The Fubini theorem for stochastic integrals with respect to L(0)-valued random measures depending on a parameter

For a stochastic integral with respect to an L(0)-valued random measure theta in the sense of Bichteler and Jacod, whose integrand from L(1,0)(theta) depends measurably on a parameter in a measurable space, we establish the measurability in this parameter. In the L(1)-valued case with a norm integrable in the parameter we prove a theorem on the rearrangement of integrals which generalizes the classical Fubini theorem. An analogous result for an L(0)-valued measure is obtained by its prelocal reduction to an L(1)-valued measure.