I'm working through Kingman's Poisson Processes and have a question about defining the mean measure. Why is it that in most cases we specify an intensity function $\lambda(x)$, where the mean measure of the process is then defined as $\mu(A)=\int_A \lambda(x)dx$ (for $A$ in our state space $S$)? Why don't we just directly specify a function defined on subsets of our state space which we take to be our mean measure? Is it just because this is too hard to do in general/ in practice? Is it because we somehow want to take into account the measure of our state space into the mean measure of our process?
For most properties of a Poisson process, the intensity density is not needed at all. It is sufficient to define a $\sigma$-finite intensity measure.
However sometimes it is convenient to have the intensity measure without atoms (e.g., for the Rényi theorem).
(To be precise, our state space is the set of nonnegative integers. While the time space is $S$.)