Alternate characterization of a spatial Poisson point process Would I be correct to assess that a spatial Poisson point process on some compact, say the $d$-dimensional sphere, can be simulated by first choosing some $n \sim \mathrm{Poisson} (\beta)$ number of points, and then uniformly distributing them over this compact? Here $\beta$ is the mean number of points generated by the PPP.
Or in other words, conditioning the total number of points to be fixed as $k$, does a spatial Poisson point process simulate the uniform distribution with $k$-points over the compact?
 A: Yes, this is true.
Homogeneous PPP over $\mathbb{R}^2$
You can use this as a simple characterization of a homogeneous Poisson Point Process (PPP) over the 2-d plane $\mathbb{R}^2$ as follows:  Chop $\mathbb{R}^2$ up into a countably infinite number of disjoint unit squares of the type $[n, n+1) \times [m, m+1)$ (for $n,m$  integers): 
$$\mathbb{R}^2 = \cup_{n,m\in \mathbb{Z}}\left( [n, n+1)\times [m,m+1)\right)$$
We can index the squares by ordered pairs of integers $(n,m)$, which represents the lower-left vertex of the square. For each square $(n,m)$, independently generate a Poisson random variable $N_{n,m}$ with parameter $\lambda>0$. Then independently throw down $N_{n,m}$ points, uniformly over that square.
Homogeneous PPP over $\mathbb{R}^k$
A similar construction can be used for $\mathbb{R}^k$ for $k$ a positive integer.  
Of course, we often talk of the case $k=1$ as arrivals over a timeline, and it often makes sense to consider only the times $t \geq 0$, in which case the Poisson process can be characterized by i.i.d. inter-arrival times $\{X_i\}_{i=1}^{\infty}$ that are exponentially distributed with parameter $\lambda>0$.
Inhomogeneous PPP over $\mathbb{R}^2$
For all $(x,y)\in \mathbb{R}^2$ define $\lambda(x,y)$ as a real-valued spatial intensity function, assumed to be nonnegative and to have finite integral over any compact region of $\mathbb{R}^2$. Now for each unit square $(n,m)$ of $\mathbb{R}^2$ independently do this:
1) Compute $\lambda_{nm} =\int_{x= n}^{n+1}\int_{y= m}^{m+1} \lambda(x,y)dxdy$. By assumption we know $0\leq \lambda_{nm} <\infty$. 
2) If $\lambda_{nm}=0$ then put no points in square $(n,m)$. 
3) If $\lambda_{nm}>0$, independently generate a random variable $N_{n,m}$ that is Poisson with parameter $\lambda_{nm}$. Then independently throw down $N_{n,m}$ points over the unit square $(n,m)$ using a location distribution with PDF $f_{X,Y}(x,y)$ given by
$$ f_{X,Y}(x,y) = \left\{ \begin{array}{ll}
\frac{\lambda(x,y)}{\lambda_{nm}} &\mbox{ $(x,y) \in [n,n+1)\times [m, m+1)$} \\
0  & \mbox{ otherwise} 
\end{array}
\right.$$
