  For a general class of Gaussian processes $W$, indexed by a sigma-algebra
$\mathscr F$ of a general measure space $(M,\mathscr F, \sigma)$, we give
necessary and sufficient conditions for the validity of a quadratic variation
representation for such Gaussian processes, thus recovering $\sigma(A)$, for
$A\in\mathscr F$, as a quadratic variation of $W$ over $A$. We further provide
a harmonic analysis representation for this general class of processes. We
apply these two results to: $(i)$ a computation of generalized Ito-integrals;
and $(ii)$ a proof of an explicit, and measure-theoretic equivalence formula,
realizing an equivalence between the two approaches to Gaussian processes, one
where the choice of sample space is the traditional path-space, and the other
where it is Schwartz' space of tempered distributions.
