A remark on a theorem of Beurling and Helson

Abstract

The following result is due to Beurling and Helson [1, 2]: let , be a finite regular Borel measure on a locally compact abelian group G, and let u = Ean,uf7+v be its Lebesgue decomposition into discrete and continuous parts,' where /, is the mass 1 at g. Then if ,u has unimodular Fourier-Stieltjes transform (i.e., |,,=1), Z 2=1. The purpose of this note is to point out an extremely direct proof of the result, which also yields additional information. Let ,* represent, as usual, the measure defined by setting u*(f) =ff(-g)(dg), fGCo(G) or, alternatively, by setting 1u*(E)=Fp(E-') for all Baire sets E. Then ,u*= 2n w-,+V*, where v* is again continuous, and (u*) Thus | , =1implies (u*,*)^= | , 2-1= 'o and IA *,u* IAO. But