  This paper is devoted to (discrete) $p$-adic dynamical systems, an important
domain of algebraic and arithmetic dynamics. We consider the following open
problem from theory of $p$-adic dynamical systems. Given continuous function
$f:Z_p > Z_p.$ Let us represent it via special convergent series, namely van
der Put series. How can one specify whether this function is measure-preserving
or not for an arbitrary $p$? In this paper, for any prime $p$ we present a
complete description of all compatible measure-preserving functions in the
additive form representation. In addition we prove the criterion in terms of
coefficients with respect to the van der Put basis determining whether a
compatible function $f:Z_p > Z_p$ preserves the Haar measure.
