Decompositions of Measures on Compact Abelian Groups

Abstract

It is shown that the set of finite regular Borel measures with natural spectra for a compact abelian group $\mathfrak{G}$ is closed under addition if and only if $\mathfrak{G}$ is discrete. If $G$ is a non-discrete locally compact abelian group, then there exists a finite regular Borel measure with natural spectrum such that the corresponding multiplication operator on $L^1(G)$ is not decomposable.