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Measures on Aut(M)

Daniel Max Hoffmann

TL;DR

The paper addresses the problem of defining a convolution of measures for non-locally-compact groups such as $\mathrm{Aut}(M)$ to study actions on type spaces. It shifts from regular Borel (or Keisler) measures to the broader class of $\tau$-additive Borel measures $\mathcal{M}_\tau$ to obtain a well-defined convolution on a group action $G\curvearrowright X$ via $(\mu * \nu)(E) = (\mu \times \nu)(\pi^{-1}[E])$. Key contributions include establishing measurability, Fubini-type identities, and inner regularity preservation, proving associativity and the semigroup structure on $\mathcal{M}_\tau(G)$, and proving joint continuity in the action map. Moreover, the framework yields a discrete dynamical system $(\mathcal{M}_\tau(X), \mathcal{M}_\tau(G))$, and in the model-theoretic setting with $X$ as a space of types this reduces to the conv$(\mathrm{Aut}(M))$-dynamics, offering a robust tool for tameness and typology studies.

Abstract

We describe a class of measures on Aut(M) for which the convolution product with Keisler measures is well-defined.

Measures on Aut(M)

TL;DR

The paper addresses the problem of defining a convolution of measures for non-locally-compact groups such as to study actions on type spaces. It shifts from regular Borel (or Keisler) measures to the broader class of -additive Borel measures to obtain a well-defined convolution on a group action via . Key contributions include establishing measurability, Fubini-type identities, and inner regularity preservation, proving associativity and the semigroup structure on , and proving joint continuity in the action map. Moreover, the framework yields a discrete dynamical system , and in the model-theoretic setting with as a space of types this reduces to the conv-dynamics, offering a robust tool for tameness and typology studies.

Abstract

We describe a class of measures on Aut(M) for which the convolution product with Keisler measures is well-defined.

Paper Structure

This paper contains 4 sections, 8 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. $\tau$-additive and Keisler measures
  3. Convolution action for $\tau$-additive measures
  4. Convolution semigroup

Key Result

Lemma 3.3

Assume that a topological group $G$ acts on a topological space $X$ via $\pi:G\times X\to X$. Let $\mu\in{\mathcal{M}}_{\tau}(G)$ and let $\nu\in{\mathcal{M}}_{\tau}(X)$. Then the formula where $E\subseteq X$ is Borel, defines a $\tau$-additive Borel probability measure on $X$.

Theorems & Definitions (17)

  • Definition 2.1: 411H in Fremlin
  • Definition 2.2: 411C in Fremlin
  • Lemma 3.3: Convolution
  • proof
  • Remark 3.4
  • proof
  • Lemma 4.1
  • proof
  • Corollary 4.3
  • Lemma 4.4
  • ...and 7 more