Translations of Measures on the Circle

Abstract

Given a measure µ on S ¹ we can associate with it two sets:

$$\begin{array}{*{20}{c}} {A\left( \mu \right) = \left\{ {t \in {S^1}:{\mu _t}\;and\:\mu \:are\:not\:mutually\:\sin gular} \right\},} \\ {H\left( \mu \right) = \left\{ {t \in {S^1}:{\mu _t}\;\:and\:\mu \:have\:the\:same\:null\:sets} \right\}.} \end{array}$$

Here µ _t denotes the measure µ translated by t: µ _t (A) = µ(tA), for any Borel subset A of S ¹.