Fourier analysis and determining sets for Radon measures on \({\mathbb{R}}^ n\) (Q1068233)

A bounded Borel subset E of \({\mathbb{R}}^ n\) is said to be a determining set for a class C of measures if for any \(\mu_ 1,\mu_ 2\in C\) the condition \(\mu_ 1(x+E)=\mu_ 2(x+E)\) for all \(x\in {\mathbb{R}}^ n\) implies that \(\mu_ 1=\mu_ 2\). The author studies determining sets for various classes C and also rephrases various old results in the language of determining sets. It is proved that no symmetric Borel set is a determining set for the class of tempered measures (measures such that the corresponding distribution is tempered) and that for \(n\geq 2\) no spherically symmetric Borel set is a determining set for the class of measures vanishing at \(\infty.\) For the class of finite measures it is well known that every (bounded) Borel set of positive Lebesgue measure is a determining set. For a certain class of measures \(\mu\) (viz. such that \(\int e^{r\| x\|}d| \mu | <\infty\) for all \(r>0)\) the author proves the following analogue at \(\infty:\) if E is a bounded Borel set and \(\mu (x+E)=0\) for all x outside a certain compact set then \(\mu\) has compact support.