Trace functions and Galois invariant p-adic measures

Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, $\overline{\mathbb{Q}}_p$ a fixed algebraic closure of $\mathbb{Q}_p$, and $\mathbb{C}_p$ the completion of $\overline{\mathbb{Q}}_p$ with respect to the $p$-adic valuation. We study trace functions associated to $p$-adic measures defined on compact subsets of $\mathbb{C}_p$ which are invariant under the action of the Galois group $G=Gal_{cont}(\mathbb{C}_p/\mathbb{Q}_p)$.