I have difficulties in understanding point process.
Here is my problem : We start with the following definition of a point process
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(E,\mathcal{E})$ a measurable space. We define a point process $\Pi$ as a random variable whose realisation are at most countable set of points. Formally this gives (if I'm right) $\Pi:\Omega\to E, \omega\mapsto\Pi(\omega)={(\pi_{i})_{i\in I}}$ where $I$ is at most countable.
To each point process we associate its counting process $N$ defined by $N(A)(\omega):= \#(A\cap\Pi(\omega))$ for $A\in\mathcal{E}$.
Then we defined a Poisson process with mean measure $\mu$ as a point process whose counting process verifies:
for all disjoint sets $A_1,..., A_k\in\mathcal{E}$, the r.v. $N(A_i)$ are independent;
for every $A \in\mathcal{E}$, $N(A)$ follows a $\mathcal{P}(\mu(A))$.
Then it is said that the measure $\mu$ is related to an intensity function $\lambda$ and when this function is constant, we say that the Poisson process is homogeneous, should this be viewed as the fact that "in mean" each $A$ has the same measure?
Then, we restrict ourselves to $(E,\mathcal{E})=(\mathbb{R_{+}},\mathcal{B}(\mathbb{R_{+}}))$ and the "points" of $\Pi$ are ordered: $$0< X_1 < X_2 <...$$ in order to write the counting process as $$N_t := N([0,t])=\sum_{i=1}^{\infty}1(X_i\leq t) \text{ for all } t \ge 0.$$
First the term points is a little but confusing since the $X_i's$ are random variables in fact ?
Second, from my understanding of a point process, for each outcomes, the set of points is not the same and has a different size, so directly consider the sequence ${X_i}_{\in\mathbb{N^{*}}}$ means that we are restrict to the case of countable set ?
Thank you a lot for your help
The intensity function is the density (if it exists) of the mean measure $\mu$ with regard to some reference measure.
When $E = \mathbb{R}_+$, the reference measure is usually Lebesgue measure. When the intensity $\lambda$ is constant, the distribution of $N(A)$ is Poisson with parameter $\lambda|A|$, where $|A|$ denotes the Lebesgue measure of $A$. It depends only of the length of $A$.
In the definition of Poisson point processes, we may restrict ourselves to sets $A$ with finite $\mu$-measure. If one looks at sets $A$ with infinite $\mu$-measure, the right convention for Poisson measure with infinite parameter is $\delta_\infty$.
If we have a finite Poisson process on $\mathbb{R}_+$ with constant intensity $\lambda>0$, then $N_\infty := N(\mathbb{R}_+)$ is almost surely infinite. Indeed, for every integer $m \ge 0$, $$P[N_t \le m] = e^{-\lambda t}\sum_{k=0}^m \frac{(\lambda t)^k}{k!}.$$ Letting $t$ go to infinity yields $P[N_\infty \le m] = 0$.