An Abstract Regularity Lemma

TL;DR

This paper generalizes Szemeredi's Regularity Lemma to abstract measure spaces, providing a broader framework and new applications where traditional proofs do not apply, thus expanding the lemma's utility.

Contribution

It introduces an abstract measure space version of Szemeredi's Regularity Lemma and demonstrates its applicability through novel, tailored applications.

Findings

Extended SRL to abstract measure spaces

Identified conditions for the original proof to hold

Presented applications unique to this approach

Abstract

We extend Szemeredi's Regularity Lemma (SRL) to abstract measure spaces. Our main aim is to find general conditions under which the original proof of Szemeredi still works. To illustrate that our approach has some merit, we outline several applications. Some of these applications seem to be tailored to our approach: in particular, we are not aware of any alternative proofs.