What is Poisson Point Process?

In general, a point process is a random variable $N$ from some probability space $(\Omega,\mathcal{F},P)$ to a space of counting measures on ${\bf R}$, say $(M,\mathcal{M})$. So each $N(\omega)$ is a measure which gives mass to points $$ \ldots < X_{-2}(\omega) < X_{-1}(\omega) < X_0(\omega) < X_1(\omega) < X_2(\omega) < \ldots $$ of ${\bf R}$ (here the convention is that $X_0 \leq 0$. The $X_i$ are random variables themselves, called the points of $N$.

The intensity of a point process is defined to be $$ \lambda_N = {\bf E}[N(0,1]]. $$

There are many different possible point processes, but the Poisson point process with intensity $\lambda$ is the one for which the number of points in an interval $(0,t]$ has a Poisson distribution with parameter $\lambda t$: $$ P[N(0,t] = k] = \frac{(\lambda t)^k e^{-\lambda t}}{k!} $$ and which is stationary. Stationarity is a little more involved to go into here, but in this context you can think of it as meaning that the measure of two different intervals of equal length is the same, thus $$ P[N(s,s+t] = k] = \frac{(\lambda t)^k e^{-\lambda t}}{k!}, \ \ \forall s. $$

From Wikipedia:

For any disjoint bounded subsets $B_1,\ldots,B_n$ and non-negative integers $k_1,\ldots,k_n$ we have that $$\Pr[\xi(B_i) = k_i, 1 \leq i \leq n] = \prod_i e^{-\lambda \|B_i\|}\frac{(\lambda \|B_i\|)^{k_i}}{k_i!}.$$ The constant $\lambda$ is called the intensity of the Poisson point process. Note that the Poisson point process is characterised by the single parameter $\lambda$.

So, let the expectation measure be $Eξ$. Then, $Eξ(\cdot)=\lambda\|\cdot\|$, where $\|\cdot\|$ is the Lesbegue measure and $\lambda$ is the intensity.