Lebesgue decomposition of probability measures:Applications to equivalence, singularity and kriging

Abstract

We use Lebesgue decomposition of two probability measures on a measurable space to obtain conditions for their equivalence and singularity in terms of the density of the absolutely continuous part of one probability measure with respect to the other. This allows us to obtain simple proofs of Kakutani’s theorem on product measures (Kakutani, 1948) and an extension of the result of Shepp (1966). In addition, using the density form of two finite-dimensional Gaussian measures, we derive analogues of major results on equivalence and singularity (Parzen, 1963; Kallianpur and Oodaira, 1963; Rozanov, 1968) for Gaussian random fields. These can be used to study the interpolation of the spatial data (Stein, 1999).