On a theorem of S. Saeki (Q1071987)

The author shows the following, which is a generalization of a theorem due to \textit{S. Saeki} [Proc. Am. Math. Soc. 90, 391-396 (1984; Zbl 0537.43010), Theorem 2]: Let G be an LCA group, and let H be a closed subgroup of G. Let \(\tilde E\) be a closed subset of \(\hat G/H^{\perp}\) that is a Riesz set in \(\hat H\cong \hat G/H^{\perp}\) (i.e., \(M_{\tilde E}(H)\subset L^ 1(H))\). Put \(E=\pi^{\perp}(\tilde E)\), where \(\pi: \hat G\to \hat G/H^{\perp}\) is the natural homomorphism. Let \(\mu\) be a measure in \(M_ E(G)\) such that \(\alpha({\bar\gamma}\mu) \in L^ 1(G/H)\) for all \(\gamma\in E\), where \(\alpha: G\to G/H\) is the natural homomorphism. Then \(\mu\) is absolutely continuous with respect to the Haar measure of G. The author obtaines another theorem by using a theorem due to \textit{L. Pigno} [ibid. 29, 511-515 (1971; Zbl 0216.148), Theorem 2]. However the theorem of Pigno is false in general [see \textit{L. Pigno}, ibid. 48, 515 (1975; Zbl 0298.43006)]. Thus it is not known whether the author's second theorem (Theorem 3) is correct or not.