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.