The F. and M. Riesz theorem on certain transformation groups. II (Q2640826)

The classical F. and M. Riesz theorem states that analytic measures on the circle group are absolutely continuous with respect to Lebesgue measure. The classical theorem has been extended in many directions by various authors. In Part I the author [ibid. 17, No.3, 289-332 (1988; Zbl 0663.43002)] extended results of \textit{F. Forelli} [Acta Math. 118, 33-59 (1967; Zbl 0171.342)] to a large class \({\mathcal C}\) (having a rather complex description) of transformation groups. The class \({\mathcal C}\) is unnecessary, for the main theorems of the author's earlier paper are now shown to hold whenever G is a compact abelian group acting on a locally compact Hausdorff space X. One key to the author's new and more natural arguments is the construction of a quotient transformation group (G,X/\(\sim)\) for which, among other things, mutual singularity of specified measures is preserved under the quotient map.