A Poisson point process on 
with a locally finite Borel intensity
measure 
is a random counting measure whose counts in
disjoint Borel sets of finite intensity are independent
random variables with the Poisson distribution.
Here 
is a positive integer, and local finiteness means
for every bounded Borel set 
. For every Borel
set 
of finite intensity, its count has expectation value 
. Repeated points are allowed at
point atoms, points 
with 
. On 
, a homogeneous process of constant intensity
function 
uses 
,
where 
denotes Lebesgue measure. Its law
is invariant under translations. It has almost
surely no repeated points because Lebesgue measure
has no point atoms.
For example, planar Poisson nuclei of intensity 
with 
generate the random Voronoi
diagram in Voronoi percolation. The number
of nuclei in a unit square has expectation
value 
,
whereas cell colors are sampled independently
of these nuclei.
