Generically stable and smooth measures in NIP theories

TL;DR

This paper explores stable-like behaviors in NIP theories, introducing generically stable measures and demonstrating their widespread presence, along with new concepts like generic compact domination and measure definability in real and p-adic fields.

Contribution

It introduces the concept of generically stable measures in NIP theories, characterizes them, and proves their ubiquity, along with establishing measure definability results.

Findings

Introduction of generically stable measures and their characterizations

Proof of the ubiquity of generically stable measures in NIP theories

Approximate definability of Borel probability measures in real and p-adic fields

Abstract

We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity. We also introduce generic compact domination. We also prove the approximate definability of arbitrary Borel probability measures on definable sets in the real and p-adic fields.