Uniform partitions of unity on locally compact groups (Q1179962)

The authors are interested in what they call uniform partitions of unity (UPU) for a locally compact group. These are pairs \((M,\phi)\) consisting of a subset \(M\) of \(G\) and a continuous, positive function \(\phi\) on \(G\) with compact support such that \(\sum_{g\in M}\phi(g^{-1}x)=1\text{ for all }x\in G\). The first result is that every locally compact group has a UPU. The proof is by reduction to the case where \(G\) is a connected Lie group. The second interesting problem is to describe the possible subsets \(M\) of \(G\) for which there exists a UPU \((M,\phi)\). The authors give such a description in the case \(G=\mathbb{R}\): The subsets \(M\) of \(\mathbb{R}\) in question are finite unions of cosets of discrete subgroups of \(\mathbb{R}\). An essential ingredient in their proof is P. J. Cohen's classification of idempotent measures on abelian locally compact groups.