Questions about a solution to a point process exercise

Question

I have questions regarding the solution to this exercise:

Exercise: Let $\eta$ be a stationary simple point process with intensity measure $\gamma \,\mathrm{d}x$ on $\mathbb{R}, \gamma >0$ such that $\mathbb{P}(\eta(\mathbb{R})=0)=0$. Let $\theta_y$ be the shift operator $$\theta_y \mu(B):=\mu(B+y)\qquad B\in \mathcal{B}(\mathbb{R}^d)$$ Let $x\in\mathbb{R}_+$ and $j\in \mathbb{N}_0$. Show that $\mathbb{P}$-almost surely $$1_{\{\eta((0,x]\leq j\}}=\int1_{\{t\leq 0,\eta((t,x])= j\}}\eta(\mathrm{d}t)$$

Solution:Suppose $\eta$ is simple, has infinitely many points on $\mathbb{R}_{-}$ and that from $x$ it is possible to rank all the points on the left side by their distance. Then there exist a unique $t^*\in(\infty,x)$ such that $\eta(\{t^*\})=1$ and $\eta((t^*,x])=j$. If $t^*>0$ both expresion are $0$, if $t^*\leq 0$ both are equal to $1$ Because $\mathbb{P}(\eta(\mathbb{R})=0)$ and $\eta$ stationarity, we know that it has infinitely many points on $\mathbb{R}$ almost surely. By stationarity we have $$\eta((0,y])=\theta_y\eta(\mathbb{R}_{-})\stackrel{d}{=}\eta(\mathbb{R}_{-})$$ but we have also that $\eta((-\infty,y])\xrightarrow{a.s} \infty$ as $y\to \infty$, therefore $\eta(\mathbb{R}_-)=\infty$.

I have the following questions:

1. Why does the first equality here $$\eta((0,y])=\theta_y\eta(\mathbb{R}_{-})\stackrel{d}{=}\eta(\mathbb{R}_{-})$$ hold? Or should this mean that $\eta((0,y])\stackrel{d}{=}\theta_y\eta(\mathbb{R}_{-})$? I still don't get why $\eta(\mathbb{R}_{-})=\infty$

2. Why can we conclude from $\mathbb{P}(\eta(\mathbb{R})=0)=0$ and stationarity that is has infinitely many points on $\mathbb{R}$ and not for example finitely many?

Answer 1

For the first question, replace $0$ with $-\infty$ and hopefully everything will then make sense. Explicitly, it should be clear that $$\eta((-\infty,y])=\theta_y\eta((-\infty,0])$$ and the right hand side is equal in distribution to $\eta((-\infty,0])$ since $\eta$ is stationary.

For the second question, it is a well-known fact that $\mathbb P(\eta(\mathbb R^d)\in\{0,\infty\})=1$ for any stationary point process $\eta$ on $\mathbb R^d$. This is known as the zero-infinity dichotomy. For a proof, see Proposition 12.1.VI in Daley and Vere-Jones' "An Introduction to the Theory of Point Process: Volume II: General Theory". A free preview is available on Google books.