On Lebesgue Decomposition Of P-Adic Distributions

Given a prime number p, let C-p be the Tate field, which is the topological completion of the algebraic closure of the field of p-adic numbers with respect to the usual p-adic absolute value. Let X be a compact subset of C-p, which is without isolated points. The goal of our paper is to give a p-adic analogoue of the classical theorem of Lebesgue-Radon-Nikodym, that is, under some hypothesis, a p-adic distribution on X with values in C-p, in the sense of Mazur, decomposes into a sum of two distributions, one of them given by the Radon-Nikodym derivative and the other a p-adic measure.