# Poisson Process Intensity Function

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.
(To be precise, our state space is the set of nonnegative integers. While the time space is $S$.)