(x_o,x_L)$, as $z$. The distribution of $z$ is well-known and is $f(z)=2z/r_e^2$ for $0<z<r_e$.
For fixed $z$, define the lunar region $A_r(z) = B(x_L,r_e) \backslash B(x_o,r_e)$. Denote its area by $\tilde{A}_r(z)$. This again is given by a well-known formula. Equation (1) appears to suggest that if we define $\mathcal{M}=\lambda\mathbb{E}_z[\tilde{A}_r(z)]$, then we have that the probability distribution of the number of points in $A_r(z)$ is given by $\text{Pois}(\mathcal{M})$.
This seems like a really nice result and I hope my interpretation is correct. If so, in an infinite homogeneous PPP of intensity $\lambda$, it seems to suggest that we can define the area of a generic spatial region $A(z)$ with an arbitrary probability density function (pdf) $f(z)$ and conclude, that irrespective of this pdf, the distribution of the number of points in the region is given by $\text{Pois}(\lambda\mathbb{E}_z[A(z)])$. This would be nice. Is it true?
This is not true. You can choose a probability arbitrarily close to $1$ that the area is $0$, and hence contains no points, and yet have an arbitrarily large expected area (by making the area corresponding to the remaining small probability arbitrarily large), which would be arbitrarily unlikely to contain no points according to the Poisson distribution.
Since the paper you link to isn't freely available, I can't say anything specific about your equation ($1$).