I'm studying the theory of finite point processes on "An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, Second Edition" by Daley and Vere-Jones. I have some difficulties in understanding an elementary concept. I will use a different notation with respect the book.
Background
From my simple point of view, a finite point process $\mathsf{X}$ is a finite set of points $X_1$, $\dots$, $X_\eta$ in $\mathbb{R}^n$, where the cardinality $\eta$ of the set $\mathsf{X}$ is random and the points $X_1$, $\dots$, $X_\eta$ contained in the set $\mathsf{X}$ are random as well.
\begin{equation}\mathsf{X}=\{X_1,\dots, X_\eta\}\end{equation}
so, this means that a finite point process is characterized by a discrete density $p_N(\eta)$ on $\mathbb{N}$, which describes the cardinality of $\mathsf{X}$, and a family of joint PDFs on $\mathbb{R}^{n\times\eta}$ of the form
\begin{equation}\mathcal{S}=\{p_{X_1,\dots,X_\eta}(x_1,\dots,x_\eta)\}_{\eta\in \mathbb{N}^+}\end{equation} which describes how the points in $\mathsf{X}$ are distributed in $\mathbb{R}^n$. Since $\mathsf{X}$ is a set, it is invariant with respect any permutations of the points $X_1$, $\dots$, $X_\eta$, for example the notations $\{X_2,X_3,X_1\}$ and $\{X_3,X_2,X_1\}$ represents the same finite point process $\{X_1,X_2,X_3\}$. This means that the joint PDFs in $\mathcal{S}$ need to be symmetric, i.e. permutation invariant, i.e. \begin{equation}p_{X_1,\dots, X_2}(x_{\sigma(1)},\dots, x_{\sigma(\eta)})=p_{X_1,\dots,X_\eta}(x_1,\dots,x_\eta)\end{equation} for every $\eta!$ possible permutations $\sigma(i)$ of the indexes $i$.
Now, define the Janossy density of order $\eta$ as \begin{equation}p_\mathsf{X}(\{x_1,\dots,x_\eta\})\triangleq \begin{cases} \eta! \, p_{X_1,\dots,X_\eta}(x_1,\dots, x_\eta)\,p_N(\eta) & \text{ if } \eta>0 \\ p_N(0) & \text{ if } \eta=0 \end{cases}\end{equation} where the Janossy of order zero is denoted as $p_\mathsf{X}(\varnothing)$.
My problem
At page 125 the book says that the quantity
\begin{equation}p_\mathsf{X}(\{x_1,\dots,x_\eta\})\text{ d}x_1\cdots\text{d}x_\eta\end{equation}
is the probability of the event
\begin{equation}\begin{aligned} &\text{there are exactly } \eta \text{ points in the process,} \\ &\text{one in each of the } \eta \text{ distinct infinitesimal regions } (x_i, x_i+\text{d}x_i) \end{aligned}\tag{E1}\end{equation}
I don't get it. This is my reasoning:
- the quantity \begin{equation}p_{X_1,\dots,X_\eta}(x_1,\dots, x_\eta)\text{ d}x_1\cdots\text{d}x_\eta\end{equation} is the probability of the event \begin{equation}\begin{aligned}&X_1\in (x_1,x_1+\text{d}x_1)\text{ and } \\&X_2\in (x_2,x_2+\text{d}x_2) \text{ and }\\ &\vdots\\ &\text{ and }X_\eta\in (x_\eta,x_\eta+\text{d}x_\eta)\end{aligned}\end{equation}
- the quantity \begin{equation}\eta!\,p_{X_1,\dots,X_\eta}(x_1,\dots, x_\eta)\text{ d}x_1\cdots\text{d}x_\eta\end{equation} is the probability of the event \begin{equation}\tag{E2}\bigcup_{\sigma}\begin{cases}&X_1\in (x_{\sigma(1)},x_{\sigma(1)}+\text{d}x_{\sigma(1)})\text{ and } \\ &X_2\in (x_{\sigma(2)},x_{\sigma(2)}+\text{d}x_{\sigma(2)}) \text{ and }\\ &\vdots\\ &\text{ and }X_\eta\in (x_{\sigma(\eta)},x_{\sigma(\eta)}+\text{d}x_{\sigma(\eta)})\end{cases}\end{equation} where the union is performed over all the $\eta!$ possible permutations $\sigma(i)$.
For me the events $(\text{E1})$ and $(\text{E2})$ are the same. I cannot see the difference.
