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

For a stationary random function ξ, sampled on a subset D of Rd, we examine the equivalence and orthogonality of two zero-mean Gaussian measures P1 and P2 associated with ξ. We give the isotropic analog to the result that the equivalence of P1 and P2 is linked with the existence of a square-integrable extension of the difference between the covariance functions of P1 and P2 from D to Rd. We show that the orthogonality of P1 and P2 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.