Simple point process

A simple point process is a special type of point process in probability theory. In simple point processes, every point is assigned the weight one.

Definition
Let $$ S $$ be a locally compact second countable Hausdorff space and let $$ \mathcal S $$ be its Borel $ \sigma $-algebra. A point process $$ \xi $$, interpreted as random measure on $$ (S, \mathcal S) $$, is called a simple point process if it can be written as
 * $$ \xi =\sum_{i \in I} \delta_{X_i} $$

for an index set $$ I $$ and random elements $$ X_i $$ which are almost everywhere pairwise distinct. Here $$ \delta_x $$ denotes the Dirac measure on the point $$ x $$.

Examples
Simple point processes include many important classes of point processes such as Poisson processes, Cox processes and binomial processes.

Uniqueness
If $$ \mathcal I $$ is a generating ring of $$ \mathcal S $$ then a simple point process $$ \xi $$ is uniquely determined by its values on the sets $$ U \in \mathcal I $$. This means that two simple point processes $$ \xi $$ and $$ \zeta $$ have the same distributions iff
 * $$ P(\xi(U)=0) = P(\zeta(U)=0) \text{ for all } U \in \mathcal I$$