  We study stochastic integrals driven by a general subordinator and establish
a zero-one law for the finiteness of the resulting integral as well as moment
estimates. As an application, we use these results to obtain structural
properties of SPDEs driven by multiplicative pure jump noise, which include (1)
a maximal inequality for a multiplicative stochastic convolution $Z_t$, (2) a
small ball probability of $Z_t$, (3) the existence of invariant measures and
accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions
to SPDEs.
