On the equivalence of probability spaces

Abstract

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.