A classification of orbits admitting a unique invariant measure

Abstract

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_{\infty }$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_{\infty }$-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 $(\mathbb{Q},<)$.

22 pages. Small changes following referee suggestions

References in corpus (2)

• On exchangeable random variables and the statistics of large graphs and hypergraphs
• Graphons, cut norm and distance, couplings and rearrangements

Cited by in corpus (1)

• Forcing with Invariant Measures