Lévy functional and jump process martingales

An arbitrary jump process is considered without any assumption about the jump times and allowing the jump times to have accumulation points of arbitrary order. Certain basic martingales q(t, A) and the related Lévy system describing the jumps are introduced, and a notion of quadratic integration with respect to the predictable quadratic variation <q, q> of the basic martingales is defined. If M_t is a (locally) square integrable martingale with respect to the family of σ-fields generated by the jump process it is shown that M_t can be represented as a stochastic integral with respect to the basic martingales and with an integrand (locally) square integrable with respect to <q, q>.