Submission date: 10 June 2015.

Abstract:

The group S_∞ acts via the logic action on the space of countable structures in a given countable language that have a fixed underlying set. We consider the number of ergodic probability measures on this space that are invariant under the logic action and are concentrated on the isomorphism class of a particular structure. We show that this number must be either zero, or one, or continuum. Further, such an isomorphism class admits a unique S_∞-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order (Q,<).

Mathematics Subject Classification: Primary: 03C98, 37L40, Secondary: 60G09, 05C80, 03C75, 05C63

Keywords and phrases: invariant measure, logic action, high homogeneity

Full text arXiv 1412.2735: pdf, ps.