Two-series theorem in additive arithmetical semigroups

Abstract. In this paper, we embed the additive arithmetical semigroup in a probability space \(\Omega:=(\beta G,\sigma(\overline{\cal A}), \overline{\delta})\) where \(\beta G\) denotes the Stone–Čech compactification of \(G\). We show that every additive function \(\tilde{g}\) on \(G\), \(\tilde{g}(a)=\sum\limits_{p^k || a} \tilde{g}(p^k)\) \((a \in G)\), can be identified with a sum \(X:=\sum\limits_{p}{X}_p\) of independent random variables on \(\Omega\). Further, we characterize the class of essentially distributed additive functions.