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.