A property of spectrums of measures on certain transformation groups

Abstract

\S 1 Introduction.Let X be a locally compact Hausdorff space.Let C_{0}(X) be the Banach space of continuous functions on X which vanish at infinity, and let M(X) be the Banach space of complex-valued bounded regular Borel measures on X with the total variation norm.Let M^{+}(X) be the set of nonnegative measures in M(X) .For \mu\in M(X) and f\in L^{1}(|\mu|) , we often write \mu(f)=\int_{X}f(x)d\mu(x) .Let X' be another locally compact Hausdorff space, and let S:\grave{X}arrow X' be a continuous map.For \mu\in M(X) , let S(\mu)\in M(X') be the continuous image of \mu under S. We denote by \mathscr{B}(X) the \sigma -algebra of Borel sets in X. \mathscr{B}_{0}(X) means the \sigma -algebra of Baire sets in X.That is, \mathscr{B}_{0}(X) is the \sigma -algebra generated by compact G_{\delta} sets in X.Let G be a LCA group with dual \hat{G} .M(G) and L^{1}(G) denote the

Locations

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