Necessary conditions on the support of RP-measures

By associating RP-measures with annihilators of A0→ – a subspace of the polydisc algebra – we provide necessary conditions for when a compact subset of Tn can contain the support of a non-zero RP-measure. Among other things we show that the support of a positive RP-measure cannot be contained in reflections of inverse images of half-planes by functions in A0→, sets of linear measure zero, and when n=2, graphs of strictly increasing functions. Furthermore, we prove that failure of a compact set S to support any positive annihilator of A0→ is equivalent to uniform approximability of all continuous functions on S by functions in A0→. For n=2 this gives a necessary and sufficient condition for the support of positive RP-measures.