Towards an Effective Theory of Absolutely Continuous Measures

TL;DR

This paper develops a constructive, metastable approach to a theorem on exchanging limits for convergent $L^1$ functions, utilizing a one-dimensional Szemeredi's regularity lemma for $L^1$ functions.

Contribution

It introduces a new metastable formulation and a one-dimensional regularity lemma for $L^1$ functions, advancing the theoretical understanding of absolutely continuous measures.

Findings

A constructive approach to limit exchange in $L^1$ functions.

A one-dimensional Szemeredi's regularity lemma for $L^1$ functions.

Enhanced theoretical framework for absolutely continuous measures.

Abstract

We give a constructive, metastable formulation of a theorem about the exchange of limits for convergent sequence $L^1$ functions. A crucial tool is a one-dimensional version of Szemeredi's regularity lemma for $L^1$ functions.