2015/09/21 by Nathanael Ackerman, Ackerman, Nathanael · 2 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Cellular Automata and Applications #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #math.LO
paper · pdf · doi:10.48550/arxiv.1509.06170
openalex publication_date 2015/09/21 · arxiv created 2021/10/10 · arxiv updated 2021/10/12 · openalex created_date 2021/10/25 · openalex updated_date 2026/07/28
In this paper we generalize the Aldous-Hoover-Kallenberg theorem concerning representations of distributions of exchangeable arrays via collections of measurable maps. We give criteria when such a representation theorem exists for arrays which need only be preserved by a closed subgroup of the symmetric group over ℕ. Specifically, for a countable structure M, with underlying set the ℕ, we introduce the notion of an "Aut(M)-recipe", which is an Aut(M)-invariant array obtained via a collection of measurable functions indexed by the Aut(M)-orbits in M. We further introduce the notion of a "free structure" and then show that if M is free then every Aut(M)-invariant measure on an Aut(M)-space is the distribution of an Aut(M)-recipe. We also show that if a measure is the distribution of an Aut(M)-recipe it must be the restriction of a measure on a free structure.