Point shift characterization of Palm measures on Abelian groups

Let N denote the space of locally finite simple counting measures on an Abelian topological group G that is assumed to be a locally compact, second countable Hausdorff space. A probability measure on N is a canonical model of a random point process on G. Our first aim in this paper is to characterize Palm measures of stationary measures onN through point stationarity. This generalizes the main result in (2) from the Euclidean case to the case of an Abelian group. While under a stationary measure a point process looks statistically the same from each site in G, under a point stationary measure it looks statistically the same from each of its points. Even in case G = R our proof will simplify some of the arguments in (2). A new technical result of some independent interest is the existence of a complete countable family of matchings. Using a change of measure we will generalize our results to discrete random measures. In case G = R we will finally treat general random measures by means of a suitable approximation.