{"paper":{"title":"Properly ergodic structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.LO","authors_text":"Alex Kruckman, Cameron Freer, Nathanael Ackerman, Rehana Patel","submitted_at":"2017-10-25T16:51:57Z","abstract_excerpt":"We consider ergodic $\\mathrm{Sym}(\\mathbb{N})$-invariant probability measures on the space of $L$-structures with domain $\\mathbb{N}$ (for $L$ a countable relational language), and call such a measure a properly ergodic structure when no isomorphism class of structures is assigned measure $1$. We characterize those theories in countable fragments of $\\mathcal{L}_{\\omega_1, \\omega}$ for which there is a properly ergodic structure concentrated on the models of the theory. We show that for a countable fragment $F$ of $\\mathcal{L}_{\\omega_1, \\omega}$ the almost-sure $F$-theory of a properly ergodi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09336","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}