Stationary countable dense random sets

We study the probability theory of countable dense random subsets of (uncou ntably infinite) Polish spaces. It is shown that if such a set is stationar y with respect to a transitive (locally compact) group of symmetries then a ny event which concerns the random set itself (rather than accidental detai ls of its construction) must have probability zero or one. Indeed the resul t requires only quasi-stationarity (null-events stay null under the group a ction). In passing, it is noted that the property of being countable does n ot correspond to a measurable subset of the space of subsets of an uncounta bly infinite Polish space.