The Mecke formula as defined in the book Lectures on the Poisson Process (page 27) is:
The text states (bottom of page 26) "This equation is a fundamental tool for analysing the Poisson process and can be used in many specific calculations." I can see that this is an interesting characterization of Poisson processes in the larger class of point processes, but I'm curious if the formula (4.2) has applications in problems of independent interest, say, in integral geometry or physics.
Question. Should one think of this Mecke formula mostly as a tool to answer the question "is some point process $\eta$ a Poisson process"? Or does it have "computational value"? I.e. is it valuable in computations where one has a Poisson process $\eta$ and a map of interest $f$?
If it has "computational value" then I'd like to know applications of this formula with "(geometrically) meaningful" maps $f$ where one side of the equation seems hard to compute and the other side is more tractable.
The text provides a multivariate generalization of this formula (page 30) and I'd be happy with applications of that too.
In Percolation Theory, Mecke's equation is needed to derive Russo's formula in the Poisson-Boolean model. If you are not into Percolation Theory, I will try to clear things up with some details.
The idea is to take random balls in $\mathbb{R}^d$ and to investigate whether you can go to infinity without ever leaving those balls or not. More formally: Take a PPP $\eta$ on $\mathbb{R}^d\times\mathbb{R}_+$ with intensity $\lambda\mathrm{d}z\otimes\nu$, where $\mathrm{d}z$ is the Lebesgue measure on $\mathbb{R}^d$ and $\nu$ is some probability measure on $\mathbb{R}_+$. If $(x,r)\in\eta$, we will add a ball with center $x$ and radius $r$. For a fixed radii distribution $\nu$, we will vary $\lambda$ and see what happens. If $\nu$ has $d$ moments, the model becomes interesting: indeed, there will be a threshold $\lambda_c$ below which it is impossible to go to infinity and above which you will almost surely find a way. Percolation Theory wants to study this transition and a very useful tool is the so-called Russo's formula which describes the derivative of the probability of some event. More presicely, take an event $A$ which depends only on balls touching a compact $K$ ($A$ is said to be local) and which is increasing, i.e. it becomes more probable when we add balls. Then Russo's formula tells us that
$ \dfrac{\mathrm{d}\mathbb{P}_\lambda[A]}{\mathrm{d}\lambda} = \mathbb{E}_\lambda\left[1_{\eta\not\in A}\int_{\mathbb{R}^d}\int_0^\infty 1_{\eta\cup\{(z,r)\}\in A}\mathrm{d}\nu(r)\mathrm{d}z\right]. $
It is immediate that there is a ling to Mecke's formula and have not seen any proof not using it. If you want to have some more information on Percolation Theory, I advise you the course notes (and videos on YouTube) from Duminil-Copin. If you want to see Russo's formula in action, you can read a very recent paper from him, Tassion and Raoufi in which they prove new results on $\lambda_c$ called Subcritical phase of $d$-dimensional Poisson-Boolean percolation and its vacant set.