A characterization of the invertible measures

A. Ülger¹

¹ Department of Mathematics Koc University Fener Yolu, 34450, Sariyer-Istanbul, Turkey

Studia Mathematica, Tome 182 (2007) no. 3, pp. 197-203

Résumé

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.

Détail

DOI : 10.4064/sm182-3-1

Keywords: locally compact abelian group its measure algebra measures lambda said equivalent there exists invertible measure varpi varpi ast lambda main result note following measure invertible widehat vert geq varepsilon widehat varepsilon equivalent measure lambda form lambda theta where theta idempotent measure