Analytic Measures on Compact Groups by K. De Leeuw and I. Glicksberg

I t is not hard to see that A and B together are equivalent to the following: The collection of Borel sets on which fi vanishes identically is invariant under rotation. This is the assertion concerning analytic measures that we extend to compact groups. We also shall state several of its consequences, including analogues of A and B. The work was inspired by, and is in part an extension of, several of the results of Helson and Lowdenslager [4; 5] . In all that follows G is a compact abelian group, ô its discrete dual, and ^ is a fixed homomorphism of ô into the group R of real numbers. The mapping yp: Ô-+R is a continuous homomorphism and 1 Supported in part by National Science Foundation Grant G147 79 and the United States Air Force Office of Scientific Research. 1 We say that a measure /* vanishes identically on a set E if p vanishes on all Borel subsets of E. 8 See however our final remarks.