Random directing measure of a Cox process ($N\mid \Lambda \sim \mathrm{PRM}(\Lambda )$)

A Cox process can be directed by a random locally finite measure, which need not have a density. Conditional counts in disjoint sets are independent Poisson variables with that measure's masses. Uniform circular offspring displacements give a directing measure supported on random circles; its unconditional mean can still be a multiple of planar Lebesgue measure. A random intensity function is the special case in which the directing measure is absolutely continuous.