Finding generically stable measures

TL;DR

This paper presents methods to construct generically stable Keisler measures in NIP theories, including symmetrization of invariant measures and sigma-additive measures, with proofs of smoothness in o-minimal theories and p-adics.

Contribution

It introduces new constructions for generically stable measures and proves their smoothness in specific theories, advancing understanding of measure stability in model theory.

Findings

Symmetrization yields generically stable measures from invariant ones.

Sigma-additive probability measures can produce generically stable measures.

Generically stable measures in o-minimal theories and p-adics are smooth.

Abstract

We discuss two constructions for obtaining generically stable Keisler measures in an NIP theory. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are smooth.