{
  "id": "2302.01519",
  "version": "v1",
  "published": "2023-02-03T03:26:20.000Z",
  "updated": "2023-02-03T03:26:20.000Z",
  "title": "Model theory of probability spaces",
  "authors": [
    "Alexander Berenstein",
    "C. Ward Henson"
  ],
  "comment": "58 pages; to appear in the volume \"Model theory of operator algebras\" as part of DeGruyter's Logic and its Application Series",
  "categories": [
    "math.LO"
  ],
  "abstract": "This expository paper treats the model theory of probability spaces using the framework of continuous $[0,1]$-valued first order logic. The metric structures discussed, which we call probability algebras, are obtained from probability spaces by identifying two measurable sets if they differ by a set of measure zero. The class of probability algebras is axiomatizable in continuous first order logic; we denote its theory by $Pr$. We show that the existentially closed structures in this class are exactly the ones in which the underlying probability space is atomless. This subclass is also axiomatizable; its theory $APA$ is the model companion of $Pr$. We show that $APA$ is separably categorical (hence complete), has quantifier elimination, is $\\omega$-stable, and has built-in canonical bases, and we give a natural characterization of its independence relation. For general probability algebras, we prove that the set of atoms (enlarged by adding $0$) is a definable set, uniformly in models of $Pr$. We use this fact as a basis for giving a complete treatment of the model theory of arbitrary probability spaces. The core of this paper is an extensive presentation of the main model theoretic properties of $APA$. We discuss Maharam's structure theorem for probability algebras, and indicate the close connections between the ideas behind it and model theory. We show how probabilistic entropy provides a rank connected to model theoretic forking in probability algebras. In the final section we mention some open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-03T03:26:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C66",
      "03C10",
      "03C45",
      "28A60"
    ],
    "keywords": [
      "model theory",
      "main model theoretic properties",
      "maharams structure theorem",
      "continuous first order logic",
      "valued first order logic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}