  The classic stochastic Fubini theorem says that if one stochastically
integrates with respect to a semimartingale $S$ an $\eta(dz)$-mixture of
$z$-parametrized integrands $\psi^z$, the result is just the $\eta(dz)$-mixture
of the individual $z$-parametrized stochastic integrals $\int\psi^z{d}S.$ But
if one wants to use such a result for the study of Volterra semimartingales of
the form $ X_t =\int_0^t \Psi_{t,s}dS_s, t \geq0,$ the classic assumption that
one has a fixed measure $\eta$ is too restrictive; the mixture over the
integrands needs to be taken instead with respect to a stochastic kernel on the
parameter space. To handle that situation and prove a corresponding new
stochastic Fubini theorem, we introduce a new notion of measure-valued
stochastic integration with respect to a general multidimensional
semimartingale. As an application, we show how this allows to handle a class of
quite general stochastic Volterra semimartingales.
