Let 
be a measure space such that
, and let 
denote the Poisson process on S
with intensity measure 
. We require to be atomless
so that the process be simple. Typically S is a bounded
region in 
and is proportional to Lebesgue measure,
but S may also be a torus or other compact set without edges.
We are interested in point processes that have densities with respect
to .
Let 
be the disjoint union of all finite Cartesian products of
S with itself, 
(k factors), including
the empty product 
, which in this context it makes sense to define
to be a set containing one element denoted 
. Let 
denote
the k-fold product of on 
for 
and let 
denote the measure on that gives mass one to the point .
Let 
denote the 
-field in inherited from
, that is the family of sets B such that 
is
an element of 
. Then defined by
is a probability measure on 
. An element 
is
visualized as a pattern of k points in S. The element
of is interpreted as the pattern with no points. Let the number
of points in x be denoted n(x), so n is a map from to
, defined by n(x) = k when .
Let h be a nonnegative function on . Then the measure 
having density h with respect to is defined by
If 
, then can be normalized to make a
probability measure P defined by 
,
. So h is an unnormalized density of P with
respect to and 
its normalizing constant.
Despite the definition of elements as ordered k-tuples we
are only interested in the interpretation of points 
as point patterns, that is as unordered sets of points. Thus we
are only interested in unnormalized densities h that are symmetric
under permutation of points in a pattern: h(x) = h(y) if x and y
would be identical if considered unordered sets. For a detailed discussion
of this issue, see Section 5.3 of Daley and Vere-Jones (1988).