  We consider a piecewise-deterministic Markov process (PDMP) with general
conditional distribution of inter-occurrence time, which is called a general
PDMP here. Our purpose is to establish the theory of measure-valued generator
for general PDMPs. The additive functional of a semi-dynamic system (SDS) is
introduced firstly, which presents us an analytic tool for the whole paper. The
additive functionals of a general PDMP are represented in terms of additive
functionals of the SDS. The necessary and sufficient conditions of being a
local martingale or a special semimartingale for them are given. The
measure-valued generator for a general PDMP is introduced, which takes value in
the space of additive functionals of the SDS. And its domain is completely
described by analytic conditions. The domain is extended to the locally
(path-)finite variation functions. As an application of measure-valued
generator, we study the expected cumulative discounted value of an additive
functional of the general PDMP, and get a measure integro-differential equation
satisfied by the expected cumulative discounted value function.
