A Bochner type theorem for compact groups (Q909186)

Let Z be a partially ordered set, and let \(\Omega\) be a subset of Z. \(\Omega\) is said to be low-complete iff for any subset Y of Z that is bounded from below by some element of \(\Omega\), there exists in \(\Omega\) \(\setminus Y\) a greatest among all lower boundaries of Y. The author obtains the following two theorems. Theorem 1. Let G be a compact abelian group with dual \(\Gamma =\hat G\). Let \(\Gamma_ 0\) be a semigroup in \(\Gamma\) such that \(\Gamma_ 0\cup (-\Gamma_ 0)=\Gamma\) and \(\Gamma_ 0\cap (-\Gamma_ 0)=\{0\}\). Let \(\Sigma\) be a nonempty subset of \(\Gamma \setminus \Gamma_ 0\) that is low-complete. Let \(\mu\) be a bounded regular measure on G that is orthogonal to \(\Gamma\) \(\setminus \Sigma\) and singular with respect to \(m_ G\), where \(m_ G\) is the Haar measure of G. Then \(\mu =0\). Theorem 2. Let G be a compact abelian group with dual \(\Gamma =\hat G\). Let \(\Xi =\{\Gamma_{\alpha}\}_{\alpha \in \Lambda}\) be a family of semigroups in \(\Gamma\) such that \(\Gamma_{\alpha}\cup (-\Gamma_{\alpha})=\Gamma\) for every \(\alpha\in \Lambda\). Let \(\delta_{\alpha}\in -\Gamma_{\alpha}\) (\(\alpha\in \Lambda)\), and put \(K=\cup_{\alpha \in \Lambda}(\delta_{\alpha}+\Gamma_{\alpha})\). Suppose that there is a semigroup \(\Gamma_ 0\) in \(\Xi\) such that \(\Gamma_ 0\cap (-\Gamma_ 0)=\{0\}\) and that \(\Gamma\) \(\setminus K\) is low-complete (with respect to the order induced by \(\Gamma_ 0)\). Let \(\mu\) be a bounded regular measure on G that is orthogonal to K. Then \(\mu\) is absolutely continuous with respect to \(m_ G\).