2012/06/30 by Nathanael Ackerman, NATHANAEL ACKERMAN, Cameron Freer +3 · 13 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Countable set #Invariant (physics) #Invariant measure #Measure (data warehouse) #Outer measure #Pointwise #Probability measure #advanced mathematical theories #math.CO #math.LO #math.PR #msc:03C75 #msc:03C98 #msc:05C63 #msc:05C80 #msc:37L40 #msc:60G09 #msc:62E10 #σ-finite measure
paper · pdf · doi:10.1017/fms.2016.15
published in Forum of Mathematics Sigma 4 (Cambridge University Press) · 46 pages, 2 figures. Small changes following referee suggestions
openalex publication_date 2016/01/01 · openalex created_date 2016/06/24 · arxiv created 2016/06/28 · arxiv updated 2016/06/29 · openalex updated_date 2026/08/06
Let L be a countable language. We say that a countable infinite L -structure M admits an invariant measure when there is a probability measure on the space of L -structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M . We show that M admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraïssé limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.