On the orthogonality of zero-mean Gaussian measures: Sufficiently dense sampling

For a stationary random function $\xi$, sampled on a subset $D$ of $\mathbb{R}^{d}$, we examine the equivalence and orthogonality of two zero-mean Gaussian measures $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ associated with $\xi$. We give the isotropic analog to the result that the equivalence of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ is linked with the existence of a square-integrable extension of the difference between the covariance functions of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ from $D$ to $\mathbb{R}^{d}$. We show that the orthogonality of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ can be recovered when the set of distances from points of $D$ to the origin is dense in the set of non-negative real numbers.