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

Criteria of measure-preserving for $p$-adic dynamical systems in terms of the van der Put basis