Lacunary sets on transformation groups (Q1326588)

The classical F. and M. Riesz theorem for measures on the torus group is the starting point for various investigations on lacunary sets on abelian groups with ordered dual, on compact groups and on certain transformation groups. See e.g. the authors' previous investigation [Hokkaido Math. J. 21, 483-489 (1992)]. In the paper under review the authors define new lacunary sets for (abelian locally compact) transformation groups \(G\) acting on a l.c. Hausdorff space \(X\), endowed with a positive quasi invariant measure \(\sigma\). In a natural way convolutions of measures on \(G\) and \(X\) respectively are defined, this leads to a definition of the spectrum of \(\mu \in M(X)\), of \(M_ \Lambda\) resp. \(L^ 1_ \Lambda\) for subsets \(\Lambda \subset \widehat{G}\), and of Riesz-, nice placed -- resp. Shapiro-sets with respect to \(\sigma\). For transformation groups a new subspace \(N(\sigma) := \{\mu \in M(X): L^ 1(G) * \mu \ll \sigma\}\) and corresponding lacunary sets are introduced. The paper is devoted to study relations between these sets, relations with localization properties [defined according to \textit{Y. Meyer}, Stud. Math. 30, 87-89 (1968; Zbl 0159.425)] and to connections between lacunarity properties of subsets \(\Lambda \subset \widehat {G}\) and the Radon Nikodym property of the corresponding \(L^ 1_ \Lambda\) spaces.