# Counterexample for Mecke equation in higher dimensions

I am currently reading the book Lectures on the Poisson Process by Gunter Last and Mathew Penrose. (The book can be found here.)

I have a question about an exercise in the book's 4th chapter (Exercise 4.7). Some background about the exercise. Two of the main results of the chapter are the Mecke equation in one dimension (Theorem 4.1) and multiple dimensions (Theorem 4.4).

Theorem 4.1 Mecke equation

Let $$\lambda$$ be an $$s$$-finite measure on $$\mathbb{X}$$ and $$\eta$$ a point process on $$\mathbb{X}$$. Then $$\eta$$ is a Poisson process with intensity measure $$\lambda$$ if and only if $$\mathbb{E}\left[ \int f \left( x,\eta \right) \eta \left( dx \right) \right] = \int \mathbb{E}\left[ f \left( x,\eta + \delta_{x} \right) \lambda \left( dx \right) \right]$$ for all $$f \in \overline{\mathbb{R}}_{+}\left( \mathbb{X} , \textbf{N} \right)$$.

Theorem 4.4 Multivariate Mecke equation

Let $$\eta$$ be a Poisson process on $$\mathbb{X}$$ with $$s-finite$$ intensity measure $$\lambda$$. Then, for every $$m \in \mathbb{N}$$ and for every $$f \in \overline{\mathbb{R}}_{+}\left( \mathbb{X}^{m} , \textbf{N} \right)$$, $$\mathbb{E}\left[ \int f \left( x_{1},\ldots,x_{m},\eta \right) \eta^{m}\left( d \left( x_{1},\ldots,x_{m} \right) \right) \right] = \int \mathbb{E}\left[ f \left( x_{1},\ldots,x_{m},\eta + \delta_{x_{1}} + \ldots + \delta_{x_{m}}\right) \right] \lambda^{m} \left( d \left( x_{1},\ldots,x_{m} \right) \right).$$ $$\qquad$$ (4.11)

Exercise 4.7 Converse to Theorem 4.4

Let $$m \in \mathbb{N}$$ with $$m > 1$$. Prove or disprove that for any $$\sigma$$-finite measure space $$\left( \mathbb{X},\mathcal{X},\lambda\right)$$, if $$\eta$$ is a point process on $$\mathbb{X}$$ satisfying (4.11) for all $$f \in \overline{\mathbb{R}}_{+}\left( \mathbb{X}^{m} , \textbf{N} \right)$$, then $$\eta$$ is a Poisson process with intensity measure $$\lambda$$. (For m = 1, this is true by Theorem 4.1.)

However, the result in one dimension is stronger than that in higher dimensions. In particular, Theorem 4.1 includes the "opposite direction," i.e., it yields an equivalent characterization of the Poisson process, while Theorem 4.4 only gives us a property of the Poisson process. Exercise 4.7 now asks if the opposite direction also holds in higher dimensions i.e. if

$$\mathbb{E}\left[ \int f \left( x_{1},\ldots,x_{m},\eta \right) \eta^{m}\left( d \left( x_{1},\ldots,x_{m} \right) \right) \right] = \int \mathbb{E}\left[ f \left( x_{1},\ldots,x_{m},\eta + \delta_{x_{1}} + \ldots + \delta_{x_{m}}\right) \right] \lambda^{m} \left( d \left( x_{1},\ldots,x_{m} \right) \right)$$ $$\forall f \in \overline{\mathbb{R}}_{+}\left( \mathbb{X}^{m} , \textbf{N} \right)$$ $$\Rightarrow$$ $$\eta$$ is a Poisson process with intensity measure $$\lambda$$.

I tried to prove the statement using the same ideas from Theorem 4.1, but that did not work. Furthermore, given how the Exercise and Theorems are phrased, I assume that the opposite direction does not hold. However, I was unable to find a counter-example.

Am I correct in assuming that the opposite direction in Theorem 4.4 does not hold? What would be a good counter-example?