Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory

Let u(t, x), t [epsilon] R, be an adapted process parametrized by a variable x in some metric space X, [mu]([omega], dx) a probability kernel on the product of the probability space [Omega] and the Borel sets of X. We deal with the question whether the Stratonovich integral of u(., x) with respect to a Wiener process on [Omega] and the integral of u(t,.) with respect to the random measure [mu](., dx) can be interchanged. This question arises, for example, in the context of stochastic differential equations. Here [mu](., dx) may be a random Dirac measure [delta][eta](dx), where [eta] appears as an anticipative initial condition. We give this random Fubini-type theorem a treatment which is mainly based on ample applications of the real variable continuity lemma of Garsia, Rodemich and Rumsey. As an application of the resulting "uniform Stratonovich calculus" we give a rigorous verification of the diagonalization algorithm of a linear system of stochastic differential equations.