Haar invariant sets and compact transformation groups (Q1916704)

Let \(G\) be a compact abelian group with dual group \(\widehat {G}\). Let \(M(G)\) and \(L^1 (G)\) be the usual measure algebra and the group algebra, respectively. \(M_s (G)\) stands for the space of singular measures in \(M(G)\). For a subset \(E\) of \(\widehat {G}\), \(M_E(G)\) denotes the space of measures in \(M(G)\) whose Fourier-Stieltjes transforms vanish off \(E\). \(E\) is said to be Haar invariant if \(\mu\in M_E (G)\) implies \(\mu_a\in M_E (G)\), where \(\mu_a\) is the absolutely continuous part of \(\mu\). The authors obtain the following Theorem. Let \((G, X)\) be a (topological) transformation group, in which \(G\) is a compact abelian group and \(X\) is a locally compact Hausdorff space. Let \(\sigma\) be a quasi-invariant Radon measure on \(X\). Let \(E\) be a Haar invariant set in \(\widehat {G}\), and let \(\mu\) be a measure in \(M(X)\) with \(\text{sp} (\mu)\) contained in \(E\), where \(\text{sp} (\mu)\) denotes the spectrum of \(\mu\). Then \(\text{sp} (\mu_a)\) is contained in \(E\), where \(\mu_a\) is the absolutely continuous part of \(\mu\) with respect to \(\sigma\).