Continuous stable regularity

TL;DR

This paper extends the stable graph regularity lemma to a continuous setting using model theory, providing new tools and examples for stable functions in metric structures.

Contribution

It introduces an analytic version of the stable regularity lemma for functions on metric spaces, utilizing continuous model theory and local Keisler measures.

Findings

Proves an analytic stable regularity lemma for functions on metric spaces.

Develops tools for ultraproducts of metric structures and linear functionals.

Provides examples of stable functions in continuous settings.

Abstract

We prove an analytic version of the stable graph regularity lemma from \cite{MaSh}, which applies to stable functions $f\colon V\times W\to [0,1]$. Our methods involve continuous model theory and, in particular, results on the structure of local Keisler measures for stable continuous formulas. Along the way, we develop some basic tools around ultraproducts of metric structures and linear functionals on continuous formulas, and we also describe several concrete families of examples of stable functions.