  This paper defines the class of c\`adl\`ag functional marked point processes
(CFMPPs). These are (spatio-temporal) point processes marked by random elements
which take values in a c\`adl\`ag function space, i.e. the marks are given by
c\`adl\`ag stochastic processes. We generalise notions of marked
(spatio-temporal) point processes and indicate how this class, in a sensible
way, connects the point process framework with the random fields framework. We
also show how they can be used to construct a class of spatio-temporal Boolean
models, how to construct different classes of these models by choosing specific
mark functions, and how c\`adl\`ag functional marked Cox processes have a
double connection to random fields. We also discuss finite CFMPPs, purely
temporally well-defined CFMPPs and Markov CFMPPs. Furthermore, we define
characteristics such as product densities, Palm distributions and conditional
intensities, in order to develop statistical inference tools such as likelihood
estimation schemes.
