On Lebesgue decomposition of $p$-adic distributions

Given a prime number $p$, let $\mathbb{C}_p$ 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 $\mathbb{C}_p$, 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 $\mathbb{C}_p$, 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.