  We derive explicit integrability conditions for stochastic integrals taken
over time and space driven by a random measure. Our main tool is a canonical
decomposition of a random measure which extends the results from the purely
temporal case. We show that the characteristics of this decomposition can be
chosen as predictable strict random measures, and we compute the
characteristics of the stochastic integral process. We apply our conditions to
a variety of examples, in particular to ambit processes, which represent a rich
model class.
