Invariant measures in simple and in small theories

TL;DR

This paper explores invariant measures in simple theories, providing examples of formulas and groups with specific measure properties, and discusses paradoxical decompositions and positive results in small theories.

Contribution

It introduces new examples of formulas and groups with particular measure behaviors and establishes positive results on definable amenability and Grothendieck rings in small theories.

Findings

Existence of formulas with measure zero despite non-forking

Definable groups that are not definably amenable

Positive results on definable amenability and Grothendieck rings in small theories

Abstract

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.