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Abstract

Let $ G$ be a locally compact abelian group and $ M(G)$ its measure algebra. Two measures $\mu $ and $\lambda $ are said to be equivalent if there exists an invertible measure $\varpi $ such that $\varpi \ast \mu =\lambda $. The main result of this note is the following: A measure $\mu $ is invertible iff $|\widehat{\mu }\vert \geq \varepsilon $ on $\widehat{G}$ for some $\varepsilon >0$ and $\mu $ is equivalent to a measure $\lambda $ of the form $\lambda =a +\theta $, where $a\in L^{1}(G)$ and $\theta \in M(G)$ is an idempotent measure.